bucky feed - LessWrong 2.0 Readerbucky’s posts and comments on the Effective Altruism Forumen-usComment by Bucky on Project Proposal: Gears of Aging
https://lw2.issarice.com/posts/h4ceuEDj7LBk8vAhp/project-proposal-gears-of-aging?commentId=SY3DiAT87NjcGxZcp
<p>What are your estimates for how many nodes / causal relationships you would need to investigate to figure out one blueprint?</p>
buckySY3DiAT87NjcGxZcp2020-05-10T19:26:35.855ZComment by Bucky on Why do you (not) use a pseudonym on LessWrong?
https://lw2.issarice.com/posts/owfXRCRuZizt3JHQy/why-do-you-not-use-a-pseudonym-on-lesswrong?commentId=QgFm4dTNFNFJvYkCa
<p>I was going to write an answer but this sums up my thought process perfectly.</p>
buckyQgFm4dTNFNFJvYkCa2020-05-08T06:41:34.827ZComment by Bucky on "God Rewards Fools"
https://lw2.issarice.com/posts/NzqhjL9nd8FSYA5QN/god-rewards-fools?commentId=wHMFFJo3oJQ7mwAbE
<p>Halloween as a counterexample? (Or possibly the exception which proves the rule?)</p>buckywHMFFJo3oJQ7mwAbE2020-05-06T08:23:06.143ZComment by Bucky on 2020 predictions
https://lw2.issarice.com/posts/orSNNCm77LiSEBovx/2020-predictions?commentId=HpuJ2Bcj6G7Gyz7gW
<blockquote>This is a math error...</blockquote><p>Good point, thanks.</p><blockquote>Lombardy had a <strong>population</strong> fatality rate of 0.2%</blockquote><p>I don't think this really means anything without knowing the fraction infected. Robbio's <a href="https://twitter.com/TabulaFour/status/1247487723470020608">antibody testing</a> a month ago showed 13-14% infected so naively this gives 1.4% IFR. Possibly some sampling bias though. On the other hand this is a small town and presumably larger towns / cities would expect higher rates.</p><p>I'm willing to accept that IFR might push a bit over 1% but that doesn't overcome the need for a massive outbreak to happen across the whole US without significant action being taken to minimise the impact to get to 3M deaths.</p>buckyHpuJ2Bcj6G7Gyz7gW2020-05-04T20:01:52.767ZComment by Bucky on 2020 predictions
https://lw2.issarice.com/posts/orSNNCm77LiSEBovx/2020-predictions?commentId=Si9e8E3GabCiJmX9R
<p><a href="https://medium.com/@gidmk/what-is-the-infection-fatality-rate-of-covid-19-7f58f7c90410">This</a> analysis suggests that even in Wuhan and NYC the IFR wasn't higher than 1%. </p><p><a href="https://www.medrxiv.org/content/10.1101/2020.04.10.20060764v1">This</a> paper put Lombardy IFR = 1.1 but with a large confidence interval (0.2 - 2.1). It predicts a higher IFR across the world than in Lombardy which is weird. That's the paper which has the highest IFR of any in the 13 included in the analysis above.</p><p>Ventilators <a href="https://www.npr.org/sections/health-shots/2020/04/02/826105278/ventilators-are-no-panacea-for-critically-ill-covid-19-patients">aren't particularly effective</a>, saving less than half of the people who go on them so even worst case ventilator shortage will less than double IFR. Not sure what other hospital equipment would become the choke point - possibly oxygen supply? Temporary general hospital beds are alot easier to get quickly than temporary ICU beds so I wouldn't anticipate this being unsolvable.</p><p>Not everyone will get infected (due to herd immunity) so 330M isn't the number to be looking at, although assuming a runaway infection we'd have R=3 so ~220M infected.</p><p>To get the 3 million deaths you would need to have the situation where almost everywhere in the US had a massive outbreak killing 1% of their population with their hospitals in meltdown and all of the government institutions doing nothing to stop it and most people on an individual level not taking precautions like masks etc.</p>buckySi9e8E3GabCiJmX9R2020-05-04T10:26:29.655ZComment by Bucky on [U.S. Specific] Criminal Justice Reform in the Time of Covid-19
https://lw2.issarice.com/posts/iAHxqocJo36pQ3CQd/u-s-specific-criminal-justice-reform-in-the-time-of-covid-19?commentId=EobiBygmTAp4wP8st
<p>Agreed, especially accounting for presumably significant overlap. I’m guessing whatever this claim is based on either has some sort of selection bias or is counting more people as a loved one than common usage</p>
buckyEobiBygmTAp4wP8st2020-05-02T19:24:05.246ZComment by Bucky on [U.S. Specific] Criminal Justice Reform in the Time of Covid-19
https://lw2.issarice.com/posts/iAHxqocJo36pQ3CQd/u-s-specific-criminal-justice-reform-in-the-time-of-covid-19?commentId=rqdaYFAfuBng3Y4G9
<p>15 loved ones?</p>
buckyrqdaYFAfuBng3Y4G92020-05-02T14:34:53.689ZComment by Bucky on 2020 predictions
https://lw2.issarice.com/posts/orSNNCm77LiSEBovx/2020-predictions?commentId=eCCxYC2g722DT4bNM
<p>1. Bay Area lockdown (eg restaurants closed) will be extended beyond June 15: </p><p>2. …until Election Day: </p><p>3. Fewer than 100,000 US coronavirus deaths: </p><p>4. Fewer than 300,000 US coronavirus deaths: </p><p>5. Fewer than 3 million US coronavirus deaths: </p><p>6. US has highest official death toll of any country: </p><p>7. US has highest death toll as per expert guesses of real numbers: </p><p>8. NYC widely considered worst-hit US city: </p><p>9. China’s (official) case number goes from its current 82,000 to 100,000 by the end of the year: </p><p>10. A coronavirus vaccine has been approved for general use and given to at least 10,000 people somewhere in the First World: </p><p>11. Best scientific consensus ends up being that hydroxychloroquine was significantly effective: </p><p>12. I [Scott] personally will get coronavirus (as per my best guess if I had it; positive test not needed): </p><p>13. Someone I [Scott] am close to (housemate or close family member) will get coronavirus: </p><p>14. General consensus is that we (April 2020 US) were overreacting: </p><p>15. General consensus is that we (April 2020 US) were underreacting: </p><p>16. General consensus is that summer made coronavirus significantly less dangerous: </p><p>17. …and there is a catastrophic (50K+ US deaths, or more major lockdowns, after at least a month without these things) second wave in autumn: </p><p>…</p><p>19. At least half of states send every voter a mail-in ballot in 2020 presidential election: </p><p>20. PredictIt is uncertain (less than 95% sure) who won the presidential election for more than 24 hours after Election Day. </p><p>POLITICS:</p><p>21. Democrats nominate Biden, and he remains nominee on Election Day: </p><p>…</p><p>26. Trump is re-elected President: </p><p>27. Democrats keep the House: </p><p>28. Republicans keep the Senate: </p><p>29. Trump approval rating higher than 43% on June 1: </p><p>30. Biden polling higher than Trump on June 1: </p><p>… </p><p>33. Boris still UK PM: </p><p>34. No new state leaves EU: </p><p>35. UK, EU extend “transition” trade deal: </p><p>36. Kim Jong-Un alive and in power: </p><p>ECON AND TECH:</p><p>37. Dow is above 25,000: </p><p>38. …above 30,000: </p><p>39. Bitcoin is above $5,000: </p><p>40. …above $10,000: </p><p>…</p><p>42. Crew Dragon reaches orbit: </p><p>43. Starship reaches orbit: </p>buckyeCCxYC2g722DT4bNM2020-05-01T20:12:34.902Z2020 predictions
https://lw2.issarice.com/posts/orSNNCm77LiSEBovx/2020-predictions
<p>EDIT: Ha, just noticed that Zvi has done something similar, I’ll be interested to check another source.</p><h1>The need for comparison</h1><p>A couple of posts (<a href="https://www.lesswrong.com/posts/DAc4iuy4D3EiNBt9B/how-to-evaluate-50-predictions">1</a>, <a href="https://www.lesswrong.com/posts/BthNiWJDagLuf2LN2/evaluating-predictions-in-hindsight">2</a>) recently have shown how difficult it is to judge predictions without a baseline to judge against - calibration testing is the only real option. </p><p>Having predictions from another source to compare against allows Brier scores or log-likelihoods to be used to see which set of predictions are best. It also allows 50% predictions to be meaningful.</p><p>It’s hard to judge predictions in hindsight without accidentally adding what you know now into the discussion (see Scott’s <a href="https://www.lesswrong.com/posts/BthNiWJDagLuf2LN2/evaluating-predictions-in-hindsight">comment</a> on Zvi’s post).</p><p>So if you want to assess how good your predictions are, it is best to put them out there in advance against a set of questions that you have some values to compare against.</p><p>So I thought I would attempt to put my own probabilities on the SSC predictions from this year (or at least those which I could be reasonably expected to know about). This means that I will be able to see not only if I am well calibrated but also whether I am able to bring in all the evidence I can think of and integrate it into a good prediction. If I get a score close to Scott's then I'll be happy. I don’t know if other people do this too although a quick googling didn’t find anything.</p><p>So as not to anchor myself on Scott’s answers (a.k.a. cheating), I deleted Scott’s estimates before going back over and doing my own and then comparing. This is more like the <a href="https://www.lesswrong.com/posts/BthNiWJDagLuf2LN2/evaluating-predictions-in-hindsight#Method_Three__The_Green_Knight_Test">Green Knight test</a> mentioned in Zvi’s post. I have added a <a href="https://www.lesswrong.com/posts/orSNNCm77LiSEBovx/2020-predictions?commentId=eCCxYC2g722DT4bNM">comment</a> to this post with the unscored list in case anyone else wants to give it a go before reading on.</p><p>I'm tempted to say that no-one is allowed to claim that either of us have made a poor prediction without having tried it off a blank list yourself - it was A LOT harder than I expected it to be! I've done calibration checking myself but putting it out publicly felt really stressful. In truth, feel free to say if you think I have any probability off - <a href="https://www.lesswrong.com/posts/Z5wF8mdonsM2AuGgt/negative-feedback-and-simulacra">simulacrum level 1</a>, agreed?</p><h1>Comparison of predictions</h1><p>So here are my predictions. I’ve kept the SSC numbering and indicated where there are any predictions I’ve skipped. Along with any personal predictions, I removed the Reade accusation questions as it isn’t something I’m familiar with (non US citizen here).</p><p>Any probabilities where our odds differed by more than a factor of 2 I have put in bold underline and added a description of my thinking. </p><h2>CORONAVIRUS:</h2><p>1. Bay Area lockdown (eg restaurants closed) will be extended beyond June 15: 80%</p><p>2. …until Election Day: 10%</p><p>3. Fewer than 100,000 US coronavirus deaths: <strong><u>5% (SSC 10%)</u></strong></p><p><em>Existing death toll officially 55,000 is an undercount and we probably need to <a href="https://www.nytimes.com/interactive/2020/04/28/us/coronavirus-death-toll-total.html">add ~50%</a> to that as a minimum (I assume this will have been made official before the end of the year). Add that to the current rate of 2,000/day which will take a while to go down and I think 100,000+ becomes almost inevitable.</em></p><p>4. Fewer than 300,000 US coronavirus deaths: 60%</p><p>5. Fewer than 3 million US coronavirus deaths: <strong><u>95% (SSC 90%)</u></strong></p><p><em>Given <a href="https://www.lesswrong.com/posts/nRX7uwT2wNvvmd2Yd/coronavirus-justified-key-insights-thread?commentId=Cs2ZCQiWYRsBSepcd">Coronavirus IFR <1%</a> then with a US population of 330 million this seems almost certain. I would have put this probability higher if there was a higher option.</em></p><p>6. US has highest official death toll of any country: 80%</p><p>7. US has highest death toll as per expert guesses of real numbers: 60%</p><p>8. NYC widely considered worst-hit US city: <strong><u>80% (SSC 90%)</u></strong></p><p><em>I umm-ed and ahh-ed between 80 and 90 on this one and am not sure I made the right choice</em>.</p><p>9. China’s (official) case number goes from its current 82,000 to 100,000 by the end of the year: 70%</p><p>10. A coronavirus vaccine has been approved for general use and given to at least 10,000 people somewhere in the First World: 40%</p><p>11. Best scientific consensus ends up being that hydroxychloroquine was significantly effective: <strong><u>60% (SSC 20%)</u></strong></p><p><em>I suspect Scott has more knowledge on this one. The only reason I went as high as I did was that tricky word significant. If it means statistical significance then there’s a fair chance that a meta-analysis might find a result given a large enough sample size, even if it isn’t really clinically significant. For clinically significant I would have been closer to Scott.</em></p><p>12. I personally will get coronavirus (as per my best guess if I had it; positive test not needed): <strong><u>10% (SSC 30%)</u></strong></p><p><em>I wasn’t sure whether to try to predict this one as I don’t have much information on whether Scott would likely get coronavirus. I decided to just outside view it – I’m predicting 10% or so infection (by predicting 300,000 or so deaths) with a decent number of those having already happened. I didn’t really think at the time about how many people Scott will see in his job when he’s no longer working from home so I may well have underestimated here. On the other hand my impression is that California is being more cautious than most states (?) so I wouldn't expect cases to be concentrated there.</em></p><p>13. Someone I am close to (housemate or close family member) will get coronavirus: <u>3<strong>0% (SSC 70%)</strong></u></p><p><em>See previous. I'm not sure how many people are included here but there is probably significant correlation between these people so I didn't raise the probability too much above 10%.</em></p><p>14. General consensus is that we (April 2020 US) were overreacting: 60%</p><p>15. General consensus is that we (April 2020 US) were underreacting: <strong><u>10% (SSC 20%)</u></strong></p><p><em>Possibly not much difference between us, I would probably have put 15% if that was an option</em></p><p>16. General consensus is that summer made coronavirus significantly less dangerous: <strong><u>40% (SSC 70%)</u></strong></p><p><em>Generally I’ve heard that warmer countries haven’t been especially well protected so far but haven’t really looked into it. Possibly I should have gone more with the prior that viruses are often worse in winter but I'm not sure if this is availability bias for cold/flu vs say Ebola/HIV for which I'm unaware of seasonal variation? Maybe I should ask someone with an MD?</em></p><p>17. …and there is a catastrophic (50K+ US deaths, or more major lockdowns, after at least a month without these things) second wave in autumn: 20% (SSC 30%)</p><p><em>Scott and I both estimate P(17|16) 40%-50% so 16 is where we had the difference.)</em></p><p>…</p><p>19. At least half of states send every voter a mail-in ballot in 2020 presidential election: 30%</p><p>20. PredictIt is uncertain (less than 95% sure) who won the presidential election for more than 24 hours after Election Day. 20%</p><h2>POLITICS:</h2><p>21. Democrats nominate Biden, and he remains nominee on Election Day: 90%</p><p>…</p><p>26. Trump is re-elected President: 60%</p><p>27. Democrats keep the House: 60%</p><p>28. Republicans keep the Senate: 60%</p><p>29. Trump approval rating higher than 43% on June 1: <strong><u>50% (SSC 30%)</u></strong></p><p><em>Eyeballing his approval ratings he was 42% or so for a while and is currently a smidge higher. Looking back on this now I was probably high here.</em></p><p>30. Biden polling higher than Trump on June 1: 70%</p><p>… </p><p>33. Boris still UK PM: 90%</p><p>34. No new state leaves EU: 90%</p><p>35. UK, EU extend “transition” trade deal: <strong><u>30% (SSC 80%)</u></strong></p><p><em>This was a tricky one to assess – I agree that they might need to do something but for someone elected on the platform of “Get Brexit Done” this would be a risky move, although given coronavirus it might be forgiven. I would expect them to have to try some kind of intermediate deal but not an extension of the current transition period.</em></p><p>36. Kim Jong-Un alive and in power: <strong><u>80% (SSC 60%)</u></strong></p><p><em>I think this one depends almost entirely on how much you believe the current reports of ill health. I haven’t paid much attention but wrote the reports off as probably just speculation but I'm not particularly attached to that conclusion.</em></p><h2>ECON AND TECH:</h2><p>37. Dow is above 25,000: 70%</p><p>38. …above 30,000: <strong><u>10% (SSC 20%)</u></strong></p><p><em>Currently 24,300. In normal growth times I think this would struggle to get there. I feel like there are too many things which have to go right to make this 20% probable but I’m no economist!</em></p><p>39. Bitcoin is above $5,000: 70%</p><p>40. …above $10,000: 20%</p><p>…</p><p>42. Crew Dragon reaches orbit: <strong><u>90% (SSC 80%)</u></strong></p><p><em>The Dragon I believe because it’s scheduled for only a month or so away. I may have been slightly overgenerous on this, in truth I was torn between 80 and 90.</em></p><p>43. Starship reaches orbit: <strong><u>20% (SSC 40%)</u></strong></p><p><em>Do any of Musk’s projects get done on schedule? I’m not complaining as the projects are tricky and he certainly doesn’t have the biggest delays (c.f.<a href="https://xkcd.com/2014/"> JWST</a>) but there is a bit of a track record here. For the <a href="https://en.wikipedia.org/wiki/Falcon_Heavy_test_flight">Falcon Heavy</a> any timeline given until the first flight ended up taking 2-3 times as long.</em></p><h2>Discussion</h2><p>We disagreed significantly on 14 questions, roughly agreeing on the remaining 21 (60% agreement rate).</p><p>As I mentioned above, trying to come up with my own figures was A LOT harder than looking at Scott’s numbers and thinking whether I agree with them or not as I had done in previous years. Reflecting on Scott’s answers now has already persuaded me that I would like to change my answers on a few questions (8, 11, 12, 16, 29), not necessarily all the way to Scott’s answers but certainly in that direction.</p><p>On the other hand there are ones where I think my probabilities are better (3, 5, 38, 43).</p><p>The most disagreed on question is 35 (extending Brexit transition deal) where we differ by a huge odds ratio of 9.3 (80% Scott vs 30% me). I am genuinely unsure on this one. Boris might have enough credibility to pull off an extension based on coronavirus but I’m fairly sure he will want to be seen to be doing something, particularly with regards to immigration and other EU rules.</p><p>Doing a bit of research now, <a href="https://uk.finance.yahoo.com/news/coronavirus-political-betting-puts-63-chance-that-brexit-transition-period-will-not-be-extended-095726708.html?guccounter=1&guce_referrer=aHR0cHM6Ly93d3cuZ29vZ2xlLmNvbS8&guce_referrer_sig=AQAAAFBMgsdHOwLyuNDysYxMUURNy74DE-bs_-eyonsORN-W8wRPBr0CzYyKlmeUxg3cGyMpIoRAW28lFcq2iUgZS_bu7EaeK8vdWOeccjKy1VXYEr1QOgbmh1nefmZ9B4MsDUB4ylKhFsPycV8wFw__ftBc6mCgt9OdwmpdDfFnEi5z">here’s</a> a news article about how the odds on Smarkets have changed on this from Scott’s level (80%) on 14th April to a bit above my level (40%) by 20th April because:</p><blockquote>Over the last few days, Whitehall said it will not accept any delay to the Brexit transition period beyond this year even if the EU offers an extension.</blockquote><blockquote>“We will not ask to extend the transition. And, if the EU asks, we will say no. Extending the transition would simply prolong the negotiations, prolong business uncertainty, and delay the moment of control of our borders,” said a spokesperson.</blockquote><p>I didn’t know about this statement before making my prediction but I do feel like my reasoning was sound. Given this statement the 40% of the market seems high and I would put the probability more like 20% but this isn't a recommendation that anyone do anything based on that!</p><h1>Suggested alternative probability options</h1><p>One thing which I struggled with was choosing between options, especially at the high/low percentages, when I’d have liked to choose something between the available options (questions 6, 8, 15, 33, 38 and 42). This is worse at the extremes because the odds ratios between 80% and 90% and between 90% and 95% are greater than 2, whereas between 50% and 60% is only 1.5 (in fact the 80-90 gap is twice as large as the 50-60 gap).</p><p>If I were making up my own levels I would try to choose a constant odds ratio between consecutive levels. If I keep the same number of levels (11) and the same maximum confidence (95%) this works out as an odds ratio of 1.8 between levels. The levels then become:</p><p>5%, 9%, 15%, 24%, 36%, 50%, 64%, 76%, 85%, 91%, 95%</p><p>With some rounding we could get:</p><p>5%, 9%, 15%, 25%, 35%, 50%, 65%, 75%, 85%, 91%, 95%</p><p>which is easier remember. With these we have a maximum 1.89 odds ratio between 75% and 85%.</p><p>Another option which doesn’t make for such nice round numbers would be:</p><p>5%, 8%, 13%, 21%, 31%, 43%, 57%, 69%, 79%, 87%, 92%, 95%</p><p>This has one more option but when you negate the <50% predictions you get the same number of groups so you’re not spreading the results any thinner for your analysis. It has the advantage that you don’t throw away any information for calibration testing as there is no 50% option. The odds ratio between adjacent options is then ~1.7.</p><p>Some might not like the lack of 50% option but I think that’s actually a feature rather than a bug – you’re being asked to at least pick a side, even if you only assign it 1.3:1 odds in favour. Obviously if you genuinely believe 50% then you can’t put your true belief but that’s true of most probabilities whatever groupings you decide on – you sacrifice resolution in order to be able to analyse calibration.</p>buckyorSNNCm77LiSEBovx2020-05-01T20:11:04.423ZComment by Bucky on Growth rate of COVID-19 outbreaks
https://lw2.issarice.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks?commentId=mk3LXG7B4s4i6f2pS
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="cases_t = cases_{t-1}\times2^\frac{1}{d_t}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">×</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.01em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mfrac" style=""><span class="mjx-box MJXc-stacked" style="width: 0.943em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 83.3%; width: 1.131em; top: -1.288em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span class="mjx-denominator" style="font-size: 83.3%; width: 1.131em; bottom: -1.024em;"><span class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.003em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span><span class="mjx-sub" style="vertical-align: -0.297em; padding-right: 0.05em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span></span></span><span style="border-bottom: 1px solid; top: -0.296em; width: 0.943em;" class="mjx-line"></span></span><span style="height: 1.927em; vertical-align: -0.854em;" class="mjx-vsize"></span></span></span></span></span></span></span></span> </p><p>The doubling times in this post only really apply for the start of an outbreak. Once control measures are put in place the doubling time changes - see this <a href="https://www.lesswrong.com/posts/ZqW5wzyLhbayHy2QC/covid-19-growth-rates-vs-interventions">post</a>. To look at this properly you'd need a more complex model - <a href="https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIR_model">SIR</a> is a good starting point but proper models are much more complex.</p>buckymk3LXG7B4s4i6f2pS2020-04-30T09:27:20.154ZComment by Bucky on Helping Lily Make Dinner
https://lw2.issarice.com/posts/PmdnyszLPXHC7d7PL/helping-lily-make-dinner?commentId=NrbehkHWDJhPXgydf
<p>If she ends up wanting to do more cooking there are <a href="https://www.amazon.co.uk/Plastic-vegetable-Lettuce-Cutting-children/dp/B071WV41LF">plastic kitchen knives</a> available. Our kids like to use them on the rare occasions that they want to help out. Probably good enough for a soft avacado but might struggle with onion!</p>
buckyNrbehkHWDJhPXgydf2020-04-27T06:37:31.052ZComment by Bucky on Growth rate of COVID-19 outbreaks
https://lw2.issarice.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks?commentId=6t2hWHiWZjmh8ZgkQ
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padding-bottom: 0.298em;">c</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span><span class="mjx-sub" style="font-size: 83.3%; vertical-align: -0.267em; padding-right: 0.06em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.81em;" class="mjx-line"></span></span><span style="height: 1.514em; vertical-align: -0.565em;" class="mjx-vsize"></span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span> </p>bucky6t2hWHiWZjmh8ZgkQ2020-04-24T22:24:50.342ZComment by Bucky on My Covid-19 Thinking: 4/23 pre-Cuomo Data
https://lw2.issarice.com/posts/ijd2TsoGrhcriAeNs/my-covid-19-thinking-4-23-pre-cuomo-data?commentId=EW2dLWJ5zhPR8HX8x
<p>After the stay-at-home orders started (~22 March) we no longer expect to see exponential growth in actual infections so the delay between infections and cases identified causes there to be a varying ratio between them.</p><p><br>Add that to the fact that the testing rate was the main thing controlling how many cases were identified which messes everything up. In late March/early April the positive rate of tests in New York was ~50% which renders the numbers fairly meaningless.</p>buckyEW2dLWJ5zhPR8HX8x2020-04-24T10:05:15.626ZComment by Bucky on April Coronavirus Open Thread
https://lw2.issarice.com/posts/qapqE86xrjQkD8eZ2/april-coronavirus-open-thread?commentId=oCXcrzNEGGqMpnWiM
<p>The magnitude of the numbers here seem wrong to represent people being infected twice.</p>
<p>From April 9-17 there were 74 newly discovered positive tests in those who had previously recovered. Over the same period there were only 203 new cases discovered. If the 74 received a new infection then they are getting infected at 2000x the rate of the general population.</p>
<p>Obviously there are a fair few reasons why they might be getting reinfected at a higher rate but I can’t think of a way it would be that much more. The reoccurrence of an existing infection would make a lot more sense.</p>
buckyoCXcrzNEGGqMpnWiM2020-04-18T23:07:24.309ZComment by Bucky on Evaluating Predictions in Hindsight
https://lw2.issarice.com/posts/BthNiWJDagLuf2LN2/evaluating-predictions-in-hindsight?commentId=uraopQr8iykFksv5e
<p>One advantage of calibration testing is that it doesn't require a market/opponent. I suspect that this is at least partly why Scott uses this method.</p>buckyuraopQr8iykFksv5e2020-04-16T21:23:58.908ZComment by Bucky on April Coronavirus Open Thread
https://lw2.issarice.com/posts/qapqE86xrjQkD8eZ2/april-coronavirus-open-thread?commentId=dqKdTJ9pesazAunFp
<p>Some potentially useful numbers I've been working on estimating:</p><p>1. The number of days lag between registered cases and deaths</p><p>2. The adjusted CFR for each country taking this into account</p><p>The method is essentially to try different lags (dividing current deaths by cases from x days ago) and see which length of lag gives a constant CFR over time (normally CFR increases with time as the growth rate of cases slows earlier than that of deaths).</p><p>Here are the results for a few countries:</p><p>China: 9 day lag, CFR=4%</p><p>USA: 7 day lag, CFR=6.5%</p><p>Italy: 4 day lag, CFR=14.5%</p><p>Spain: 2 day lag, CFR=10.5%</p><p>Germany: 10 day lag, CFR=3.5%</p><p>France: 10 day lag, CFR=24%</p><p>Switzerland: 6 day lag, CFR=4%</p><p>UK: 4 day lag, CFR=18.5%</p><p>I'm not sure about these, especially UK but they do create nice constant values for CFR over a period of 2-4 weeks (UK only 10 days) which suggests a predictable pattern, despite variation in testing.</p><p>The France result is also not quite as consistent as the others and is surprisingly high so I don't quite trust it either. I could make a case that for an estimate of 7 day lag and 20% CFR.</p>buckydqKdTJ9pesazAunFp2020-04-13T21:55:03.315ZComment by Bucky on Seemingly Popular Covid-19 Model is Obvious Nonsense
https://lw2.issarice.com/posts/QuzAwSTND6N4k7yNj/seemingly-popular-covid-19-model-is-obvious-nonsense?commentId=skh7X3ANBiZ8B9Jbw
<p><a href="https://www.lesswrong.com/posts/duxy4Hby5qMsv42i8/the-real-rules-have-no-exceptions">The real rules have no exceptions</a></p>
<p>In Newton’s case the real rule (or at least the practical rule) is the meta-rule of when Newton is good enough and what to use when it isn’t. Without that knowledge you can’t form a meta-rule and you don’t know when to believe the model and when not to. You can maybe assess it probabilistically but I wouldn’t want to place much on the result.</p>
buckyskh7X3ANBiZ8B9Jbw2020-04-12T23:21:38.980ZComment by Bucky on Seemingly Popular Covid-19 Model is Obvious Nonsense
https://lw2.issarice.com/posts/QuzAwSTND6N4k7yNj/seemingly-popular-covid-19-model-is-obvious-nonsense?commentId=DY5i2qqGQ6e54ycS4
<p>Italy seems to me to have stalled in decreasing R at about R=0.9. China and South Korea both got down to R=0.5. I have a concern that the UK has stalled at about R=1.3 (25% confidence) but I suspect that a few days more data may disprove this.</p><p>The US appears to still be on a downwards trajectory (currently just above R=1) but where exactly it stops will make a huge difference to the final tally. If I were to be making a model then this is the main place where I would focus my attention to give reasonable confidence intervals.</p>buckyDY5i2qqGQ6e54ycS42020-04-12T20:38:12.609ZComment by Bucky on How to evaluate (50%) predictions
https://lw2.issarice.com/posts/DAc4iuy4D3EiNBt9B/how-to-evaluate-50-predictions?commentId=bGXtsZkJ3xAnMGauk
<p>The <a href="https://en.wikipedia.org/wiki/Bayes_factor">Bayes factor</a> calculation which I did is the analytical result for which BIC is an approximation (see this <a href="https://www.lesswrong.com/s/onCRFFN7rGXTg3jyc">sequence</a>). Generally BIC is a large N approximation but in this case they actually do end up being fairly similar even with low N.</p>buckybGXtsZkJ3xAnMGauk2020-04-11T06:46:01.949ZComment by Bucky on How to evaluate (50%) predictions
https://lw2.issarice.com/posts/DAc4iuy4D3EiNBt9B/how-to-evaluate-50-predictions?commentId=EGdvkqcoHYjh3JfLJ
<blockquote>For example, Scott ending up ~60% right on the things that he thinks are 50% likely suggests that he's throwing away some of his signal </blockquote><p>If we compare two hypotheses:</p><p>Perfect calibration at 50%</p><p>vs</p><p>Unknown actual calibration (uniform prior across [0,1])</p><p>Then the Bayes factor is 2:1 in favour of the former hypothesis (for 7/11 correct) so it seems that Scott isn't throwing away information. Looking across other years supports this - his total of 30 out of 65 is 5:1 evidence in favour of the former hypothesis.</p>buckyEGdvkqcoHYjh3JfLJ2020-04-10T21:57:17.153ZComment by Bucky on On R0
https://lw2.issarice.com/posts/yoff8f999fica5W2d/on-r0?commentId=8C2B7AFPMWivWDLWG
<p>This is great, I particularly like the grocery delivery idea.</p>
<p>As I understand it the R0 variance is a big reason (the main reason?) for the flu vaccine being given to children at least in the UK - they have the potential to have a very high R0. According to <a href="https://www.google.co.uk/amp/s/amp.theguardian.com/education/2020/apr/06/school-closures-have-little-impact-on-spread-of-coronavirus-study">this study</a> worrying about kids infections may not be helpful for COVID-19 but this seems like the right kind of thing to consider.</p>
<blockquote>
<p>If the doubling time is 2.5 days and the serial interval is 5 days then R0 should be about 4, so each serial interval can let us double twice.</p>
</blockquote>
<p>If I understand this correctly I think this would mean R0=3 as1in 4 people infected by the end of 5 days was already infected at the beginning.</p>
<p>Edit: I think I got this last bit wrong as only 3/4 of the people infected at the beginning are still infectious (according to the simplified model I’m imagining) so the original value of R0=4 is correct</p>
bucky8C2B7AFPMWivWDLWG2020-04-09T19:54:16.256ZComment by Bucky on April Coronavirus Open Thread
https://lw2.issarice.com/posts/qapqE86xrjQkD8eZ2/april-coronavirus-open-thread?commentId=mgEL2Hy29urdbo4x2
<p>I share your rough estimates of IFR in your other comment here although I was concerned about how high IFR might be with overwhelmed hospitals.</p><p>Sampling bias at its worst here would mean that IFR is 3 times more than those calculations (i.e. 1.5-2%). If this is the worst case in Lombardy where the hospitals are overwhelmed then it is something of a relief to me that higher rates are unlikely.</p>buckymgEL2Hy29urdbo4x22020-04-08T15:40:27.436ZComment by Bucky on What is the impact of varying infectious dose of COVID-19?
https://lw2.issarice.com/posts/FftgfqqdRAx5ssiAp/what-is-the-impact-of-varying-infectious-dose-of-covid-19?commentId=y6Apr5ibj7v9KW8sX
<p>It isn't clear - that's a good point and would suggest that the upper bound might actually be higher than it appears at first glance. If we take <a href="https://www.lesswrong.com/posts/qapqE86xrjQkD8eZ2/april-coronavirus-open-thread?commentId=Pm9sB9RQTCTFnAXdF">10%</a> of infections being hospital based (which might not be accurate as that statistic is from South Korea and the above paper is in China outside Hubei) then 16% of the outside-the-home transmission might be hospital based.</p><p>I should say that only 284 of the 468 transmission events are included in either household and non-household. I don't know what the other 40% of cases were but I guess the researchers weren't able to identify the relationship from the public data that they were using. It does appear that this undefined 40% has a lower serial interval than either of the two defined groupings as the serial interval of all cases together is lower 3.96 [3.53, 4.39].</p>buckyy6Apr5ibj7v9KW8sX2020-04-07T12:54:50.840ZComment by Bucky on What is the impact of varying infectious dose of COVID-19?
https://lw2.issarice.com/posts/FftgfqqdRAx5ssiAp/what-is-the-impact-of-varying-infectious-dose-of-covid-19?commentId=3XKqrkzCS2s5smEDe
<p>If initial viral load makes a difference one would expect to see shorter time from infection to diagnosis/hospitalisation in cases which are transmitted within households. There is suggestive evidence in <a href="https://www.medrxiv.org/content/medrxiv/early/2020/03/06/2020.02.19.20025452.full.pdf">this paper</a> which includes data on the serial time for household (4.03 [3.12, 4.94]) and non-household (4.56 [3.85, 5.27]) secondary infections. The number in square brackets are the 95% CI.</p><p>This is fairly weak evidence that there is a difference and also gives some weak indication as to what the maximum effect of initial viral load might be.</p><p>The raw data from <a href="https://www.medrxiv.org/content/medrxiv/early/2020/03/06/2020.03.03.20029983.full.pdf">this paper</a>, for example, might be used to give more information on this and also severity which is more what we're interested in - the Tianjin data appears to be fairly complete albeit with only 135 cases.</p><p>EDIT: added link to 2nd paper</p>bucky3XKqrkzCS2s5smEDe2020-04-06T22:26:29.393ZComment by Bucky on An alarm bell for the next pandemic
https://lw2.issarice.com/posts/37ijFTYSaTNKrsSF4/an-alarm-bell-for-the-next-pandemic?commentId=WrbMrThXALXCcrjY7
<p>I think even a few days has the potential to be extremely valuable if it can be pulled off. If worldwide reactions had happened a few days sooner then half of the cases could have been avoided. LW ringing an alarm bell a few days earlier might not have had an effect on policy but its important to note how big the potential gains are.</p><p>As you say in the OP, the next time any pandemic comes along the worldwide response is likely to be better. So my main question is how do we generalise this advice for other severe dangers.</p><p>To me one of the main issues if the speed at which things happen. Most things which happen gradually give enough time for people to react without disastrous consequences - COVID only gives a few days before your problem is doubled. This would be fairly high on my checklist specifically for a future pandemic - low doubling times - but for general alarm bell ringing speed of problem development should also be up there.</p><p>*insert obligatory FOOM comment here...*</p>buckyWrbMrThXALXCcrjY72020-04-06T19:53:02.939ZComment by Bucky on An alarm bell for the next pandemic
https://lw2.issarice.com/posts/37ijFTYSaTNKrsSF4/an-alarm-bell-for-the-next-pandemic?commentId=RFpX5E4mqNbtoS5Qq
<p>Did you estimate how early using this would have caused an alarm to be raised for COVID-19?</p><p>I think the top 3 the harm questions were confirmed in <a href="http://weekly.chinacdc.cn/en/article/id/e53946e2-c6c4-41e9-9a9b-fea8db1a8f51">this paper</a> on 11th Feb but maybe there were other papers before this or we could have inferred from public data?</p><p>2,000 deaths was 18th Feb.</p><p>Escaping a lockdown attempt would probably be ~21st Feb in South Korea (the virus didn't really escape China lockdown - it had escaped before the lockdown)</p><p>Indirect transmissibility I'm not quite sure about a date?</p><p>Pre-symptomatic transmission again I'm not sure - from the papers in jimrandomh's <a href="https://www.lesswrong.com/posts/9GyKccaJdLEbdhyTi/a-significant-portion-of-covid-19-transmission-is">post</a> maybe early-mid Feb we had a good hint.</p>buckyRFpX5E4mqNbtoS5Qq2020-04-06T16:15:34.981ZComment by Bucky on COVID-19 growth rates vs interventions
https://lw2.issarice.com/posts/ZqW5wzyLhbayHy2QC/covid-19-growth-rates-vs-interventions?commentId=WFhLW7GP9BH8WRheG
<p>Yes, we definitely expect to see a lag between growth rates of cases and deaths, it is odd that even when this seems to be present it is only a couple of days to a week. I think this may be partly due to delays in diagnosis. 17.8 days is between onset of symptoms to death. However there is normally a lag between onset of symptoms and diagnosis (onset to hospitalisation I think is generally a bit less than a week) but even this still leaves a theoretical 10+ day lag.</p><p>That is all based on relative numbers within a country. Comparing CFR (case fatality rate) values between countries is notoriously unreliable due to testing capability. Looking at naive CFR I think the UK are about to overtake Italy as having the worst CFR in this set of 10 despite being earlier in their epidemic. This is either due to being worse at testing or better at diagnosing deaths as being COVID related (some countries aren't counting deaths which don't occur in hospital - <a href="https://www.reuters.com/article/us-health-coronavirus-italy-deaths-insig/death-at-home-the-unseen-toll-of-italys-coronavirus-crisis-idUSKBN21N08X">source</a>). CFR in the US is low compared to where other countries were at similar points in their epidemic so I guess it won't reach 10% but it is likely to reach 5%.</p>buckyWFhLW7GP9BH8WRheG2020-04-06T08:34:05.890ZComment by Bucky on What will the economic effects of COVID-19 be?
https://lw2.issarice.com/posts/owk7eBzwjPHNQbNeL/what-will-the-economic-effects-of-covid-19-be?commentId=iD4ycbjDbvjTmdn5J
<p>The ILO (international labour Organization, a UN agency) has a <a href="https://www.ilo.org/wcmsp5/groups/public/---dgreports/---dcomm/documents/briefingnote/wcms_738753.pdf">report</a> on this.</p>
<p>Some key findings:
Estimated increase in unemployment of 5-25 million - c.f. 22 million for 2008-9 crisis</p>
<p>These based on assumptions of 2-8% drop in global gdp</p>
<p>Value add from Chinese Industrial was down 13.5% in Jan/Feb</p>
buckyiD4ycbjDbvjTmdn5J2020-04-02T20:11:09.043ZComment by Bucky on How special are human brains among animal brains?
https://lw2.issarice.com/posts/d2jgBurQygbXzhPxc/how-special-are-human-brains-among-animal-brains?commentId=eYtQycW3fC5NkWo5Z
<p>You might be interested in <a href="https://www.lesswrong.com/posts/XjuT9vgBfwXPxsdfN/might-humans-not-be-the-most-intelligent-animals">this post</a> which explores similar territory.</p>buckyeYtQycW3fC5NkWo5Z2020-04-01T14:19:04.646ZComment by Bucky on April Coronavirus Open Thread
https://lw2.issarice.com/posts/qapqE86xrjQkD8eZ2/april-coronavirus-open-thread?commentId=Pm9sB9RQTCTFnAXdF
<p>South Korea, as always, are a treasure trove on information - they <a href="https://www.cdc.go.kr/board/board.es?mid=a30402000000&bid=0030">publish</a> details every day which includes major outbreak clusters, some of which are hospitals. Of the non-cult related cases where they have managed to identify the source of the infection, hospital based infections account for 20%. If you include cases where they haven't identified the source then it's more like 10% which is probably a fairer reflection as hospital clusters probably mainly do get identified.</p><p>(They changed their reporting layout on March 25th and the new version doesn't quite contain as much information so I've based this on the <a href="https://www.cdc.go.kr/board/board.es?mid=a30402000000&bid=0030">24th</a>)</p>buckyPm9sB9RQTCTFnAXdF2020-04-01T14:10:14.612ZComment by Bucky on April Coronavirus Open Thread
https://lw2.issarice.com/posts/qapqE86xrjQkD8eZ2/april-coronavirus-open-thread?commentId=Ey6Ley4m7WxzmHzzD
<p>I think there's a decent amount of correlation with between lockdown dates and entering linear growth. Below are the lockdown dates and starts of the linear phase for some of the worst hit countries.</p><p>China 23rd Jan -> 5th Feb</p><p>S. Korea 20th Feb -> 1st March (This wasn't a mandated government lockdown but people did seem to <a href="https://www.reuters.com/article/us-china-health-southkorea-cases/like-a-zombie-apocalypse-residents-on-edge-as-coronavirus-cases-surge-in-south-korea-idUSKBN20E04F">stay inside</a> in the worst hit areas)</p><p>March:</p><p>Italy 9th -> 21st</p><p>Spain 15th -> 26th</p><p>Germany 16th -> 27th</p><p>France 17th -> not yet linear (last 2 days have been high)</p><p>Switzerland 20th -> 21st</p><p>US 22nd (NY) -> not yet linear</p><p>UK 23rd -> approaching linear? Possibly already there</p><p>These are remarkably consistent at 10-14 days, apart from Switzerland (very fast) and France (looked like it had gone linear at about the normal time but has increased again). </p><p><a href="https://chart-studio.plotly.com/~Bucky13/9">This graph</a> shows the same data but is annotated with containment steps taken by each country (it isn't averaged over 3 days so the exact numbers don't match up but the same pattern applies).</p>buckyEy6Ley4m7WxzmHzzD2020-04-01T13:17:00.229ZComment by Bucky on Peter's COVID Consolidated Brief for 29 March
https://lw2.issarice.com/posts/Enqwi2MyixX97SJ22/peter-s-covid-consolidated-brief-for-29-march?commentId=DTC3rPmwHmzinReBo
<blockquote>
<p>The obvious conclusion is that Japan just isn’t testing anyone. This turns out to be true – they were hoping that if they made themselves look virus-free, the world would still let them hold the Tokyo Olympics this summer.</p>
</blockquote>
<p>I think this really needs to substantiated before claiming it is true (I realise this is a quote but still).</p>
<p>Personally I think people are looking at the wrong denominator for Japan - Japan’s tests / population is low but their tests / positive test is high (20:1 or so, S Korea is 30:1, Western nations are <10:1).</p>
buckyDTC3rPmwHmzinReBo2020-03-29T21:07:02.128ZComment by Bucky on Iceland's COVID-19 random sampling results: C19 similar to Influenza
https://lw2.issarice.com/posts/68ZG5SYcRQ5q8F7QR/iceland-s-covid-19-random-sampling-results-c19-similar-to?commentId=j7uh2Rj7jsc935Hjw
<p>What’s your take on the South Korean data?</p>
<p>They were testing thoroughly (30 negative tests for every 1 positive) all the way through their outbreak so either they were useless at choosing who to test (seems unlikely as they got the outbreak under control pretty fast) or they were finding nearly everyone. Their CFR was 1.3%.</p>
buckyj7uh2Rj7jsc935Hjw2020-03-28T21:18:31.061ZComment by Bucky on COVID-19 growth rates vs interventions
https://lw2.issarice.com/posts/ZqW5wzyLhbayHy2QC/covid-19-growth-rates-vs-interventions?commentId=7A6QhDKea8g5APQQn
<p>Comparing confirmed cases to deaths should identify that confounder if it’s there. Interestingly the US is one of the countries which showed up as possibly confounded at the beginning. More recently I suspect this is less of an issue.</p>
<p>My analysis suggests about a 40% decrease in R due to hygiene and social distancing. R0 is ~3 for COVID-19 so this bring R down to ~1.8 which means the virus is still growing fairly fast. For flu R0 is ~1.3 so after these measures R is ~0.8 and therefore is shrinking.</p>
bucky7A6QhDKea8g5APQQn2020-03-28T20:27:37.272ZComment by Bucky on COVID-19 growth rates vs interventions
https://lw2.issarice.com/posts/ZqW5wzyLhbayHy2QC/covid-19-growth-rates-vs-interventions?commentId=M5p2SBQLoL5svtZXY
<p>Yes, holding at a high number is tricky and not particularly desirable. If you get doubling time above 6 days then it’s likely that you’ll start decreasing cases.</p>
<p>I think that the most important thing if trying to hold at a low level whilst relaxing restrictions is ensuring that the doubling time is longer than the incubation time (which is the main lag in your control loop). That way if you have made an error the virus isn’t too far gone before you start to notice and contact tracing for containment remains viable.</p>
buckyM5p2SBQLoL5svtZXY2020-03-28T10:16:26.296ZComment by Bucky on COVID-19 growth rates vs interventions
https://lw2.issarice.com/posts/ZqW5wzyLhbayHy2QC/covid-19-growth-rates-vs-interventions?commentId=CNyu6vDpX7KvXT2xc
<p>A couple of additional points to leggi:</p>
<p>Elizabeth <a href="https://www.lesswrong.com/posts/owk7eBzwjPHNQbNeL/what-will-the-economic-effects-of-a-3-week-quarantine-be-3?commentId=MTjShuuvNbGkKiSpr">calculates</a> roughly 25% of people are in essential roles. These people are less able to reduce numbers of contacts.</p>
<p>At least initially many people don’t take social distancing seriously so the effects are likely to ramp up over time.</p>
<p>In that case it makes sense that initially doubling times increase over 5 and over time they keep increasing.</p>
<p>In China the distance was enforced and Koreans took it seriously right away so it didn’t take long for their doubling times to increase.</p>
buckyCNyu6vDpX7KvXT2xc2020-03-28T08:36:12.954ZCOVID-19 growth rates vs interventions
https://lw2.issarice.com/posts/ZqW5wzyLhbayHy2QC/covid-19-growth-rates-vs-interventions
<p>It’s been a couple of weeks since I <a href="https://www.lesswrong.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks">posted</a> regarding the growth rates of COVID-19 cases in various countries. Now, after these countries have implemented various control measures it is more clear how each measure effects growth rate. This post looks at the 10 countries with the highest number of confirmed cases.</p><h1>Death growth rates roughly match confirmed case growth rates</h1><p>Firstly, to check whether the confirmed case growth rates roughly reflect the actual growth rates, they can be compared to death rates (suggested in a <a href="https://www.lesswrong.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks?commentId=sufWySv4cKZS7YACv">comment</a> by Unnamed on my previous post). This doesn’t show what the ratio is between the number of cases and the number of detected cases, it only shows whether the fractional growth per day in detected cases roughly represents the actual growth rate of the virus. </p><p>Obviously the death growth rate doesn’t perfectly indicate the spread of the virus but it has different biases so if the two are similar then they probably at least somewhat reflect the actual growth rate.</p><p>A graph to compare growth rates of confirmed cases & deaths of the 10 worst affected countries is <a href="https://chart-studio.plotly.com/~Bucky13/7">here</a>. (Select country at the top)</p><p>(Note: For the y-axis I have used doubling time of infectious cases, which I’ve defined as 10 days fairly arbitrarily. The actual length of time doesn’t matter too much to the results but it’s important to have some limit otherwise the recent China and South Korea results particularly make less sense. This makes the definition of doubling time for deaths a bit odd but it's good enough for the purposes of comparing the rates.)</p><p>Generally the rates match well. Iran and USA are the main ones where there is significant divergence (and both look more reasonable recently), the others all seem sensible.</p><p>In theory there should be a lag between cases and deaths which we seem to see in Italy and Spain but the data is too noisy to say for sure.</p><h1>Growth rates vs interventions</h1><p>I’ve created <a href="https://chart-studio.plotly.com/~Bucky13/8">4 different displays</a> of the cases data:</p><p>1. Cumulative confirmed cases</p><p>2. New cases per day</p><p>3. Doubling time for infectious cases</p><p>4. Fractional change in infectious cases per day</p><p>This last one is analagous to the effective reproduction number (<span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="R"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span></span></span>) and could be converted to that if I knew the mean infectious period. I could take a stab at this but I think it's best to leave it as it is. It's useful to know that 1 on this axis is the same as <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R=1"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span>, even if there would need to be a scaling factor for the other points to convert them to <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span></span></span>. </p><p>I’ve annotated the graphs with what anti-COVID actions each country has taken and when. Apologies for anywhere I’ve got these wrong, if you see any massive errors for your country I’ll try to update.</p><p>Apologies also for the overlapping writing - mouse over the relevant points if it gets confusing and click the legend to toggle countries. Double-click to toggle between only that country or all countries.</p><p>There has been some form of lockdown for most countries but the exact extent differs between countries. I haven't attempted to distinguish between them.</p><p>There is expected to be a ~5 day delay between actions taken and effects being seen in the confirmed cases statistics as people are usually tested when they are symptomatic.</p><h2>Uninhibited growth has a doubling time of 2-3 days</h2><p>Refer to my previous post. I don’t really have much to add here, only that my initial calculations of doubling time had a small error so the doubling times are actually slightly lo (i.e. growth faster) than I initially reported.</p><h2>Growth with improved hygiene and social distancing has a doubling time of 3-5 days</h2><p>I also mentioned in that post that it seemed as though the doubling time for each country was increasing over time. This seems to me to represent additional simple precautions starting to be taken - such as improved hygiene and social distancing (short of a lockdown). </p><p>4-5 days is probably the best that can be achieved by these methods. Many countries have put these in place but none have been able to slow the spread of the virus sufficiently without taking additional actions.</p><h2>Growth of virus with partial lockdown has doubling time >4 days</h2><p>Different countries have enacted different strictness levels in their lockdowns. These haven't been in place long enough to know exactly what's happening but they have had an effect and in Italy's case especially this has started to strongly increase doubling times.</p><h2>Growth with virus under control has halving time as low as 2-5 days</h2><p>The indication of having <span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="R<1"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;"><</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span> (i.e. active cases decreasing) is that the doubling time becomes negative (and represents a halving time). This is probably seen better in the daily growth factor graph where a value <1 indicates shrinkage.</p><p>We have 2 examples of countries which have had significant outbreaks and brought them under control – China and South Korea. In both cases the doubling time starts climbing and keeps going until the active cases starts to decrease. Under full control the halving time of active cases was 2-5 days.</p><p>We don’t currently have any countries with a large number of cases where the doubling time is >6 days and holds steady for a prolonged period. The possible exception is Iran but I have less confidence in the data there due to the mismatch between growth rate of confirmed cases and deaths and in the last few days it looks like the growth rate is increasing again. </p><p>I suspect that having a sustained high doubling time is possible if <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span></span></span> is just above 1 but so far either a country is not doing enough (doubling time of 2-5 days) or they are doing enough and the cases are about to start decreasing. If <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R_0"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span> is large to start with it’s hard to find that perfect amount of intervention which takes <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span></span></span></span> to 1 so that the number of new cases stay manageable. Possibly as China and South Korea loosen their restrictions they are starting to find that point.</p><h1>Country summaries</h1><p>The above is based on looking at the performances of various countries as described below.</p><h2>China</h2><p>China successfully applied a quarantine in Wuhan which reduced a rapidly growing epidemic to a handful of new cases per day. This quarantine was very strict compared to other countries on this list and the halving rate was 2-5 days. Other, less strict quarantines are likely to shrink more slowly.</p><p>More recently (11th March), the restrictions in Wuhan <a href="https://www.csmonitor.com/World/Asia-Pacific/2020/0311/In-Wuhan-a-cautious-return-to-work-after-coronavirus-ebbs">were eased</a> to allow citizens to go back to work. Since 18th March the virus has started growing again, so far averaging a doubling rate of 7 days or so. So far they are performing <a href="https://medium.com/@tomaspueyo/coronavirus-the-hammer-and-the-dance-be9337092b56">the dance</a> successfully.</p><p>This seems to me the most likely next stage for Western countries. Exactly what rate the virus is managed to before it needs to be suppressed again is unclear. China at the moment should have plenty of tests and protective equipment so whatever they achieve is likely to be fairly close to the best possible scenario. Successful contact tracing could allow it to pause indefinitely without a full lockdown.</p><h2>Italy</h2><p>Doubling times have been increasing as the government has implemented additional control measures.</p><p>Lombardy (main outbreak in Italy) was locked down fairly early on. This increased the doubling time to 3-4 days. Later lockdowns which eventually covered the entire country increased this further such that the number of new cases per day appears to be levelling off. Most recently the Lombardy lockdown was tightened to decrease spread rate further. I don’t think it will be long before the number of live cases starts to decrease.</p><h2>USA</h2><p>The growth rate in the USA shows the least evidence of slowing down. The growth rate in deaths is less so there may be something confounding the data on confirmed cases, such as increasing coverage of testing.</p><p>Many states, including the main centres appear to have implemented lockdowns in recent days so these should start having an effect shortly. Some counties in the Bay area implemented a lockdown earlier but John Hopkins have started aggregating by state and any effects haven’t shown up in the California figures yet.</p><h2>Spain, Germany, France, Switzerland, UK</h2><p>These have all followed fairly similar paths. Schools have closed between 2k and 5k cases (Switzerland ~1k). Lockdowns have happened between 5k and 10k cases.</p><p>Some countries seem very keen to say there is no lockdown (e.g. Germany, Switzerland) and their actions are correspondingly less strict. However they do entail a large curtailment of freedoms even if they are less strictly enforced.</p><p>France seems to have been most strict with their measures in enacting fines for violators although I don’t know how effective these are.</p><p>If I borrow VipulNaik's <a href="https://www.lesswrong.com/posts/pBPiZQYBF9niRAMSq/coronavirus-the-four-levels-of-social-distancing-and-when">taxonomy</a>, all of them are somewhere between level 2 and level 3 lockdown.</p><p>The UK and Switzerland are a bit behind the other countries in terms of cases but their actions have similarly lagged so haven't taken advantage of their initial advantage.</p><h2>Iran</h2><p>Iran is a strange one in that their total number of new cases per day has been fairly flat for a couple of weeks. I’m not sure whether they have achieved a perfect <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R_0=1"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span> or whether their data is a bit funny – their deaths data don’t really reflect their confirmed cases although more recently they start to match more closely.</p><h2>South Korea</h2><p>South Korea are my favourite COVID-19 dealing country.</p><p>They essentially had it under control until the infamous patient 31 infected a large number in his church who then went on to infect more until there were >5k cases associated with the church (more than half of the total number of cases in the country).</p><p>Despite the government never imposing any particularly strict orders, the entire city of Daegu was <a href="https://www.reuters.com/article/us-china-health-southkorea-cases/like-a-zombie-apocalypse-residents-on-edge-as-coronavirus-cases-surge-in-south-korea-idUSKBN20E04F">deserted</a> within a couple of days of the patient 31 outbreak being confirmed. Between that and the intensive contact tracing and testing program the outbreak was quickly brought under control so that the hospitals weren’t overrun and the fatality rate was kept down to 1.3%.</p><p>In 2015 South Korea experienced the second worst outbreak of MERS. There were 168 confirmed infections and 38 people died. <a href="https://thebulletin.org/2020/03/south-korea-learned-its-successful-covid-19-strategy-from-a-previous-coronavirus-outbreak-mers/">This article</a> has an interesting summary of how the lessons from that outbreak fed into the COVID-19 response.</p><p>The halving time during the reduction phase was 3-5 days.</p><p>Arguably they have now entered their dance phase as <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R_0\approx1"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">≈</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span>.</p><h2>Japan</h2><p>I haven’t included Japan on the graphs as nothing much has happened there which is pretty amazing. Japan is probably what South Korea would look like if there had been no patient 31.</p><p>They have taken precautions similar to Western countries before the latter implemented stricter lockdown. However they have managed to contain every cluster of cases before any have got out of control.</p><p>There has been a lot of talk about Japan not doing enough testing and that their numbers are artificially suppressed. My prior for this is pretty low – this seems like an unlikely thing for a government to do, especially as it wouldn’t take long before the truth came out as the death toll rose. </p><p>As for evidence against that hypothesis, Japan have done a lot of testing compared to the number of cases - 19 out of 20 tests come back negative (even if the absolute numbers are low). If they are deliberately suppressing their numbers then they’re doing a really good job at testing the wrong people.</p><p>Japan’s cases started to get serious in mid-Feb. I think it’s clear that they managed to avoid any out of control outbreaks until at least the beginning of March, otherwise there would be so many cases by now that it would be obvious. If they can keep the virus in control for 3 weeks then they can probably keep it in control for a couple more up until now. If a cluster does get out of control in Japan then I expect it to go the same way as South Korea.</p><p>Of course the Japanese government could be lying about everything but again if they are I would expect better evidence from citizens/journalists by now. </p><h1>Summary</h1><p><u>Doubling times</u></p><p>Unmitigated spread: 2-3 days</p><p>Improved hygiene and basic social distancing: 3-5 days</p><p>Lockdown with work allowed: 5+ days (possibly cases decreasing)</p><p><u>Halving times (single sample each)</u></p><p>Full lockdown: 2-5 days</p><p>Flexible lockdown + Epic contact tracing: 3-5 days</p>buckyZqW5wzyLhbayHy2QC2020-03-27T21:33:25.851ZComment by Bucky on Breaking quarantine is negligence. Why are democracies acting like we can only ask nicely?
https://lw2.issarice.com/posts/pfRX8CkFYbYhgHdfS/breaking-quarantine-is-negligence-why-are-democracies-acting?commentId=GSjy5vHE2JFBJx68s
<blockquote>
<p>You can't successfully sue Bob for giving you COVID unless you can prove it more likely than not that your COVID came from Bob. That's basically impossible.</p>
</blockquote>
<p>In South Korea where the contact tracing is working this seems like it would be possible. Patient 31 has been more-or-less determined to have lead to 5k+ people getting infected.</p>
<p>If the numbers come down in the US such that the authorities are able to contain via contact tracing then this would become reasonable there too.</p>
buckyGSjy5vHE2JFBJx68s2020-03-25T20:53:36.192ZComment by Bucky on What will the economic effects of COVID-19 be?
https://lw2.issarice.com/posts/owk7eBzwjPHNQbNeL/what-will-the-economic-effects-of-covid-19-be?commentId=rpM3LWksq9FrzDQKJ
<p>Here's my rough sketch of what might happen to large manufacturing companies due to shutdowns within the supply chain.</p><h2>Costs of other people's shutdowns</h2><p>Different countries / companies are likely to have different shutdown periods.</p><p>Supply chains are highly international so shutdowns in one country have huge knock-on effects. Most companies will aim to be dual sourced on critical components which require long qualification periods but this isn't always simple. Even if dual sourced, bringing up the second source to cover the loss from the other won't happen overnight, if at all.</p><p>For instance, say you buy a critical component from Italy. It's dual sourced but on an 80:20 ratio so if Italy shuts down then your the potential output drops by 80%. </p><p>In reality stock levels will often sit at 30 days so for a shorter shutdown there is some slack. However, 30 days is an average. With enough different components you're likely to be low-ish on stock for at least a couple of components and may be forced to drop output.</p><p>The other side of this is customers. Some customers will shut their production lines, others stay open. If your product in fairly homogenous this might be fine and match your incoming component levels. With a wider product range you may find that your supply line fails in the place where you most need the components.</p><p>The above will apply to a lesser extent where companies have to slow production.</p><h2>Cost of your shutdown</h2><p>Whilst you're shut down you aren't producing anything. This obviously has the direct impact of reducing turnover to 0 for however long you're closed. This is almost certainly the biggest impact. If the government steps in to cover this (at least partly, e.g. the UK government will pay 80% of wages) then large manufacturers shouldn't have an issue. Even if they have a temporary cash flow problem, the banks are likely to step in to help out.</p><p>Costs like renting the space that you're in might still need to be paid. However if you're struggling then whoever you're renting off doesn't really have the option to rent out to someone else in the short term so there may be some renegotiating being done (I'm less confident about this point - I'm not sure how the contracts would work out).</p><p>When you shutdown you probably already have lots of goods on their way to you by sea. My guess is that if possible people will try to get these accepted into the factory although I know some deliveries are being turned away if they have come through a high risk country.</p><p>When you reopen, everything will be a mess and the first week will be chaos (although with people working from home maybe this can be minimised with good planning). For a month or so things will be a bit muddled so efficiency won't be optimal.</p><p>You'll have some customers chasing you to get product immediately. I guess there'll be a huge demand for airfreight - maybe this is how the airlines can recuperate some of their losses?</p><p>There may need to be some working with customers and suppliers on contractual terms of payment etc. In normal circumstances these are very tightly controlled but I would anticipate that most companies will be able to take the practical approach and overcome the bureaucracy which is inherent in such negotiations, due to the exceptional circumstances. Companies which are unable/unwilling to do this are likely to suffer additional damage.</p><h2>Smaller companies</h2><p>The above mostly applies to smaller manufacturers but to a lesser degree.</p><p>They are likely to be lower on the list of priorities for banks to sort out emergency loans which could cause a number to go out of business. This may be the target of additional government intervention.</p><p>Supply chains are likely less complex and so have fewer critical point to go wrong. They are also probably able to switch suppliers more easily if required.</p><p>They will manage to get things sorted out more easily before and afterwards.</p><p>Smaller companies have less leverage in negotiating new contracts. In purchasing this is offset by probably being able to be more flexible. In sales this is harder if they are selling to larger companies.</p><h2>Overall economy</h2><p>So multiply the above throughout the economy and you get a large variation across companies depending on how the individual supply chains which they are a part of are hit. Everyone will kind of muddle through as best they can but things will be far from efficient for as long as there are significant parts of the world in lockdown, even for companies which aren't in lockdown.</p><p>The obvious cost of lockdown (lack of productivity) is likely to be the most important and other considerations are likely to be large but considerably smaller.</p><p>***</p><p>I wrote the above and then realised that this was based on the assumption that overall demand for your product will be the same a year after lockdown as it was a year before. For many industries this is probably true but others (e.g. some luxury goods?) might not bounce back fully or might bounce back into a different shape than before. This is a completely different question that I'm not sure how to answer.</p>buckyrpM3LWksq9FrzDQKJ2020-03-25T13:50:51.808ZComment by Bucky on Using smart thermometer data to estimate the number of coronavirus cases
https://lw2.issarice.com/posts/yqbqeCzCuisWkMezo/using-smart-thermometer-data-to-estimate-the-number-of?commentId=w3DwuDAwT9NwCL9BB
<p>I think it would be good practice to add the results of your sensitivity analysis into your summary here. If I'm understanding your sheet correctly the range found from the sensitivity to uncertainty in baseline rate of fever gives COVID-19 numbers of 0.9 - 7.0 million cases?</p>buckyw3DwuDAwT9NwCL9BB2020-03-23T09:45:29.623ZComment by Bucky on Preprint says R0=~5 (!) / infection fatality ratio=~0.1%. Thoughts?
https://lw2.issarice.com/posts/eLhJ96bLuNsA6qNvw/preprint-says-r0-5-infection-fatality-ratio-0-1-thoughts?commentId=44rtdjdWhQPzm22ep
<p>If there is a 1:1 symptomatic:asymptomatic ratio and 2,000,000 odd infections then there are 1,000,000 symptomatic people out there and only 40,000 identified. Of that 1,000,000 we expect 200,000 to require hospitalisation and 50,000 to require ICU.</p>
<p>If this was true I would expect someone to have noticed.</p>
<p>There might be another explanation for the figures that I’m missing but, as I said, I think it’s up to them to explain what they think is going on.</p>
bucky44rtdjdWhQPzm22ep2020-03-21T16:47:24.084ZComment by Bucky on Preprint says R0=~5 (!) / infection fatality ratio=~0.1%. Thoughts?
https://lw2.issarice.com/posts/eLhJ96bLuNsA6qNvw/preprint-says-r0-5-infection-fatality-ratio-0-1-thoughts?commentId=4CYTQihNcGfA6Mfgm
<p>Diamond princess is important because they did 100% testing so it gives us an idea of asymptomatic : symptomatic ratio. The result was roughly 1:1, nothing like 50:1 or whatever this paper suggests. The science study with 6:1 is at least plausible if you account for symptomatics who weren't identified.</p><p>If South Korea hadn't managed to test the majority of their cases then it is unlikely that they would have managed to reduce their infection rate so dramatically - their quarantine measures aren't massively strict although I think the population are self-enforcing good practice pretty well. I doubt that Wuhan death rates could be below South Korean rates due to the acknowledged overcrowding in Wuhan. Again, 0.6% is kind of plausible, the model here (0.1%) isn't.</p>bucky4CYTQihNcGfA6Mfgm2020-03-20T22:31:24.672ZComment by Bucky on Preprint says R0=~5 (!) / infection fatality ratio=~0.1%. Thoughts?
https://lw2.issarice.com/posts/eLhJ96bLuNsA6qNvw/preprint-says-r0-5-infection-fatality-ratio-0-1-thoughts?commentId=bTXmLa3NQtWnAHa2t
<p>Like others I doubt the infection and fatailty rates because of South Korea and Diamond princess (if the author knew about how much this result conflicts with those datasets then its up to them to argue why the new paper is better).</p><p>R0=5 isn't completely unbelieveable. If the doubling time without containment measures is <a href="https://www.lesswrong.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks">2 days </a>and the infective period is 12 days (i.e. 5 days incubation period and a week afterwards) then R0=5. Unfortunately based on the rather unbelievable infection and fatality rates I don't think this paper really adds any evidence for this - it suggests the model is fatally flawed.</p>buckybTXmLa3NQtWnAHa2t2020-03-20T20:33:13.022ZComment by Bucky on Preprint says R0=~5 (!) / infection fatality ratio=~0.1%. Thoughts?
https://lw2.issarice.com/posts/eLhJ96bLuNsA6qNvw/preprint-says-r0-5-infection-fatality-ratio-0-1-thoughts?commentId=R3R5wwKtC5QTniM9f
<p>I was particularly bemused by quoting cumulative infections to 7 significant figures where the 95% confidence interval spanned a factor of 2. This did not fill me with confidence...</p>buckyR3R5wwKtC5QTniM9f2020-03-20T20:22:25.926ZComment by Bucky on Growth rate of COVID-19 outbreaks
https://lw2.issarice.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks?commentId=tr3d6XLy3R9qZj8kL
<p>Over the past week or so Australia has lost containment and is running at a doubling time of 3-3.5 days. I don't know whether that correlates with higher concentration of cases in big cities - my prior would be that most imported cases would arrive in the big cities in the first place but I haven't checked this.</p>buckytr3d6XLy3R9qZj8kL2020-03-19T15:44:04.626ZComment by Bucky on Covid-19 Points of Leverage, Travel Bans and Eradication
https://lw2.issarice.com/posts/Ddgry4k64oBZYfrHy/covid-19-points-of-leverage-travel-bans-and-eradication?commentId=y3hEHtDWBbce57CQ3
<p>It doesn't matter hugely whether hospitals are overloaded by 3-5x for a long time or 20x for a relatively short time. In the former about 75% of people are unable to get treatment, in the latter 95%. 3-5x is better but isn't as much better as it might seem.</p><p>The UK government has claimed that hygiene/social distancing can flatten the curve by 20% but 20% doesn't make much difference unless you happen to be just over capacity beforehand.</p>buckyy3hEHtDWBbce57CQ32020-03-19T15:14:19.810ZComment by Bucky on [UPDATED] COVID-19 cabin secondary attack rates on Diamond Princess
https://lw2.issarice.com/posts/w49gTfuQEZxRDS6jM/updated-covid-19-cabin-secondary-attack-rates-on-diamond?commentId=Pa8xh4nPhQhn9CXTG
<blockquote>If I assume the rate at which people in single-person cabins get infected (8%) is the rate of infection outside the cabin, and that the higher rate of infection in two-person cabins is caused entirely by within-cabin secondary transmission, then it looks like each person would have to infect their partner an average of 1.5 times each. This also tells us that the transmission rate between elderly couples sharing a bed is likely to be extremely high, and also that people in single-person cabins must be different in some way--perhaps they spent less time in the ship's common areas. </blockquote><p>This was my original thought too. However, as the 8% is based on only 6 positive cases it isn't a very precise figure.</p><p>As an example, the maximum likelihood for any pair of variables for my models comes at background infection rate of 0.133, secondary attack rate=0.55 with no tertiary attack (I didn't mention this in the OP for fear of people taking the 0.55 to be especially relevant). In this case the probability of getting 6 or fewer infections in 1-berth cabins would be 0.11 - unlikely but not massively so.</p><p>The corresponding probabilities for 2, 3 and 4-berth cabins are 0.68, 0.14 and 0.50. Those 4 numbers seem fairly random, suggesting that there's no need to stipulate base rates which vary based on cabin size to explain the data.</p><p>In truth I suspect that there may be differences in the base rate between cabin sizes but wouldn't have known in advance which size would have had a higher base rate. With only 4 data points even using 2 variables in the model is pushing it - if I used anymore I could have explained almost anything!</p><p>***</p><p><em>Edit: Section below is no longer endorsed</em></p><p><em>Regarding the effect of quarantine measures, only 115 of the 536 passenger infections analysed had onset after the quarantine started. Figure 1 <a href="https://www.niid.go.jp/niid/en/2019-ncov-e/9417-covid-dp-fe-02.html">here</a> suggests to me that almost all of the infections occurred before quarantine and onset was delayed by incubation period.</em></p>buckyPa8xh4nPhQhn9CXTG2020-03-19T10:30:41.241ZComment by Bucky on [UPDATED] COVID-19 cabin secondary attack rates on Diamond Princess
https://lw2.issarice.com/posts/w49gTfuQEZxRDS6jM/updated-covid-19-cabin-secondary-attack-rates-on-diamond?commentId=uzjLSm6ugkaYYftiY
<p>attack rate = 1 within a cabin would be everyone catches it at some point (but not necessarily immediately) provided that someone brings it in in the first place - its a rate per sick person rather than per unit time. I don't have data on whether this is the case although I doubt it. </p><p>Technically I suppose having 18 cases in 4-berth cabins does rule that out. My model isn't sophisticated enough to catch something like that - I look at average illness rate as an input to the binomial distribution, I never check whether the total number is likely. Adding that complexity might help narrow down the true secondary attack rate.</p><p>I've added a log graph.</p>buckyuzjLSm6ugkaYYftiY2020-03-18T23:34:36.243Z[UPDATED] COVID-19 cabin secondary attack rates on Diamond Princess
https://lw2.issarice.com/posts/w49gTfuQEZxRDS6jM/updated-covid-19-cabin-secondary-attack-rates-on-diamond
<p>Update 19/03/20: Inspired by johnswentworth's <a href="https://www.lesswrong.com/posts/w49gTfuQEZxRDS6jM/covid-19-cabin-secondary-attack-rates-on-diamond-princess?commentId=W5thdiRuXHMS8TBuT">comment</a>, I implemented a multinomial distribution on the 4-berth cabin result. Taking this additional information into account the model shows reduced likelihood of secondary attack rates of >0.9.</p><h2>Introduction</h2><p>Jimrandomh recently <a href="https://www.lesswrong.com/posts/QdfD3bbMYAbCLv4aB/covid-19-s-household-secondary-attack-rate-is-unknown">showed</a> how we have no real idea about the household secondary attack rates of COVID-19.</p><p>The Diamond Princess <a href="https://www.niid.go.jp/niid/en/2019-ncov-e/9417-covid-dp-fe-02.html">data</a> showed that the proportion of passengers infected with COVID-19 increased with cabin occupancy. </p><p>It occurred to me that this data could be used to infer the cabin secondary attack rates.</p><h2>Data</h2><p>I eyeballed the data in figure 2 in the report linked above.</p><p>There were 6 COVID-19 cases in single passenger cabins which looks like ~8% infection rate so there were ~75 passengers in single cabins.</p><p>For double cabins the numbers are 485/2425 = 20%.</p><p>For triple cabins 27/129 = 21%.</p><p>For 4-berth 18/60 = 30%.</p><p>(all numbers are per person, rather than per cabin)</p><p>These numbers add up to 2,689 total passengers which is slightly more than 2,646 actually included but this is close as eyeballing is likely to get me.</p><h2>Method</h2><p>I implemented a model with 2 variables:</p><p>1. The background rate of infection without sharing a cabin (just from being on the ship).</p><p>2. An additional rate of infection for each infected person an individual shared a cabin with.</p><p>Given those two variables I was able to create predicted infection rates for each size of cabin by calculating the probability of the number of initial cases in a cabin (before secondary attack) and then the probability of each result after applying secondary attacks. </p><p>I created 2 models, one where I only included secondary attack and another where the victim of the secondary attack could in turn cause a tertiary attack on any remaining healthy members of the cabin. Tertiary attack may not have been possible (or somewhat suppressed) by the quarantine and/or other factors.</p><p>Importantly the secondary attack rate as used by me here is “probability of contracting COVID-19 for each person in the cabin who had COVID-19”. So if you live with 2 infected people then you have a higher probability of contracting than if you just lived with 1. In 4-berth cabins having even one person infected gives a high probability of at least one of the remaining people being infected at which point the other 2 have a higher chance (when allowing for tertiary attack). </p><p>Even with a relatively low attack rate per person, it ends up being likely that many people in a 4-berth cabin will end up infected. For instance with a 0.3 secondary attack rate there is a >30% chance of all 4 people getting it from a single incoming case. A 0.5 secondary attack rate brings this up to >70% chance</p><p>These models were used to create likelihoods for the results actually witnessed via a binomial distribution.</p><p>As this model isn’t computationally expensive I just brute-force calculated the likelihood over a number of possible values of the 2 variables. I then integrated across the background rate to give the likelihood function of the secondary attack rate.</p><h2>Results</h2><p>The likelihoods of the secondary attack rates for the two models are shown in the figure below. I’ve also included a combined likelihood based on equal confidence in both models.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1584627279/blog%20posts/SecondaryAttack2.png" class="draft-image " style="width:85%"></figure></span><p>And on a log axis:</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1584627284/blog%20posts/SecondaryAttackLog2.png" class="draft-image " style="width:86%"></figure></span><p>This is slightly frustrating – there is a large range of secondary attack rates which fit the data adequately. </p><p>The most noticeable thing is that a very low secondary attack rate appears to be ruled out. Only 7% of the likelihood is below 0.15 and 3% below 0.1. This goes against the results from the papers analysed in jimrandomh's post (0.1 and 0.15)</p><p>The large range of possible values is caused in large part by the relatively small sample size for all except 2-berth cabins.</p><h2>Discussion</h2><p>There are some potential confounders here, for instance 2-berth cabins are probably mainly couples whereas 4 berth are relatively more likely to include children. I don't expect these effects to be very large (couples and their children will all have close contact) but hopefully someone will point out any potential larger confounders in the comments if there are any. </p><p>It is also not certain that cabin secondary attack rates convert directly to household secondary attack rates although my personal expectation is that they wouldn't be too far off.</p><p>Most of these secondary attack values are very bad news for larger households. Plenty of <a href="https://www.lesswrong.com/posts/9GyKccaJdLEbdhyTi/a-significant-portion-of-covid-19-transmission-is">presymptomatic transmission</a> means that if one person gets it then at least one more person will likely get it before anyone is aware that they have. So if someone does become symptomatic then isolating from each other is likely to be as important as being careful around the patient.</p><p>Isolating from each other when no-one has symptoms is likely a very costly exercise as it would need to be maintained for months but the bigger the household the more benefit is to be gained from taking care.</p><p>My impression from looking at the virus growth rate data from various countries is that massively improving hygiene and implementing social distancing can increase the doubling time by a factor of 2 (I hope to write this up in the coming days). If it can similarly halve secondary attack rate then this could be hugely important in large households to prevent a single case infecting the entire house.</p><p>Note that as jimrandomh said, leaving a household with a sick patient in order to avoid contracting COVID-19 is a bad idea.</p><blockquote>If people tried to move out when their housemates got sick, they wouldn't lower their own risk much, but they would spread it wherever they moved to. </blockquote><h2>Conclusion</h2><p>Cabin secondary attack rates of COVID-19 on the Diamond Princess were not able to be confirmed precisely. It is unlikely that the rate was very low (<0.2) and as a result additional infections are likely, especially in larger cabins.</p><p>If this can be extrapolated to households then particularly larger households may struggle to prevent additional infections after the first household member is infected.</p>buckyw49gTfuQEZxRDS6jM2020-03-18T22:36:06.099ZComment by Bucky on Bucky's Shortform
https://lw2.issarice.com/posts/8Sqbwb3XTkYAT7ZSF/bucky-s-shortform?commentId=Lkc7vwRQi7Q66ccEs
<p>Hmm, I don’t think that’s what an upvote generally represents. An upvote is more of a general “I’d like to see more like this” rather than a specific “I researched this point and found it to be correct”.</p>
buckyLkc7vwRQi7Q66ccEs2020-03-17T23:43:19.346ZComment by Bucky on Good News: the Containment Measures are Working
https://lw2.issarice.com/posts/rJcEZhiCcmkvD2M7h/good-news-the-containment-measures-are-working?commentId=9rHD6szSkWcMMPb3g
<p>The most interesting case to me is South Korea who have managed to achieve containment without mass lockdowns - instead they have an aggressive testing policy such that only 3% of tests are positive.</p>bucky9rHD6szSkWcMMPb3g2020-03-17T13:48:12.752ZComment by Bucky on Reasons why coronavirus mortality of young adults may be underestimated.
https://lw2.issarice.com/posts/oZYZj8y8GeP7GwC9f/reasons-why-coronavirus-mortality-of-young-adults-may-be?commentId=sFDyyM8wDxK2eWZMD
<p>Note that severe =/= critical (I think that post confuses severe with critical in the conclusions)</p><p>In the three Chinese studies which are quoted, severe includes e.g. cases of shortness of breath and high fever which, whilst possibly hopsitalisable under normal circumstances, not obviously fatal if hospitals are full. Severe doesn't imply "requiring mechanical ventilation or other intensive care."</p><p>A better estimate for ICU cases based on that evidence would be ~0.4% as (severe + critical) / severe = 4 in the 44k person China data.</p><p>Of course some severe but not critical cases might become critical if not treated, so death rate without any treatment would be between the two.</p>buckysFDyyM8wDxK2eWZMD2020-03-17T13:05:02.422ZComment by Bucky on Coronavirus: Justified Practical Advice Thread
https://lw2.issarice.com/posts/LwcKYR8bykM6vDHyo/coronavirus-justified-practical-advice-thread?commentId=pTqADixNvjppxgXYu
<p>Looks like <a href="https://www.medrxiv.org/content/10.1101/2020.03.09.20033217v2.full.pdf">v2</a> of the paper has corrected the error.</p>
buckypTqADixNvjppxgXYu2020-03-16T23:44:25.371ZWhy such low detected rates of COVID-19 in children?
https://lw2.issarice.com/posts/YKt5qc6NNkE7p4Z4K/why-such-low-detected-rates-of-covid-19-in-children
<p>In China <a href="http://weekly.chinacdc.cn/en/article/id/e53946e2-c6c4-41e9-9a9b-fea8db1a8f51">0.9%</a> of COVID-19 cases were in people <10. In the Chinese population 11.9% of people are <10.</p><p>In Italy its <a href="https://en.wikipedia.org/wiki/2020_coronavirus_pandemic_in_Italy">0.5%</a> and 8.4%.</p><p>In South Korea it's <a href="https://www.cdc.go.kr/board/board.es?mid=a30402000000&bid=0030">1%</a> and 8.4%.</p><p>10-19 year olds are also very underrepresented although less severely so.</p><p>This seems odd. </p><p>I've seen suggestions that this is down to younger people not being symptomatic and hence being less likely to be tested. </p><p>Contra this, these countries have implemented ALOT of tests (China: unknown but apparently can do >1.5M/week, S. Korea: 275k, Italy: 125k). I can imagine children being tested less, but this much less? If there are 97 negative tests in 100 positive tests (as in South Korea), I'm struggling to believe that there are hundreds of children out there who are asymptomatic COVID-19 carriers whom the testing agencies are just missing.</p><p>On <a href="https://www.niid.go.jp/niid/en/2019-ncov-e/9417-covid-dp-fe-02.html">Diamond Princess</a>, with 100% testing, 33% of <20s who contracted Coronavirus were symptomatic, albeit with a small sample (2/6, in under 10s it was 0/1) which isn't hugely far off the average of 49%. 20-29 year olds had a larger sample and symptomatics were 89%. It would be strange to see such a dramatic change and I am no less confused than before.</p><p>So, what gives?</p>buckyYKt5qc6NNkE7p4Z4K2020-03-16T16:52:02.508ZGrowth rate of COVID-19 outbreaks
https://lw2.issarice.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks
<p><em>Edit 14/03/2020: The top two graphs are now available as interactive versions <a href="https://chart-studio.plot.ly/~Bucky13/1">here</a> (thanks to Ruby for helping with getting this uploaded). The labels on the right are clickable to remove or add countries (double click selects only that country or all countries). The buttons at the top change the y-axis (annoyingly the y-axis range buttons auto-set to a linear scale) and the slider at the bottom zooms the x-axis.</em></p><p><em>Note that the doubling times are actually lower than in the post below due to an error in my original spreadsheet. I've also added the last few days worth of data to the graphs.</em></p><p>COVID-19 has now broken out in a number of countries. This enables us to compare spread rates across to get a better idea of what to expect.</p><p>Below is a graph of cumulative cases in each country. In an attempt to normalise the x-axis, I have plotted from the day that the total number of cases in the country passed 40 (40 was just because the earliest China data that I had started at 42).</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583786584/CountriesCumulative.png" class="draft-image " style="width:100%"></figure></span><p>The most obvious thing is that most countries follow a fairly consistent pattern of growth in the first week and a bit.</p><p>The outliers are Singapore, Japan and Australia (plus Hong Kong, not shown). These countries have lots of cases yet have not seen a corresponding fast exponential growth in cases. I'm not sure why these particular countries have bucked the trend or whether there is something odd about their reporting (I looked for this but didn't find anything).</p><p>I haven't considered how many cases are recovered as it was hard to get reliable results and for most locations recovered cases are minimal. Something weird is happening with the number of recoveries in Iran which has over 2,000, despite only passing that number of cases within the last 6 days.</p><h1>Doubling time</h1><p>We can convert the above graph into doubling time:</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583789702/DoublingTime.png" class="draft-image " style="width:93%"></figure></span><p>I've removed the outlier countries for clarity. The doubling time is fairly consistently 2-3 days. It seems to increase slightly over time.</p><h1>China growth rate</h1><p>I wrote a post <a href="https://www.lesswrong.com/posts/9iYytZ96YZQxufQbo/quadratic-models-and-un-falsified-data">previously</a> about analysing the growth rate of COVID-19 in China.</p><p>If we look at the graph above, the Chinese rate is roughly constant over the first 11 days, after which the growth rate decreases.</p><p>So the first 11 days would fit nicely to an exponential growth model, but what changed? On day 7 (23rd Feb) the quarantine was started. A decrease in growth rate starting a few days later makes sense based on what we know about incubation period.</p><p>Let's assume that the model follows an exponential distribution to start with and then after the quarantine starts to be effective it starts to obey a <a href="https://en.wikipedia.org/wiki/Gompertz_function">Gompertz function</a> which is like an exponential function with a limit to the total number of cases (thanks to <a href="https://www.lesswrong.com/posts/DhsQTTMzwuYLEhjxw/at-what-point-does-disease-spread-stop-being-well-modeled-by?commentId=gHgPSm5RWv7e8TQvQ">clone of saturn</a> for the pointer here). </p><p>I've set both the number of cases and the new case rate to be the same for the two distributions at the point that the Gompertz takes over. This is to minimise free variables so I only have 4 instead of 6.</p><p>Getting the best fit parameters for this model I get:</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583793429/ModelFit.png" class="draft-image " style="width:95%"></figure></span><p>This seems like a fairly good fit. It might be possible to get a better fit with an alternative sigmoid function but this is good enough for my purposes.</p><h1>Conclusion</h1><p>I'm fairly confident that, left unchecked, COVID-19 will increase at a doubling time of 2-3 days. When containment in breached in a location this is the rate that the growth occurs at over the first few week or so.</p><p>When effective measures are put in place this decreases. An effective quarantine may be able to convert the growth into a sigmoid function with a limit on the failure rate.</p><p>Some locations (Japan, Singapore, Australia and Hong Kong) have managed to avoid exponential growth despite having a large number of cases.</p><h2>Appendix 1 - Linear growth charts</h2><p>Suggested by <a href="https://www.lesswrong.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks?commentId=Sf7voHrZxpqeLdyZZ">Raemon</a>.</p><p>All cases</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583833157/blog%20posts/lin50000.png" class="draft-image " style="width:100%"></figure></span><p>Y-axis limited at 8,000 cases per country</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583833155/blog%20posts/lin8000.png" class="draft-image " style="width:100%"></figure></span><p>Y-axis limited at 1,000 cases per country, X-axis limited to first 10 days</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583833160/blog%20posts/lin1000.png" class="draft-image " style="width:100%"></figure></span><h2>Appendix 2 - Deaths vs cases</h2><p>Suggested by <a href="https://www.lesswrong.com/posts/KJBQ7GiyvFTBnSEEC/growth-rate-of-covid-19-outbreaks?commentId=sufWySv4cKZS7YACv">Unnamed</a>.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583835967/blog%20posts/Deaths_vs_cases.png" class="draft-image " style="width:100%"></figure></span>buckyKJBQ7GiyvFTBnSEEC2020-03-09T23:16:51.275ZQuadratic models and (un)falsified data
https://lw2.issarice.com/posts/9iYytZ96YZQxufQbo/quadratic-models-and-un-falsified-data
<p>On 5th Feb a commenter on Reddit <a href="https://www.reddit.com/r/dataisbeautiful/comments/ez13dv/oc_quadratic_coronavirus_epidemic_growth_model/">posted</a> that the coronavirus cases in China were following a suspiciously accurate quadratic curve, implying that China was making up their data and weren’t even bothering to hide it particularly well.</p><p>This set off my bullshit sensors fairly strongly so I wanted to check it out.</p><p>Having looked at the data myself, there may be other reasons why the Chinese data are not accurate but I think the quadratic pattern provides no evidence in favour of falsification.</p><p>I doubt people are making many decisions based on that post but possibly looking at where the statistics went wrong may be beneficial.</p><h1>R^2 </h1><p>It turns out that the claim of a “near perfect” model was based on a very high <span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="R^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span> value (0.9995).</p><p> <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span> is often chosen to summarise how precisely a regression fits the data – it tells you how much of the variance in the data is explained by the equation. Surely if only 1 part in 2,000 isn’t explained by the model that indicates that the data is fabricated?</p><p>The first thing to note is that "variance" is a technical term which isn't the same thing as the natural understanding of "variation". If you don't appreciate this then <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R^2=0.9995"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.9995</span></span></span></span></span></span> will seem more impressive than it is. In particular, variance isn't the deviation from the mean, but the <u>squared </u>deviation. In the more natural understanding, the quadratic model explains 44 parts in 45 of the deviation from the mean, not 1,999 in 2,000. This is still pretty good but seems like less evidence for fabrication.</p><p>An alternative explanation of the <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span> value is that it compares two models: </p><p>1. The model that you are trying to fit to the data</p><p>2. The model where the value of y is expected to be the same for all values of x (the mean of the y values)</p><p>A high <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span> value is telling us that model 1 is a much better fit that model 2.</p><p>However in our case we already know model 2 is going to be a terrible, terrible, terrible fit to our data. The y value used in the regression is total cases so far. So model 2 represents some number of cases having already been identified at the beginning of the time period in question and no more cases occurring during the time period.</p><p>So saying that model 1 accounts for 44 parts in 45 of the deviation between the data and model 2 doesn’t really tell me much – model 2 is a lost cause.</p><h1>New cases per day</h1><p>The problem here is the chosen y-axis. Instead of choosing total number of cases by a certain day, it would be better to choose new cases per day. This removes excess correlation between data points.</p><p>If we do this then instead of fitting a quadratic curve we need to be fitting a straight line (we’re taking the derivative with respect to x) but I'll still call it quadratic for the sake of consistency. Model 2 changes from representing a constant total number of cases to a constant number of new cases every day. This still isn’t a particularly likely model but is certainly an improvement.</p><p>Plotting and regressing we get:</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583453055/blog%20posts/per_day.png" class="draft-image " style="width:70%"></figure></span><p>Our <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span> value has gone down to 0.96. This is still high and suggests that quadratic growth is a fairly good model for the data but isn’t suspicious. For instance, I can also fit a power law (again 2 free parameters) to the data and get <span><span class="mjx-chtml"><span class="mjx-math" aria-label="R^2"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">R</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span></span></span></span></span> = 0.966. </p><p>So within the training set our quadratic model (linear new cases per day) explains a comparable level of variance as a power law model of new cases does.</p><h1>Looking outside the dataset</h1><p>The obvious thing to do is check whether the pattern was there outside the dataset. </p><p>It wasn’t.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1583453227/blog%20posts/Beyond.png" class="draft-image " style="width:75%"></figure></span><p>It is clear that the pattern breaks down shortly after it was noticed (day 15 on this chart).</p><p>In addition we can look at the pattern before the training set. Again as soon as we go outside of the dataset used to create the graph the pattern completely breaks down. This is not surprising as at this point the quadratic model predicts a negative number of new cases per day.</p><p>(Looking a deaths instead of cases shows a similar story. For deaths the pattern keeps going for a little longer but even then the power law fit matches the data better.)</p><p>So to make the case for China falsifying the data quadratically you have to also say that the start date for them doing it was ~20th Jan and the end date more-or-less straight after the pattern was noticed. Presumably this would be justified by China having been caught out and changing from then on.</p><p>Or possibly, this is just how the virus develops. Now that we have some developments in other countries it is possible to compare spread rates. Where the disease has got out of containment there is a remarkably consistent pattern of growth which matches the China rate very closely. (I'm planning to write a more detailed post on this.)</p>bucky9iYytZ96YZQxufQbo2020-03-08T23:34:58.128ZBucky's Shortform
https://lw2.issarice.com/posts/8Sqbwb3XTkYAT7ZSF/bucky-s-shortform
bucky8Sqbwb3XTkYAT7ZSF2020-03-08T00:08:23.193ZRugby & Regression Towards the Mean
https://lw2.issarice.com/posts/joyLkEAdkKeSsm5BK/rugby-and-regression-towards-the-mean
<p>With the rugby world cup final approaching this Saturday I figured I'd publish this which I wrote towards the end of last year but never submitted.</p><h1>Introduction</h1><p>A couple of weeks ago England played New Zealand in Rugby Union. For those who don’t follow the sport New Zealand (the All Blacks) are comfortably the best team in the world. They have been top of the world rankings since 2009. They win 92% of their matches. With a population of 4.8m, it’s fair to say that New Zealand’s <a href="https://putanumonit.com/2015/11/10/003-soccer1/">national rugby affiliation</a> must be off the charts. </p><p>At half time England were 15-10 up. An interesting statistic was rolled out. Since their World Cup win in 2011 they have only lost 4 out of 20 of the matches in which they were behind at half time. At the end of the match the All Blacks had turned it around again, winning 16-15, making it 4/21 losses.</p><p>This sounds very impressive but I wondered how much of this effect could be put down to <a href="https://en.wikipedia.org/wiki/Regression_toward_the_mean">regression towards the mean</a>. After all, a team that wins 92% of their matches might well be good enough to win 81% of matches when they are losing at half time without postulating an extra performance boost in such circumstances.</p><p>I thought it would be fun to investigate.</p><h1>Method</h1><p>I gathered all of their results and corresponding half time scores since the 2011 World Cup. I only included data from teams who could realistically win a half against the All Blacks in order not to bias the results with high scores against weaker teams. I included Argentina, Australia, England, France, Ireland, Scotland, South Africa and Wales. This gave 85 results.</p><p>If anything, this will be biased towards having more wins when losing at half time. Losing at half time would imply that the opposition is more likely to be towards the top end of performance. However the predicted second half performance is based on all matches involving the relevant teams. So the calculation isn’t perfect but I couldn’t think of a better way to do it without reducing my dataset which would reduce experimental power.</p><p>The dataset isn’t big enough to allow for lowering my p-value threshold much by multiple hypothesis testing. I have two formal p-values:</p><p> Do New Zealand generally perform better in the second half than the first?</p><p> Do New Zealand perform better than normal in the second half when they are losing at half time?</p><p>I’m looking for p<0.025 for these 2. I calculate other p-values they are labelled as exploratory.</p><h1>Results</h1><h2>First half vs second half</h2><p>Analysing first half vs second half in all games, the mean and standard error of the All Blacks’ points difference over the opposition were:</p><p>First half: <span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="\mu_f = +5.9"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em;">μ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">5.9</span></span></span></span></span></span> <span><span class="mjx-chtml"><span class="mjx-math" aria-label="\sigma_f = 10.8"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.519em; padding-bottom: 0.519em; padding-right: 0.06em;">f</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">10.8</span></span></span></span></span></span> </p><p>Second half: <span><span class="mjx-chtml"><span class="mjx-math" aria-label="\mu_s = +10.0"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em;">μ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">10.0</span></span></span></span></span></span> <span><span class="mjx-chtml"><span class="mjx-math" aria-label="\sigma_s = 11.25"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">11.25</span></span></span></span></span></span> </p><p>This difference in relative performance is significant (p=0.012) – the All Blacks perform better in the second half than the first half. This matches the commonly held impression that the All Blacks’ fitness is better than most teams so they can outperform them in the second half. This is important for future calculations.</p><p>With those first half numbers and a normal distribution, we would expect the All Blacks to lose 28% of their first halves – i.e. 23.6/85. The fact that they have only lost 21 is slightly better than this but not significantly (p=0.082, exploratory).</p><h2>Improved performance when losing at half time</h2><p>Combining the actual data for the amount that they were down at half time in those 21 matches with a normal distribution for their average performance in the 2nd half would predict 7.4/21 losses. Their actual performance (4 losses) is better than this but not significantly (p=0.068).</p><p>Certainly regression towards the mean explains a big fraction of the extra comebacks. </p><p>Naively it seems like you should make a comeback in fewer than half the matches that you are losing at half time. Even if we optimistically say 10/21 comebacks are expected then we can explain away 3.6 additional comebacks by appealing to RttM and there are only 3.4 comebacks left to be explained by improved performance.</p><p>However those 3.4 extra comebacks are still quite impressive - the All Blacks lost 46% fewer games when behind at half time than you would expect even accounting for RttM - but the sample size is too small for it to confidently imply an actual improved performance.</p><h2>Overall performance</h2><p>If I combine the normal distributions from the two halves I get a predicted win rate of 83% - not as good as the 92% achieved. This is significant (p=0.016) but I'm not counting it as it was exploratory. </p><h2>Further exploration</h2><p>So the All Blacks’ performance in second halves where they were losing at half time is better than their average but there is insufficient data to conclude that this is not just coincidence.</p><p>I put together a graph showing the points difference between the All Blacks and their opponents to investigate. </p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1543009526/blog%20posts/New_Zealand.png" class="draft-image " style="width:94%"></figure></span><p>What stands out is the large spike in games which the All Blacks finished with a points advantage of 0-4 after having lost the first half. Of the 21 lost first halves, the All Blacks drew/won the match by a small amount 10 times.</p><p>These extra matches won more than explain the increase in comebacks above the expected level. They also explain nearly all of the difference between overall expected win percentage based on normal distributions and actual win percentage.</p><p>It appears that the All Blacks have won a lot of games by a tight margin over the last 8 years. Is this a coincidence? If not, do they win tight matches because they are great or are they great because they win tight matches?</p><h1>Footnote</h1><p>Looking back at this now, I wasn't harsh enough on the 2nd question. This wasn't a randomly selected question, it was a statistic specifically chosen by the commentator because it sounds impressive.</p><p>If the null hypothesis is true, p-values should be uniformly distributed between 0 and 1. Assuming there were 10 such statistics which the commentator could have used, it wouldn't be surprising that the most juicy-looking one had p<0.10 even if none of the purported effects were real. Therefore p=0.068 is really thoroughly unimpressive and I would conclude more strongly against the second question being confirmed by the data.</p><p>None of this applies to the first question (better in second half than first half in general) as this wasn't a cherry-picked question - I only asked it because it was important for the second question. It also has a reasonable causal mechanism (relative fitness levels) so I'd be inclined to believe it.</p>buckyjoyLkEAdkKeSsm5BK2019-10-30T16:36:00.287ZAge gaps and Birth order: Reanalysis
https://lw2.issarice.com/posts/YnXd7zfGGZfMD9QtA/age-gaps-and-birth-order-reanalysis
<p>This post follows on from my <a href="https://www.lesswrong.com/posts/uZEeqmeFjs3nmawn7/age-gaps-and-birth-order-failed-reproduction-of-results">previous post</a> detailing some areas where I was unable to reproduce Scott’s <a href="https://slatestarcodex.com/2019/05/14/age-gaps-and-birth-order-effects/">analysis</a> of how the age gap between siblings modifies the SSC Birth order effect. I suggest you read that post first but here’s the summary:</p><blockquote>I attempted to reproduce Scott’s analysis of <a href="https://slatestarcodex.com/2019/05/14/age-gaps-and-birth-order-effects/">Birth order effect vs Age gap</a>. I found that:</blockquote><blockquote>There appeared to be an error in graphs 2 & 3 where people with one sibling were counted when they shouldn’t have been (graph 2) or were counted twice (graph 3)</blockquote><blockquote>Comparing oldest children to youngest children causes a bias in the results which can be prevented by comparing oldest children to 2nd oldest children</blockquote><blockquote>I was unable to reproduce Scott’s result on people reporting 0 year age gap – I get a non-significant 58% older siblings compared to Scott’s 70%. I was unable to discover the cause of the difference.</blockquote><h1>Summary</h1><p>I reanalysed how sibling age gap modifies the SSC birth order effect. I found that:</p><p>The birth order effect is relatively steady for the first 4-8 years of age gap at about 70% respondents being the firstborn vs secondborn. For larger age gaps the effect reduces. There is insufficient evidence to conclude how long this reduction takes or whether the effect is completely removed at very large age gaps.</p><p>2 other trends were noted in the data but evidence for them was not strong:</p><ul><li>The reduction may not be the same (or might disappear) for larger families </li><li>Birth order effect may be lower at 1 year age gap vs 2-7 year age gap</li></ul><p>Considering competing theories on the cause of the Birth order effect, two theories fit the data well: </p><ul><li>Intra-family dynamics </li><li>Decreased parental investment</li></ul><p>And three theories fit the data poorly: </p><ul><li>Changed parental strategies </li><li>Maternal antibodies </li><li>Maternal vitamin deficiencies</li></ul><h1>Introduction</h1><p>The original reason for me looking at this data was to analyse whether the data support a sudden drop between years 7 and 8 or whether there is an alternative explanation which fits the data.</p><p>I will note here that I’m not a trained statistician and am using this as practice of Bayesian model comparison, inspired by johnswentworth’s recent <a href="https://www.lesswrong.com/s/onCRFFN7rGXTg3jyc">model comparison</a> sequence. I’d say I’m 80% confident in my broad conclusions, less so in the specifics - I'd be fairly confident there are a couple of errors lurking in here somewhere.</p><h1>Analysis: All family sizes combined</h1><h2>Is there a sudden drop after 7 years?</h2><p>Getting back to the data, here’s the result that I’m going to focus on, comparing 1st to 2nd children in all family sizes:</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567591794/Birth%20order%20effect/Post%202/My_3_new.png" class="draft-image center" style="width:76%" /></figure></span><p>Eyeballing the graph makes the sudden drop after 7 years look like the most natural explanation. However, we had no reason, a priori, to think that a 7 year age gap would have any special significance – a drop could have happened after 1 or 10 years for all we knew. </p><p>If we model a sudden drop after 6 or 8 years the model starts to match the data significantly less well, any further away from 7 than that and the model performs really poorly. Although a general “sudden drop” model has a high maximum likelihood at 7 years, the overall model likelihood is lower due to the lower likelihoods for other drop years.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567679163/Birth%20order%20effect/Post%202/Sudden_Drop_Lines.png" class="draft-image center" style="width:75%" /></figure></span><h2>General slope model</h2><p>Imagine a model which is similar to a sudden drop model but the drop is ramped down over a number of years. The model is defined by 4 parameters – percentage oldest sibling before the ramp (<span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{0}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></span></span>), percentage oldest sibling after the ramp (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span>), at what age gap the ramp starts (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{s}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></span></span></span></span>) and over how many years the ramp occurs (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span></span></span></span></span>). </p><p>The sudden drop model is nested within this model - where <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}=0"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span>.</p><p>A gentler slope doesn’t match the data as closely as a sudden drop but is less harshly penalised over a range of ramp start locations. The graph below shows what some <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}=4"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">4</span></span></span></span></span></span> years ramps might look like.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567679163/Birth%20order%20effect/Post%202/Ramp_Lines.png" class="draft-image center" style="width:86%" /></figure></span><h2>Ramp timing and length</h2><p>To find out which ramp lengths fit the data best I integrate (numerically) across the first 3 parameters in this model (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{0}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></span></span>, <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span>, <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{s}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></span></span></span></span>) to find which value of the 4th parameter (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span></span></span></span></span>) predicts the data the best – how sudden is the drop?</p><p>(Notes:</p><p>For this analysis I haven’t grouped the 10+ year age gaps together but used the actual values for the age gaps.</p><p>For all calculations in this post I assume a uniform prior across a reasonable range for each parameter.)</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567609389/Birth%20order%20effect/Post%202/Ramp_Length.png" class="draft-image center" style="width:87%" /></figure></span><p>Surprisingly, the likelihood is fairly flat over a large range of slope lengths – everything between 0 and 10 years is within a Bayes factor of 1.15 of each other.</p><p>To see what’s happening, let’s integrate over the first two parameters (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{0}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></span></span> and <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span>) and plot likelihood against ramp length (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span></span></span></span></span>) and start (<span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{s}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></span></span></span></span>).</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567610453/Birth%20order%20effect/Post%202/Ramp_Length_time.png" class="draft-image center" style="width:80%" /></figure></span><p>This shows a maximum value at <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}=0"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span>, <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{s}=7"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">7</span></span></span></span></span></span> – the sudden drop after 7 years which is so visually noticeable in the data.</p><p>However, if you follow the line along <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}=0"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span> (back of the graph), there is only a small range of <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{s}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></span></span></span></span> values which have a high likelihood. Looking instead along <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}=5"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">5</span></span></span></span></span></span>, the maximum likelihood is lower (~33% lower), but there is a larger range of <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{s}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></span></span></span></span> values which provide a fairly high likelihood. The decrease in maximum likelihood is almost exactly cancelled out by the increase in the width of the distribution.</p><p>So a sudden drop predicts the data approximately as well as a more gradual drop. </p><p>We can also integrate across <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span></span></span></span></span> to find the posterior probability of the various <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{s}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">s</span></span></span></span></span></span></span></span></span></span>values.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567609901/Birth%20order%20effect/Post%202/Ramp_Start.png" class="draft-image center" style="width:79%" /></figure></span><p>I'm going to describe this as the ramp starting between 4 and 8 years.</p><h2>Percentage oldest children before and after drop</h2><p>I also integrated over <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span></span></span></span></span> and <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{m}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span></span></span></span></span></span></span></span></span> in order to see how likelihood varied with <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{0}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></span></span> and <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span>.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567673676/Birth%20order%20effect/Post%202/Effect_before_and_after.png" class="draft-image center" style="width:78%" /></figure></span><p><span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{0}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></span></span> is very precisely defined between 0.70 and 0.71. </p><p> <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span> can take a large variety of values, between ~0.49 & 0.62 (90% CI). </p><p>In reality, the Birth order effect might decrease relatively fast to start with and then more slowly as oldest and second oldest children approach parity. This is probably the kind of thing which we would expect in real life but which can't be recreated with the ramp model.</p><p>I created an exponential decay model (with a delay in the decay starting) to test whether this might be the case and it got a slightly higher overall likelihood than the general ramp model (Bayes factor 1.5). The start of the decline was in the region 3-8 years, similar to the ramp model. The maximum likelihood half-life was 5 years although this could be anywhere between 1.2-11 years (90% CI).</p><h2>Expected values</h2><p>Using these models I calculated expected values for Birth order effect vs age gap.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567763322/Birth%20order%20effect/Post%202/Expected.png" class="draft-image center" style="width:100%" /></figure></span><p>This looks fairly sensible to me. There is a gradual start to the slope, becoming steeper into about year 8 and then shallowing out as we get closer to parity between older and younger siblings. </p><p>At larger age gaps the two models diverge which is due to a combination of the differing priors implied by the models and the sparsity of data points in this region - the likelihood isn't sufficient to overcome the prior.</p><h2>Comparison to constant birth effect model</h2><p>I also compared the general ramp model to a constant Birth order effect model. The ramp model was preferred over the constant model by a Bayes factor of ~1,000.</p><p>A constant model is actually nested within the ramp model where <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{0}=p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span> (and <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{r}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">r</span></span></span></span></span></span></span></span></span></span>, <span><span class="mjx-chtml"><span class="mjx-math" aria-label="t_{m}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.372em; padding-bottom: 0.298em;">t</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span></span></span></span></span></span></span></span></span> become meaningless). This is illustrated by the red line on the <span><span class="mjx-chtml"><span class="mjx-math" aria-label="likelihood"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">k</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">l</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">i</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">h</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">o</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span></span></span></span></span> vs <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{0}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span></span></span></span></span></span></span> & <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span> graph where the low likelihood can be seen.</p><h1>Analysis: Different sized families</h1><p>I mentioned in my previous post that it appeared that the drop was present in sibships of 2 but not in sibships of 3+.</p><p>Breaking this down further, we can compare this effect for sibships of 2, sibships of 3 and sibships of 4+ (any further breakdown causes the sample sizes to get too small).</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567591802/Birth%20order%20effect/Post%202/Family_Size.png" class="draft-image center" style="width:100%" /></figure></span><p>(The very low value at 7 year age gap for 4+ children is only a sample size of 11 so don’t take it too seriously!)</p><p>Here it appears that the drop-off in birth effect for large age gaps between first and second children happens in sibships of 2 or 3 but doesn’t happen in sibships of 4+.</p><p>Although the number of samples in the 4+ group with >7 year age gap is only 64, the difference between 2-3 and 4+ sibships is significant at p<0.05 (two-tailed t-test).</p><p>This seems an odd phenomenon. Would having extra siblings cause the birth order effect between the oldest 2 siblings to remain high for large age gaps?</p><p>Seeing something weird like this in my data causes me to ask “how many things might I have spotted during my work on this project, if they had coincidentally shown a weird looking result?” – when adjusting for post-hoc multiple hypothesis testing I should adjust not just for the tests that I did but also for the tests I didn’t do just because nothing looked odd.</p><p>In this case the answer is quite a lot so p<0.05 is probably not strict enough and my best bet would be that this data occurred by coincidence.</p><p>That's all a bit hand-wavey so I tried to calculate the Bayes factor comparing:</p><p>A general ramp model for all family sizes </p><p>vs</p><p>A general ramp model for families of 2 & 3 children combined with a shallower (or no) ramp for families of 4+ children (Only <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p_{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span></span></span></span></span> was changed between the family sizes) </p><p>The latter was preferred by a factor of 5. If I were to include other numbers of children when the change might have happened or possibility that the change happens gradually as family size got bigger then this factor would change but that would start getting way too complicated for me!</p><p>I still don't really believe this an actual effect but if someone has an explanation of what might cause this then I'm all ears.</p><h1>Possible lower birth order effect for 1 year age gap</h1><p>One other thing which I noticed is the lower Birth order effect for age gaps of 1 year as compared to gaps of 2-7 years (0.66 vs 0.71 oldest siblings). A quick calculation suggests Bayes factor comes out at 2 in favour of the Birth order effect being lower at 1 year age gap compared it being constant across 1-7 year age gaps.</p><p>Note in this case that although the Bayes factor isn't huge, it seems like this is the kind of thing which might actually happen (some of the potential causes would give this a decent prior - see section below for more discussion) so I'm much less inclined to just write this one off.</p><h1>Comparing Explanations for Birth order effect</h1><p>Scott <a href="https://slatestarcodex.com/2019/05/14/age-gaps-and-birth-order-effects/">lists</a> 5 potential causes of the Birth order effect:</p><p>1. Intra-family competition</p><p>2. Decreased parental investment</p><p>3. Changed parenting strategies</p><p>4. Maternal antibodies</p><p>5. Maternal vitamin deficiencies</p><p>I‘be renamed 1 to "Intra-family dynamics" to include non-competitive interactions between siblings. A few people have mentioned other sibling dynamics which might cause a Birth order effect (e.g. <a href="https://slatestarcodex.com/2019/05/14/age-gaps-and-birth-order-effects/#comment-753722">here</a>). The predictions of age gap effect from competitive vs non-competitive causes seem similar to me so I'll lump them together.</p><p>My thoughts for what each of the 5 potential causes would predict regarding age gap are given below. The conclusions for each potential cause end up being very similar to Scott’s (after all that work!) except that there is no need to postulate anything especially significant about 7 years and that there may be a slight increase in birth order effect between 1 and 2 years age gap.</p><h2>Intra-family dynamics</h2><p>Prediction: Birth order effect remains roughly constant with small age gaps, with less effect as the gap gets larger.</p><p>Assessment: Findings match prediction well. 4-8 years seems reasonable for levels of interactions between siblings to start decreasing. </p><p>Potentially, for a small age gap, a very advanced younger sibling might act more like an older sibling meaning that the 1 year age gap birth effect would be lower. This feels slightly forced to me (I would think any such effect would be fairly small) but am curious what others think.</p><h2>Decreased parental investment</h2><p>Prediction: Birth order effect increases as age gap increases - the longer a firstborn is the only child the longer they benefit from 100% of their parents’ attention. If the earliest years are the most important then birth order might not change after that critical period. Once older children are able to look after themselves, birth order effect might come down with larger age gaps.</p><p>Assessment: The increase in birth order effect between 1 and 2 years would match the theory, if parental investment is mostly important in the first two years. If older children start being able to look after themselves after 4-8 years then this would explain the drop in birth order effect after this time.</p><p>The match between the theory and result is good, although there are a couple of degrees of freedom to help match the prediction to the data. 4-8 years seems reasonable for children starting to look after themselves better but 2 years seems on the low side for a prediction of how long having extra attention is beneficial. Maybe between 2-5 years the two effects roughly cancel out?</p><h2>Changed parenting strategies</h2><p>Prediction: Age gap has minimal effect on Birth order effect.</p><p>Assessment: Prediction matches data poorly. It is possible that parental strategies start to reset towards firstborn strategies after longer age gaps but I wouldn’t have put much of my probability mass on that option. There is a 5 year gap between my youngest children and I definitely didn’t reset towards firstborn strategies, I suspect this would have still been true even for a much larger gap.</p><h2>Maternal antibodies</h2><p>Prediction: Age gap has minimal effect on Birth order effect. Generally you don’t need top-ups of vaccines so presumably antibodies stick around indefinitely? Or is it your body’s ability to make more? Anyway, Scott thinks this is unlikely and he’s a doctor so I’ll take his word for it.</p><p>Assessment: Prediction matches data poorly. My biology knowledge is too poor to know how likely a decrease in effectiveness after 4-8 years would be in this case.</p><h2>Maternal vitamin deficiencies</h2><p>Prediction: Very small age gaps have large effect. Birth order effect decreases rapidly for age gaps <3 years – my estimate for how long it might take to rebuild vitamin stockpiles.</p><p>Assessment: Prediction matches data poorly. 4-8 years seems way too long for vitamin stockpiles to <em>start</em> to build back up. </p><h1>Conclusions</h1><p>The SSC 2019 survey data support a constant, high, birth order effect (~2.4 oldest siblings for every 1 second oldest sibling) for age gaps <4-8 years. This is followed by a decline to a lower birth order effect at an undetermined rate. The decline does not necessarily completely remove any birth order effect although this may be the case for very large age gaps.</p><p>The data provide some evidence that:</p><ul><li>The reduction may not be the same (or might disappear) for larger families (4+ children)</li><li>Birth order effect may be lower at 1 year age gap vs 2-7 year age gap</li></ul><p>However the evidence for both of these points is relatively slim.</p><p>Intra-family dynamics and decreased parental investment predict the results well. </p><p>Changed parental strategies, maternal antibodies and maternal vitamin deficiencies do not predict the results well. </p>buckyYnXd7zfGGZfMD9QtA2019-09-07T19:33:16.174ZAge gaps and Birth order: Failed reproduction of results
https://lw2.issarice.com/posts/uZEeqmeFjs3nmawn7/age-gaps-and-birth-order-failed-reproduction-of-results
<h1>Summary</h1><p>I attempted to reproduce Scott’s analysis of <a href="https://slatestarcodex.com/2019/05/14/age-gaps-and-birth-order-effects/">Birth order effect vs Age gap</a>. I found that:</p><p>1. There appeared to be an error in graphs 2 & 3 where people with one sibling were counted when they shouldn’t have been (graph 2) or were counted twice (graph 3)</p><p>2. Comparing oldest children to youngest children causes a systematic bias in the results. This can be prevented by comparing oldest children to 2nd oldest children</p><p>3. I was unable to reproduce Scott’s result on people reporting 0 year age gap – I get a non-significant 58% older siblings compared to Scott’s 70%. I was unable to discover the cause of the difference.</p><p>I have reanalysed the data based on points 1 & 2 in a separate <a href="https://www.lesswrong.com/posts/YnXd7zfGGZfMD9QtA/age-gaps-and-birth-order-reanalysis">post</a>.</p><h1>Previously in Birth order effect</h1><p>In the 2018 Slate Star Codex survey Scott asked some questions about what order in the family respondents were born. He <a href="https://slatestarcodex.com/2018/01/08/fight-me-psychologists-birth-order-effects-exist-and-are-very-strong/">found that</a> eldest children were massively overrepresented.</p><p>Following on, <a href="https://www.lesswrong.com/posts/tj8QP2EFdP8p54z6i/historical-mathematicians-exhibit-a-birth-order-effect-too">historical mathematicians</a> and <a href="https://www.lesswrong.com/posts/QTLTic5nZ2DaBtoCv/birth-order-effect-found-in-nobel-laureates-in-physics">Nobel winning physicists</a> were found to exhibit the same property. </p><p>In the 2019 SSC survey Scott included questions about age gaps between respondents and their adjacent siblings. He <a href="https://slatestarcodex.com/2019/05/14/age-gaps-and-birth-order-effects/">analysed the results</a>, finding that:</p><blockquote>This study found an ambiguous and gradual decline [in Birth order effect] from one to seven years [Age gap between siblings], but also a much bigger cliff from seven to eight years.</blockquote><h1>Failed reproduction of Scott’s graphs</h1><p>I had originally intended to analyse the data to see if I could draw any further conclusions. However, when running the analysis I found that I was unable to reproduce Scott’s results.</p><p>Scott includes 3 graphs.</p><p>The first – comparing % of sample oldest child vs age gap for people with 1 sibling – I was able reproduce almost exactly (Scott also has access to respondents’ data who asked not to be included in the public data so we aren’t exactly the same. There may be other differences too but these are small). </p><p>The second – comparing how many oldest vs youngest children there are in the sample for people with more than 1 sibling – I was unable to reproduce. Actually, I was able to reproduce the graph but only if I also included people with 1 sibling. </p><p>This is actually what the third graph was supposed to show, but it looks like the third graph double counts the people with 1 sibling. </p><p>The graphs are below, with Scott’s on the left and my reproductions on the right. Note the similarity between Scott’s graph 3 and my graph 2.</p><p>(My version of graph 2 has a different y-axis to all of the other graphs as the range is larger)</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567586842/Birth%20order%20effect/Post%201/6_combined.png" class="draft-image " style="width:100%" /></figure></span><p>As an intuitive way of seeing that there is something wrong – Scott’s third graph lists 7,613 samples included. There are only 8,171 people in the whole survey but the third graph should be ruling out any only children and any children between first and last in their family. It seems that there should be more than 558 people in that group (actually there are 767 only children, not even counting any middle children).</p><p>(Scott’s results were <a href="https://athenaegalea.tumblr.com/post/184870253552/slatestarscratchpad-sometime-tomorrow-ill-be">double checked</a> by Tumblr user athenaegalea but this only involved looking at the data for the first graph which I also agree with).</p><p>Correcting this mistake is important as the 7 year age gap drop in birth effect is predicated on graphs 1 & 2 both showing such a drop. In my reproduction there is no such drop in graph 2 which suggests the 7 year age gap sudden drop may not be a thing after all.</p><h1>Problem with comparing oldest and youngest children</h1><p>Below I have plotted the data from graphs 1 & 2 on the same axis (switching to line graphs to make trends a bit easier to see). I have changed the y-axis to be a ratio of oldest:youngest instead of a percentage to highlight the strangeness of the results.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567586529/Birth%20order%20effect/Post%201/My_1_2_combined.png" class="draft-image center" style="width:72%" /></figure></span><p>The 1 sibling data still show a relatively constant birth effect across different age gaps until 7 years gap. 8 years and above then shows a drop in Birth order effect size.</p><p>The >1 sibling data show a very different trend. At 1 year age gap there are over 7x more firstborns than lastborns. This decreases rapidly as age gap increases. Above 9 years age gap there are actually more youngest children than oldest (although the sample size is relatively small here).</p><p>This seems odd – the two data sets should be reflecting roughly the same process but with different family sizes – the graphs should be similar, or at least closer than this! Does having additional siblings qualitatively change how age gap modifies the Birth order effect?</p><hr class="dividerBlock"/><p>Above we are comparing oldest siblings with youngest siblings and looking at the relative age gaps. In doing so we are implicitly assuming that the age gaps between 1st and 2nd children should be statistically similar to the age gaps between penultimate and last children in the general population.</p><p>This does not match with my experience. In families with >2 children that I know, age gaps between later children tend to be larger than between earlier children. </p><p>Checking this in the data, I looked at people who are neither first nor last children and compared the age gaps to their next older sibling and next younger sibling. On average the age gap to their younger sibling was 0.55 years longer than the gap to their older sibling.</p><p>This effect would explain the incredibly high birth order effect with 1 year age gap seen in my graph 2. If many oldest and few youngest children in the general population have 1 year gap to their neighbouring sibling and we add this to the SSC Birth order effect then the overall effect in the SSC sample will be huge. </p><p>It would also explain how the birth order effect in graph 2 appears to decrease dramatically for families with large age gaps – in the population as a whole, more youngest than oldest children in the overall population will have >5 year gap to their neighbouring sibling. This overall population effect cancels out some (or all for >9 year gap) of the Birth order effect from the SSC population.</p><hr class="dividerBlock"/><p>I can't think of a way to quantify and/or cancel out this effect whilst comparing oldest to youngest children but fortunately we don't have to.</p><p>Instead of comparing oldest and youngest children, we could compare first and second children. If we only look at first and second children and compare age gaps downwards and upwards respectively then we should be looking at the same underlying distribution of age gaps in the general population.</p><p>One advantage of comparing oldest to youngest siblings is that this represents the largest Birth order effect (see Scott’s 2018 <a href="https://slatestarcodex.com/2018/01/08/fight-me-psychologists-birth-order-effects-exist-and-are-very-strong/">analysis</a>). However, as most of the effect happens between first and second siblings, the effect should still be large enough to detect using these samples. </p><hr class="dividerBlock"/><p>Redoing my previous graph based on first and second children rather than first and last gives something which makes a lot more sense - there isn’t much difference between 1 sibling and >1 sibling. The >1 sibling data set doesn’t have the sudden drop after 7-years but does suggest a slight downwards trend around the same time.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567586529/Birth%20order%20effect/Post%201/My_1_2_combined_new.png" class="draft-image center" style="width:77%" /></figure></span><p>Recreating Scott’s graph 3 (i.e. including all family sizes) gives a drop in birth order effect at 7 years but not as low as with just the 1 sibling data – ~59% oldest children vs 2nd oldest, compared to ~54%.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1567586529/Birth%20order%20effect/Post%201/My_3_new.png" class="draft-image center" style="width:71%" /></figure></span><h1>Failed reproduction of birth order effect with 0 year age gap</h1><p>I also failed to reproduce Scott’s finding regarding people reporting a zero year age gap. He finds that:</p><blockquote>Weirdly, among people who reported a zero-year age gap, 70% are older siblings.</blockquote><p>but I was unable to produce a result like this.</p><p>First I removed the respondents who reported 0 year age gaps in a direction in which they reported 0 siblings. This removed over half of the reported 0 year age gaps.</p><p>Of those remaining, I get a non-significant 58% older siblings (53 vs 39). There were also 3 respondents who indicated that they were in the middle of a multiple birth.</p><p>In this case I’m not sure how I get such different results from Scott. Even if I don’t remove the responses as I detailed above I don’t get anywhere near Scott results (I actually get ~60% <u>younger</u> sibling – with so many oldest children there are more opportunities to put a 0 years in the upwards direction where it should have been left blank).</p><p>So I’m very confused about why I’m getting such different results.</p><hr class="dividerBlock"/><p>In my <a href="https://www.lesswrong.com/posts/YnXd7zfGGZfMD9QtA/age-gaps-and-birth-order-reanalysis">next post</a> I reanalyse the data based on 1st and 2nd oldest children.</p>buckyuZEeqmeFjs3nmawn72019-09-07T19:22:55.068ZWhat are principled ways for penalising complexity in practice?
https://lw2.issarice.com/posts/R6xaH3dxs3Xi4fkv6/what-are-principled-ways-for-penalising-complexity-in-1
<br/><p>Previously I <a href="https://www.lesswrong.com/posts/Q9hDFkvCSwi6cwPGy/how-is-solomonoff-induction-calculated-in-practice-1">asked about</a> Solomonoff induction but essentially I asked the wrong question. Richard_Kennaway pointed me in the direction of an answer to the question which I should have asked but after investigating I still had questions.</p><p>So:</p><p>If one has 2 possible models to fit to a data set, by how much should one penalise the model which has an additional free parameter?</p><p>A couple of options which I came across were:</p><p><a href="https://en.wikipedia.org/wiki/Akaike_information_criterion">AIC</a>, which has a flat facter of e penalty for each additional parameter.</p><p><a href="https://en.wikipedia.org/wiki/Bayesian_information_criterion">BIC</a>, which has a factor of √n penalty for each additional parameter.</p><p>where n is the number of data points.</p><p>On the one hand having a penalty which increases with n makes sense - a useful additional parameter should be able to provide more evidence the more data you have. On the other hand, having a penalty which increases with n means your prior will be different depending on the number of data points which seems wrong. </p><p>So, count me confused. Maybe there are other options which are more helpful. I don't know if the answer is too complex for a blog post but, if so, any suggestions of good text books on the subject would be great.</p><p>EDIT: johnswentworth has written a <a href="https://www.lesswrong.com/s/onCRFFN7rGXTg3jyc">sequence</a> which expands on the answer which he gives below.</p>buckyR6xaH3dxs3Xi4fkv62019-06-27T07:28:16.850ZHow is Solomonoff induction calculated in practice?
https://lw2.issarice.com/posts/Q9hDFkvCSwi6cwPGy/how-is-solomonoff-induction-calculated-in-practice-1
<p>Solomonoff induction is generally given as the correct way to penalise more complex hypotheses when calculating priors. A great introduction can be found <a href="https://www.lesswrong.com/posts/Kyc5dFDzBg4WccrbK/an-intuitive-explanation-of-solomonoff-induction">here</a>.</p><p>My question is, how is this actually calculated in practice?</p><p>As an example, say I have 2 hypotheses:</p><p>A. The probability distribution of the output is given by the same normal distribution for all inputs, with mean <span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="\mu"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em;">μ</span></span></span></span></span></span> and standard deviation <span><span class="mjx-chtml"><span class="mjx-math" aria-label="\sigma"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.001em;">σ</span></span></span></span></span></span>.</p><p>B. The probability distribution of the output is given by a normal distribution depending on an input <span><span class="mjx-chtml"><span class="mjx-math" aria-label="x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span> with mean <span><span class="mjx-chtml"><span class="mjx-math" aria-label="\mu_0+mx"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.519em;">μ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span> and standard deviation <span><span class="mjx-chtml"><span class="mjx-math" aria-label="\sigma"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em; padding-right: 0.001em;">σ</span></span></span></span></span></span>. </p><p>It is clear that hypothesis B is more complex (using an additional input [<span><span class="mjx-chtml"><span class="mjx-math" aria-label="x"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">x</span></span></span></span></span></span>], having an additional parameter [<span><span class="mjx-chtml"><span class="mjx-math" aria-label="m"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span></span></span></span></span>] and requiring 2 additional operations to calculate) but how does one calculate the actual penalty that B should be given vs A?</p><br/>buckyQ9hDFkvCSwi6cwPGy2019-06-04T10:11:37.310ZBook review: My Hidden Chimp
https://lw2.issarice.com/posts/v482HmeeeXNM4kfm3/book-review-my-hidden-chimp
<p> </p><p>tl;dr: A narrowly framed introduction to dual process theory for ~5-10 years olds.</p><p>I love a dual process theory metaphor as much as the next rationalist (researching this post I came across <a href="https://www.lesswrong.com/posts/NJL5FYe6KkRtjekeG/are-you-the-rider-or-the-elephant#jf7HEYggkJS5GqTNd">King Louie and the Apes</a> which I quite like). The Blue minimising robot series completely changed how I thought about myself and cleared up a lot of the confusions that I had about myself (e.g. what I would now call akrasia). I would say that, personally, dual process theory is the most useful thing that I have learnt under the umbrella of rationality.</p><p>I have frequently wondered how different my life would have been had I understood myself better at an early age. How would one go about explaining this to a 6 year old?</p><p>I have two children (6 and 4) and had tried to explain a bit about the 2 systems to them but with limited success. I particularly thought it would help my eldest as he can struggle behaviourally and experiences extreme emotions but he didn’t fully internalise what system 1 / system 2 meant. Then I heard that Prof Steve Peters had released a children’s book based on his chimp model and I was intrigued enough to get it for him as a Christmas present.</p><p>[note that there is a companion book to My Hidden Chimp called The Silent Guides which is aimed more at parents but which I haven’t read]</p><h1>The Author</h1><p>I’m not sure how well known Peters is outside the UK but over here he is a minor celebrity. For most people I suspect that if I mentioned Peters by name they wouldn’t know who I was talking about. If I said “You know, that sports psychiatrist who works with cyclists, the chimp guy” then they probably would.</p><p>He is probably most famous for his work with British cycling as part of their programme of incremental gains. Most recently was in the news for being <a href="https://www.bbc.com/sport/cycling/46970230">credited by Victoria Pendleton</a> for preventing her suicide. </p><p>He has also worked with Ronnie O’Sullivan which was where I first noticed his work. Prior to working with Peters, O’Sullivan struggled with drink, drugs and depression but in recent years has turned this around, giving Peters a lot of the credit.</p><p>He is also an age group world champion in 100, 200 and 400m.</p><p>I’m not his publicist, honest...</p><h1>The Chimp Model</h1><p>The <a href="https://chimpmanagement.com/the-chimp-model/">chimp model</a> is Peters’ dual process metaphor. Different writers and models seem to me to have subtly different focuses - e.g. Robin’s Elephant and rider focuses on motivation, Kahneman’s system 1 and system 2 on decision making. The chimp model in My Hidden Chimp focuses on fast, emotional responses.</p><p>An adult explanation of the chimp model is given in Peters’ previous book The Chimp Paradox. The Chimp model is expressed as the human (frontal lobe, a.k.a blue brain) and the chimp (limbic system, a.k.a red brain). My Hidden Chimp goes into less depth but explains how a chimp has a very small blue blue brain so it mainly decides things with its red brain, hence calling it your chimp. The human (blue brain, “You”) is the you that you really want to be.</p><p>[The Chimp Paradox (which I also haven’t read) goes into a triple process theory by adding the computer which stores and acts on memories. This isn’t mentioned in My Hidden Chimp, I presume because it is a simplified version for children.]</p><h1>The Book</h1><h2>Structure</h2><p>In the first third of the book, the chimp model is introduced, explained and made relevant with examples. The remainder of the book describes 10 habits which help us to manage our chimp and how to build the habits into our life.</p><h2>Teaching methods</h2><p>As you might expect from a children’s book, it is full of brightly coloured pictures.</p><p>The examples in the book are 4 children and their chimps in various different circumstances where the human thinks one thing and the chimp is thinking something else. For instance you want to go for a walk but your chimp wants to stay at home and do nothing or you know that you’re safe in bed but the chimp is scared of the dark. The examples given are wide ranging so that readers will recognise themselves in some of the characters. The chimps are given names (readers name their own chimp too) and are often presented humourously.</p><p>This is probably the best thing about the book. We had many moments of self-recognition as we read it, as did other people who I’ve heard talk about it.</p><p>In addition to the examples, there are exercises throughout the book, for instance listing times where you’ve acted in a way that you wish you hadn’t. These help with applying the lessons and to building commitment to act on the lessons.</p><h2>Section 1 - Chimp model</h2><p>The first section of the book gives a brief introduction to dual process theory and the chimp model in particular. It introduces the 4 characters and their chimps and gives examples of what they’re thinking.</p><p>The book tries to avoid some problems otential pitfalls, for instance explaining that your chimp is part of you so you can’t just blame it when you do something naughty. You have to make sure it behaves itself like you would with a pet.</p><h2>Section 2 - Habits</h2><p>The second section explains 10 habits which help manage your chimp. These are some of the basic building blocks that you’ve probably told your children about 100 times (learning to share, learning that no means no, saying sorry). Talking about them in the context of our hidden chimp works well to show why they find them hard to do but also why they are important.</p><h2>Age range</h2><p>The book doesn’t give a recommended age range but a lower bound would be the ability to understand and act on dual process theory when explained in an age relevant way. The upper bound would probably be not feeling patronised by the childish cartoon drawings. 5-10 is a fair guideline.</p><p>The book states that it can be read by the child on their own or with an adult. I would suggest that (for younger readers especially) the book should be read with an adult as I feel misunderstanding the content could have negative effects.</p><h1>Potential Criticisms</h1><p>Here I cover not only some of my own criticisms but also some criticisms which I expect other rationalists might have, even if I think the book deals fairly well with most of the issues.</p><h2>Explanation of the chimp</h2><p>I found the explanation of the chimp slightly confusing. The original explanation makes it sounds superficially as though it is the part of you which makes you do naughty things. This pivots to the part of you that is “acting without your permission”. This is slightly vague but the book makes up for this with repeated extensional definition, giving examples to indicate the kind of thing which it is talking about. Extensional definition seems like the right way to go about explaining it to a child so overall I think this works fairly well.</p><p>I should note that re-reading the book it is fairly clear what it means from the beginning yet somehow I (and others) found it confusing on first reading - go figure.</p><p>It’s important to mention that Peters’ chimp is not exactly analogous to my mental model of system 1 - my impression is that the chimp is a more narrow framing. The Chimp includes immediate, emotional, visible responses. It doesn’t include other opaque subconscious processes such as those which deal with long term motivation. It is possible that the Chimp Paradox deals with these other issues but the focus of MHC is firmly on short term emotional actions. </p><p>This probably makes sense as a children’s book as when we are young we tend to be more short term and react emotionally so learning to deal with these parts of our mind seems like the priority.</p><h2>Identifying with system 2</h2><p>This leads on to another potential problem. I suspect that <a href="https://www.lesswrong.com/posts/NJL5FYe6KkRtjekeG/are-you-the-rider-or-the-elephant">some</a> <a href="https://www.lesswrong.com/posts/XmqqkfY8XAJ6LkwdP/akrasia-is-confusion-about-what-you-want">people</a> might be unhappy with the book’s framing of your system 2 being the you that you really want to be and your system 1 being the part of you that acts without your permission. I’ve seen similar criticism of the Chimp Paradox.</p><p>Personally, I found that this worked well as a children’s book and would worry that a more nuanced framing would confuse matters. I intend to go into the rest of the story when my children are older, a bit like I’d teach a child Newtonian mechanics first and wait until they’re older to explain relativity.</p><p>I would understand if others took a different point of view (I’m not 100% convinced mysel) so I’ve just described it and will let you decide how happy you would be with the framing given. </p><p>The fact that the chimp is a more narrow framing than the elephant might help assuage some worries but I’m not sure. There is a concerted effort to describe how our chimps can sometimes be helpful – e.g. our chimp helps us to have fun and warns us of danger. The often-humourous presentation of the chimps also helps with appreciating the chimp. </p><p>However, the book also presents us as having to put our chimps in a box when they are misbehaving. The examples given in the book are of circumstances where the chimp is having a negative effect on our lives (not sharing, telling lies to parents, being unkind to friends). I’ll grant that there are edge cases where these actions might be beneficial but, especially for a child, the advice to prevent our chimps from doing these things overall seems like the right way to go.</p><h2>Taking the metaphor literally</h2><p>Dual process theory is a model of something much more complicated. We are not two people or a single person with a limbic system attached. Our brain is a big mass of neurons and connections of which we only understand a very general picture. Peters deals with this by stating carefully that this is just an interesting way to think of how our brains work and not reality. However, for most of the book he just writes as if the model is literally true. This can seem slightly odd but I suspect that it is better than the alternative of constantly qualifying every statement. I just ended up having to remind my son that it was a model every time I thought he might be taking it too literally.</p><h2>Book not matching life</h2><p>One particularly interesting experience was talking about the habit of sharing. Peters presents this as our human wanting to share and our chimp not wanting to share. My son pointed out that actually he and his chimp and were in agreement that they didn’t want to share. I don’t know how common this would be but I’d tend to agree with my son that my system 2 by default (especially when I was younger) doesn’t particularly want to share. </p><p>I suspect that there’s not much a book can do about this and that it’s a case of adding your own explanation to the book where it doesn’t exactly meet what the child needs. In this case, I reminded my son about “cooperate or defect?” (as we call Prisoners Dilemna) and how cooperating often means a better result in the end. This is part of the reason I’d recommend the book being read with an adult - customisation.</p><h2>Other</h2><p>The main critical comments from online reviews seem to be:</p><p>Too young for my child (~12 is too old)</p><p>The 10 habits are things you’re teaching your child anyway - nothing original</p><p>Kindle version is poor</p><h1>Did it work?</h1><p>Yes.</p><p>Obviously I have to add the usual caveats that it’s only a sample of 1, perhaps its regression towards the mean, maybe the other things I’ve been working on with my son have made a difference.</p><p>Even given that, I am confident that he has understood dual process theory as presented in the book and started to recognise when his chimp “takes over”. As a result, he is more able to e.g. stop himself when he gets angry or to override his instincts not to share. Reading it with him has also given us a common framework to talk about what went wrong when he has lost control. </p><p>Before reading the book I could often see how his mind was working when he did lose control and could tell that he was desperately searching for ways that he could still be in the right. Then afterwards you could see how confused he was with himself and he couldn’t figure out why he’d done what he did. Now he feels like he understands what went wrong and how he can do better next time.</p><p>He has avoided the obvious pitfall of just blaming his chimp and, if anything, is better at taking responsibility for his actions.</p><p>So throughout this review I have had to prevent myself from just saying “THIS BOOK IS AMAZING AND ANYONE WITH A 5-10 YEAR OLD CHILD SHOULD BUY IT IMMEDIATELY!”.</p><p>I realise that it is likely that it may not work quite as well with every child. The book could have been written with my son in mind so if it was going to work for anyone it was going to work for him.</p><p>You can make up your own mind if you think it will work for your family.</p><p> </p>buckyv482HmeeeXNM4kfm32019-03-04T09:55:32.362ZWho wants to be a Millionaire?
https://lw2.issarice.com/posts/Svaj4nSJCzoRZkvjT/who-wants-to-be-a-millionaire
<p>tldr; Using Toy models, the Kelly criterion, prospect theory, Bayes and amateur psychology in an unnecessarily detailed analysis of Who wants to be a Millionaire. I strongly suspect there will be errors but hopefully not too many.</p><h1>Motivating example</h1><p>I was watching <a href="https://en.wikipedia.org/wiki/Who_Wants_to_Be_a_Millionaire%3F">Who wants to be a Millionaire?</a> a couple of nights ago for the first time in about 20 years after its recent recommissioning in the UK.</p><p>One contestant’s general knowledge got her to £250,000, 2 correct answer away from £1,000,000. She had £125,000 guaranteed and still had 2 lifelines (ask the audience and phone a friend).</p><p>Question: The Norwegian explorer Roald Amundsen reached the South Pole on 14th December of which year?</p><p>A: 1891</p><p>B: 1901</p><p>C: 1911</p><p>D: 1921</p><p>She thought about/discussed this at length (there was no time limit and the total time spent on the question was about 20 minutes!). She knew that Amundsen beat Scott to the south pole and was confident that Scott was Victorian, which ruled out C & D. She pointed out that if it was 1911 then the 100 year anniversary would have been 2011 and she felt she would have remembered there being something about it in the news.</p><p>This didn’t help her choose between A & B so she asked the audience (she was confident none of her friends would know). The results were (from memory, maybe slightly off):</p><p>A: 28%</p><p>B: 48%</p><p>C: 24%</p><p>D: 0%</p><p>What would you do?</p><hr class="dividerBlock"/><p>The result that stood out to me was the 24% who said C. After everything that she said about how confident she was that it isn’t 1911, who are these people voting C? It turns out they’re the people who knew the right answer.</p><p>Unfortunately, the contestant went with the majority, said B, and left with only £125k. Not too bad really, even if <a href="https://www.cambridge-news.co.uk/news/cambridge-news/who-wants-millionaire-new-years-15621585">the name Roald Amundsen is haunting her</a> and popping up everywhere she goes.</p><p>I’ll admit that even though I suspected that the answer was C based purely on the ask the audience result, I don’t think I would have been confident enough to go for it based only on that.</p><hr class="dividerBlock"/><p>If I knew that b was the right answer, would I have been surprised that it got 48% of the vote? No, that would make sense.</p><p>If I knew that b was the wrong answer, would I have been surprised that it got 48% of the vote? Maybe a little, it is quite high, but not out of the question, especially this late in the game.</p><p>If I knew that c was the right answer, would I have been surprised that it got 24% of the vote? No, that would make sense; it’s a tough question so if most of the people who pressed c actually knew the answer then 24% sounds about right.</p><p>If I knew that c was the wrong answer, would I have been surprised that it got 24% of the vote? Yes, very, you have to be really committed to an answer to stick with it after the someone who is clearly good at general knowledge has ruled it out and given a couple of reasons why she thinks it’s wrong. Anything more than 10% would be a surprise, 24% would be really weird. This is especially true as D got 0% of the vote.</p><p>Let’s dig into some more detail.</p><h1>Modelling ask the audience</h1><h2>Simple model</h2><p>A simple model for Ask the Audience would be to expect that those who know the right answer would press the correct button and those who didn’t would guess equally spread between the 4 answers. </p><p>If we estimate that 20% of the audience actually know the answer, this gives 20:20:20:20 from the guessers, with an additional 20 for the correct answer. We get 40:20:20:20 and the correct answer is obvious. Even with a bit of random noise in the results the correct answer should be clear, provided enough people actually know the answer.</p><p>On late game questions, one would expect that fewer people will know the answer as few people win the jackpot. However, the game requires consecutive answering of questions to win the jackpot so each individual question doesn’t need to be too hard to prevent people from winning the jackpot (provided there are no <a href="https://www.youtube.com/watch?v=i1GC02ya2Vg">indiscreet coughs</a>). </p><p>Consider only people who get to the last 5 questions. Even if they, on average, know the answer to (or correctly guess) 50% of the time, only 1 in 32 will actually get to the jackpot (provided the questions are on sufficiently different subjects). 33% knowledge gives 1 in 243 wins. In the original UK series there were 5 winners in 1200 contestants (1 in 240) but as some contestants weren't good enough to reach the final 5 questions, 33% is a lower bound.</p><p>The average audience member probably isn’t as good as the best contestants but they have applied to be an audience member of WWTBAM so probably have a good interest in general knowledge (or are there with someone who does!). I think that 15-20% of the audience on average knowing the correct answer is probably not a bad starting point.</p><h2>Salience</h2><p>Imagine you’re an audience member faced with a question you don’t know the answer to. Possibly one of them stands out to you for some reason or other. Maybe it’s a name you recognise or a date which seems like a good enough guess.</p><p>Unfortunately, the girl next to you is thinking the same answer for a similar reason, as is the guy a couple of seat further down. 20% of the audience who guess the same thing as you for roughly the same reason. Instead of 40:20:20:20, the results are now 35:35:15:15.</p><p>This is bad news for the contestant as two answers are indistinguishable as the same number of people came up with a particular wrong answer due to a consistent reason as actually knew the answer. In reality this is rarely the case, with one effect being larger. </p><p>In this circumstance, most people will either take the money and run or go with the highest scoring answer. 20 years ago when I used to watch the program I had the heuristic that in the final 5 questions, it’s a better option to take the second highest scoring answer from ask the audience.</p><p>A better solution than to go consistently with the highest or second highest answer would be to consider which effect sizes you would expect – people actually knowing the answer and salience. Then, when you see the results, consider how surprising they would be based on the hypothesis that A, B, C or D is the correct answer.</p><p>Estimating salience is tricky but as an upper bound a brief look on the <a href="http://millionaire.wikia.com/wiki/Ask_the_Audience">millionaire wiki</a> gives an example of the audience voting 81:19 in favour of the wrong answer! This <a href="https://millionaire.fandom.com/wiki/Audience%27s_Results">page</a> gives even more extreme examples.</p><h2>Using 50:50 and ask the audience for the same question</h2><p>On that first page, a number of other examples are given of people using 50:50 and ask the audience lifelines on the same question. It is notable that all 4 used their 50:50 lifeline first, followed by ask the audience.</p><p>Superficially this makes sense – you want to give the audience as much information as possible to help them make the best choice. </p><p>However in reality there is probably a fairly binary split in the audience members – those who know and those who are guessing. You don’t care what the guessers think as its very hard for them to provide you with enough evidence to justify answering the question.</p><p>The only thing you actually care about is identifying the people who actually do know. If you refrain from using your 50:50 until after using ask the audience, the 50:50 serves the additional purpose of removing from the statistics a section of the audience who don’t know., increasing your signal to noise ratio. </p><p>In addition, if there is a highly salient wrong answer then you have a 2/3 chance of removing it. Say we have 35:35:15:15 ratio due to knowledge and salience effects as described above. Using the 50:50 has a 2/3 chance of leaving 35:15 and 1/3 chance of leaving 35:35. </p><p>35:15 gives an obvious best response under the model. Using the same model and removing the same questions but using the 50:50 lifeline first you would have 60:40 in favour of the correct answer. This is much less strong evidence and you would only need a relatively small bit of salience in favour of the wrong answer to tip the scales in the wrong direction.</p><p>If 35:35 is left you’re still stuck but you wouldn’t be in any better situation if you’d selected 50:50 first.</p><h2>Salience influence by the contestant</h2><p>One of the classic things to shout at the T.V. during Millionaire is “Don’t discuss the answers before using your Ask the Audience lifeline!” </p><p>It is often said that if you talk about which answers you think are most likely then you will influence the audience towards those answers. This means that you don’t get the audience’s true opinion on the subject. This seems to have happened in the Amundsen question.</p><p>The obvious thought comes to mind that one might use that influence deliberately but I’ll come back to that later.</p><p>(I’m working on the assumption that the rules prevent you from telling the audience what to do if they’re not sure!)</p><h1>How confident do I need to be?</h1><h2>3 alternative models</h2><p>Millionaire includes 2 safety nets, such that once a contestant reaches these nets they are guaranteed to win at least that amount. Once a safety net has been passed the contestants are no longer having to bet their entire bankroll on each answer.</p><p>I’m going to invoke the <a href="https://www.lesswrong.com/posts/BZ6XaCwN4QGgH9CxF/the-kelly-criterion">Kelly criterion</a> here even though I know that the assumptions of the derivation are not met. Adjust up or down according to taste.</p><p>(Interestingly one of the Kelly assumptions is that you will get unlimited future opportunities to wager with an edge. In Millionaire you get another opportunity to wager iff you take the wager offered and win. Additionally, Kelly relates to the optimal amount to bet, rather than whether to accept a set sized wager.)</p><p>If we rearrange Kelly and apply it to Millionaire (where the prize doubles for each successful question answered (or very close)) then we arrive at a formula for how confident one should be in order to guess.</p><div><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="p_k > 1 – \frac{1}{2^{q+1}-1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">k</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: 0.004em; padding-bottom: 0.298em;">–</span></span><span class="mjx-mfrac MJXc-space1"><span class="mjx-box MJXc-stacked" style="width: 3.687em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.687em; top: -1.368em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span class="mjx-denominator" style="width: 3.687em; bottom: -1.054em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em; padding-right: 0.014em;">q</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.687em;" class="mjx-line"></span></span><span style="height: 2.421em; vertical-align: -1.054em;" class="mjx-vsize"></span></span></span></span></span></div><p> where q is the number of questions since your last safety net.</p><p>This implies that for each question past the last safety net you need about an extra bit of evidence before you are justified in answering the question.</p><p>Kelly might not be the best option for choosing a required probability. Let's say I value money logarithmically and calculate expected utility. In order to justify answering I require:</p><div><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="p_{eu} > \frac{q}{q+1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-texatom" style=""><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">e</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">u</span></span></span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.382em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.382em; top: -1.226em;"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="width: 2.382em; bottom: -0.95em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em; padding-right: 0.014em;">q</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.382em;" class="mjx-line"></span></span><span style="height: 2.176em; vertical-align: -0.95em;" class="mjx-vsize"></span></span></span></span></span></div><p>This is considerably less stringent, particularly as q increases.</p><p>We can compare this to prospect theory. I'll consider a loss to be twice as painful as a similar gain is pleasureable and anchor on how much money I'll get if I don't answer. I will let the decision weights equal the probabilities as they are not extreme. I then require:</p><div><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="p_p > \frac{2^q-1}{1.5 \times 2^q-1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.284em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.284em; top: -1.425em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em; padding-right: 0.014em;">q</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-denominator" style="width: 5.284em; bottom: -0.895em;"><span class="mjx-mrow"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1.5</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">×</span></span><span class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em; padding-right: 0.014em;">q</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 5.284em;" class="mjx-line"></span></span><span style="height: 2.32em; vertical-align: -0.895em;" class="mjx-vsize"></span></span></span></span></span></div><p>This is even less stringent for high q, tending towards <span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="\hspace{1mm}^
2/_3"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mspace" style="width: 0.278em; height: 0px;"></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.439em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span></span></span> for high q.</p><p>As prospect theory is descriptive rather than prescriptive this isn't necessarily how you should behave, but may represent how people do behave.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1549021556/blog%20posts/Confidence_required.png" class="draft-image center" style="width:80%" /></figure></span><p>In the original UK version of WWTBAM, the host used to present the contestant with a cheque for the amount that they had just reached (for question ~11 onwards), before taking it away and saying "but we don't want to give you that" and proceeding to the next question. I used to assume that it was just showmanship, now I wonder whether it was a cunning plan to encourage anchoring to the current amount and making the contestant more likely to guess.</p><h2>Working through the odds</h2><p>In the example given, the contestant was 1 question past a lifeline so should require <span><style>.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
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</style><span class="mjx-chtml"><span class="mjx-math" aria-label="p>\hspace{1mm}^
2/_3"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span class="mjx-mspace" style="width: 0.278em; height: 0px;"></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.439em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span></span></span> minimum in order to answer (I'll stick with Kelly for the moment).</p><p>Let’s say she had assigned probabilities of roughly 50:50:0:0 before she asked the audience. So in order to answer she only needed a single bit of evidence for one answer over the other.</p><p> <span><span class="mjx-chtml"><span class="mjx-math" aria-label="\frac{p(E|a)}{p(E|b)} = \frac{p(a|E)}{p(b|E)} \frac{p(b)}{p(a)} = \frac{^2/_3}{^1/_3} \frac{0.5}{0.5} = \frac{2}{1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.158em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.052em; top: -1.706em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.052em; bottom: -0.999em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.158em;" class="mjx-line"></span></span><span style="height: 1.913em; vertical-align: -0.707em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.158em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.052em; top: -1.706em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.052em; bottom: -0.999em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em; padding-right: 0.026em;">E</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 2.158em;" class="mjx-line"></span></span><span style="height: 1.913em; vertical-align: -0.707em;" class="mjx-vsize"></span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.421em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.01em; top: -1.706em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.446em; padding-bottom: 0.298em;">b</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.01em; bottom: -0.999em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">a</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.421em;" class="mjx-line"></span></span><span style="height: 1.913em; vertical-align: -0.707em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.143em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.617em; top: -1.951em;"><span class="mjx-mrow" style=""><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char"></span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.435em; padding-left: 0px; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 83.3%; vertical-align: -0.38em; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.617em; bottom: -1.17em;"><span class="mjx-mrow" style=""><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char"></span></span></span><span class="mjx-sup" style="font-size: 83.3%; vertical-align: 0.347em; padding-left: 0px; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 83.3%; vertical-align: -0.38em; padding-right: 0.06em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.143em;" class="mjx-line"></span></span><span style="height: 2.207em; vertical-align: -0.827em;" class="mjx-vsize"></span></span><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.045em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.478em; top: -1.394em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.5</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.478em; bottom: -0.687em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.5</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.045em;" class="mjx-line"></span></span><span style="height: 1.472em; vertical-align: -0.486em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 0.495em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.7em; top: -1.372em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.7em; bottom: -0.665em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 0.495em;" class="mjx-line"></span></span><span style="height: 1.441em; vertical-align: -0.47em;" class="mjx-vsize"></span></span></span></span></span></span> </p><p>The audience voting 28:48 in favour of option b maybe did represent a bit of evidence for b over a if you don't expect a high salience effect (she hadn’t distinguished between the 2 in her deliberations).</p><p>Coming at the same question and evidence, I started with 25:25:25:25 odds on which answer was correct because I had no real clue. In order for one of the probabilities to rise to <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p>\hspace{1mm}^2/_3"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span class="mjx-mspace" style="width: 0.278em; height: 0px;"></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.439em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span></span></span>, I would need 6:1 evidence in favour of that answer over all of the other answers combined.</p><p>I didn’t really assign any particular salience to any answer before the contestant started talking – none of the answers particularly drew me in and I couldn’t think why they would draw other people in. There might be a small effect with people being more likely to guess the middle 2 answers but not enough for me to adjust confidently. </p><p>However, after she ruled out C & D fairly confidently, I was looking at those straight away when the results came in, expecting that if one was more than 10% points higher than the other then that would be fairly good evidence in favour of that answer.</p><p>The actual result was about as clear as you could get: C=24% & D=0%. Assuming it is correct and useful, I think it comfortably assigns better than 6:1 odds in favour of C over the other answers. </p><h2>How confident am I in my model?</h2><p>However, I didn’t have any specific evidence that the model was correct, just my amateur psychology at work. If I let m be the model being true (and me having interpreted the results correctly), say that my model assigns 90% of its probability mass to option c and that if my model is wrong then I still have 25:25:25:25 odds, I can calculate how confident I need to be in my model to achieve <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p(c)>2/3."><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3.</span></span></span></span></span></span> </p><div><span class="mjx-chtml MJXc-display"><span class="mjx-math" style="width: 100%;" aria-label="^2/_3 < p(c) = p(m) p(c|m) + p(¬m) p(c|¬m)\\
\hspace{20mm}= p(m) \big(p(c|m) - p(c|¬m)\big) + p(c|¬m)\\
p(m) > \frac{^2/_3 - p(c|¬m)}{ p(c|m) - p(c|¬m)} \approx 0.64"><span class="mjx-mrow" style="width: 100%;" aria-hidden="true"><span class="mjx-stack" style="width: 100%; vertical-align: -4.212em;"><span class="mjx-block" style="text-align: center;"><span class="mjx-box"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char"></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.439em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;"><</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-block" style="text-align: center; padding-top: 0.442em;"><span class="mjx-box"><span class="mjx-mspace" style="width: 0px; height: 0px;"></span><span class="mjx-mspace" style="width: 5.569em; height: 0px;"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mstyle"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">(</span></span></span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mstyle"><span class="mjx-mrow"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">)</span></span></span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-block" style="text-align: center; padding-top: 0.442em;"><span class="mjx-box"><span class="mjx-mspace" style="width: 0px; height: 0px;"></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 7.829em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 7.829em; top: -1.735em;"><span class="mjx-mrow"><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-mi"><span class="mjx-char"></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.439em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="width: 7.829em; bottom: -1.09em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">−</span></span><span class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 7.829em;" class="mjx-line"></span></span><span style="height: 2.826em; vertical-align: -1.09em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">≈</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.64</span></span></span></span></span></span></span></span></div><p>So should I have been willing to take the gamble? That depends on whether I thought that the model was more than 64% likely to be accurate.</p><p>I would have assigned prior probability of <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p(m)\approx0.4"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">≈</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.4</span></span></span></span></span></span> based purely on it seeming sensible vs my lack of expert knowledge and the fact that any effect might be smaller than I anticipated. The fact that answer d received 0% of the vote increased my confidence in the model and effect size but only up to maybe <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p(m) = 0.5"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.5</span></span></span></span></span></span> at best. I think that this counts as my gut instinct roughly matching with what my maths has come up with – I shouldn’t take the bet but it’s a close thing. </p><p>Having seen that the model worked on this occasion, I should update:</p><div><span class="mjx-chtml MJXc-display"><span class="mjx-math" style="width: 100%;" aria-label="\frac{p(m|c)}{p(¬m|c)} = \frac{p(c|m)}{p(c|¬m)} \times \frac{p(m)}{p(¬m)}
\\
\hspace{1cm}= \frac{0.9}{0.25} \times \frac{0.4}{0.6} = 2.4
"><span class="mjx-mrow" style="width: 100%;" aria-hidden="true"><span class="mjx-stack" style="width: 100%; vertical-align: -2.911em;"><span class="mjx-block" style="text-align: center;"><span class="mjx-box"><span class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.737em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.737em; top: -1.59em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="width: 3.737em; bottom: -1.09em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.737em;" class="mjx-line"></span></span><span style="height: 2.68em; vertical-align: -1.09em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 3.737em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.737em; top: -1.59em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="width: 3.737em; bottom: -1.09em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.737em;" class="mjx-line"></span></span><span style="height: 2.68em; vertical-align: -1.09em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">×</span></span><span class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.026em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.026em; top: -1.59em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span class="mjx-denominator" style="width: 3.026em; bottom: -1.09em;"><span class="mjx-mrow"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">¬</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 3.026em;" class="mjx-line"></span></span><span style="height: 2.68em; vertical-align: -1.09em;" class="mjx-vsize"></span></span></span></span><span class="mjx-block" style="text-align: center; padding-top: 0.442em;"><span class="mjx-box"><span class="mjx-mspace" style="width: 0px; height: 0px;"></span><span class="mjx-mspace" style="width: 2.784em; height: 0px;"></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.978em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.978em; top: -1.368em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.9</span></span></span><span class="mjx-denominator" style="width: 1.978em; bottom: -0.778em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.25</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.978em;" class="mjx-line"></span></span><span style="height: 2.146em; vertical-align: -0.778em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.298em;">×</span></span><span class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.478em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.478em; top: -1.379em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.4</span></span></span><span class="mjx-denominator" style="width: 1.478em; bottom: -0.778em;"><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.6</span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.478em;" class="mjx-line"></span></span><span style="height: 2.157em; vertical-align: -0.778em;" class="mjx-vsize"></span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2.4</span></span></span></span></span></span></span></span></div><div><span class="mjx-chtml MJXc-display" style="text-align: center;"><span class="mjx-math" aria-label="p(m|c) = 0.71"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">(</span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">m</span></span><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">|</span></span></span></span><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.298em;">c</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">)</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.077em; padding-bottom: 0.298em;">=</span></span><span class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">0.71</span></span></span></span></span></div><p>So now that I’ve seen the model work on this occasion I should be willing to bet based on a similar situation arising in future. However, if it was 2 or more questions since my last lifeline (or the salience model provided less decisive evidence) I should wait for more evidence for the model before being willing to bet. Again, this roughly matches my intuition, looking back, my prior was possibly a bit high.</p><h1>Deliberately influencing the audience</h1><p>I mentioned beforehand that an interesting strategy would be for the contestant to influence the audience deliberately to encourage it to vote in a particular way.</p><p>Imagine you were able to get all of the audience who didn’t know to vote for a single answer which you knew (or were fairly confident) was wrong. The most likely way to influence people in a particular way would be to pretend that you thought that this was the correct answer.</p><p>If it works, this should produce a very high vote for that question plus a 15-20% vote for the correct answer and a small vote for the remaining 2 options. The low vote on the other 2 which you influence away from would be good evidence that your influence attempt was successful.</p><p>If you get almost everyone in the audience voting for the influenced answer and no spike on any of the other 3 that either means that it’s the correct answer after all or that very few people know the actual answer.</p><p>I’m not sure how easy it would be to apply this level of influence towards a single answer. I suspect that audience members like to feel as though they are making some form of choice so it would probably be wise to leave open the possibility of at least one other option in your deliberations so that people can feel like they’re deciding on your favoured choice.</p><p>This might be made easier if one of the answers is particularly salient anyway. You shouldn’t need to do as much pushing to get more people to choose this answer.</p><p>If you have your 50:50 lifeline left then this will help you if your results are inconclusive. I suspect that if one maximised the use of ask the audience and 50:50 combined then it would be a rare occasion that you wouldn’t be able to get to at least <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p>\hspace{1mm}^2/_3 "><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span class="mjx-mspace" style="width: 0.278em; height: 0px;"></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">2</span></span></span></span><span class="mjx-msubsup"><span class="mjx-base"><span class="mjx-texatom"><span class="mjx-mrow"><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.446em; padding-bottom: 0.593em;">/</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.439em; padding-right: 0.071em;"><span class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">3</span></span></span></span></span></span></span></span> in favour of one answer. </p><hr class="dividerBlock"/><p>If I plan to influence the audience when I use my lifeline, I need to maximise my effect. Looking at <a href="https://www.influenceatwork.com/principles-of-persuasion/">Cialdini’s 6 principles</a>, I think authority and consensus (a.k.a. social proof) are most likely to helpful here.</p><p>If people are going to be influenced by my statements then they need to believe that I am an authority on the question at hand. This can be done in at least 2 ways prior to asking the audience:</p><p>1. Establishing that you have good overall general knowledge</p><p>2. Establishing you ability to work through tricky questions to get to the correct answer</p><p>In order to persuade people away from some answers and towards others I need to give them a reason for changing. The reason doesn’t have to be true, just believable and I have to be able to come up with this quickly. </p><p>As for consensus, I think that when I choose to use my ask the audience I should say “I’m pretty sure the answer is D and that when I see results I’m going to feel like I wasted a lifeline but I just want to be sure as there’s a small chance it might be C”. Even though the audience members don’t know what the consensus is, an expectation of the consensus is created.</p><p>My main worry is that people might realise that everyone is going to vote the same way and then try to be helpful by selecting their original thought. However, I would expect the number of people who did this to be relatively low so I hope I'd be safe.</p><h1>Summary</h1><p>Superficial readings of ask the audience results are dangerous.</p><p>If you're going to use ask the audience and 50:50 on the same question, ask the audience first.</p><p>For each additional question past your latest safety net, approximately 1 more bit of evidence is required to justify answering (Kelly). Alternatively, <span><span class="mjx-chtml"><span class="mjx-math" aria-label="p > \frac{q}{q+1}"><span class="mjx-mrow" aria-hidden="true"><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em;">p</span></span><span class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.225em; padding-bottom: 0.372em;">></span></span><span class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 1.37em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.938em; top: -1.342em;"><span class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em; padding-right: 0.014em;">q</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.938em; bottom: -0.859em;"><span class="mjx-mrow" style=""><span class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.225em; padding-bottom: 0.446em; padding-right: 0.014em;">q</span></span><span class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.298em; padding-bottom: 0.446em;">+</span></span><span class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.372em; padding-bottom: 0.372em;">1</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.296em; width: 1.37em;" class="mjx-line"></span></span><span style="height: 1.557em; vertical-align: -0.608em;" class="mjx-vsize"></span></span></span></span></span></span> for expected utility. Don't trust your gut.</p><p>Watch lots of episodes beforehand to test your ability to predict what people who don't know the answer will guess.</p><p>If possible, influence the audience so that you are better able to perform this prediction in your game.</p><p>Even if you do all this, it will, at best, get you 1 question further in the quiz - your performance is still dominated by you general knowledge and the luck of the draw as to whether you get the questions which match your areas of knowledge.</p><h2>Bonus material</h2><p>Watch out if you play in Russia (<a href="https://tvtropes.org/pmwiki/pmwiki.php/Series/WhoWantsToBeAMillionaire">from tv tropes</a>):</p><blockquote>Audiences of the Russian version are infamous for deliberately giving the wrong answer out of spite, especially to certain aggravating celebrities. </blockquote><p>Similarly, either this <a href="https://millionaire.fandom.com/wiki/Henri">French audience</a> were really stupid or complete bastards.</p><p></p>buckySvaj4nSJCzoRZkvjT2019-02-01T14:02:52.794ZExperiences of Self-deception
https://lw2.issarice.com/posts/oDADigGrmZuXMHHmX/experiences-of-self-deception
<p>It seems to me that self-deception can describe two different things - conscious and unconscious self-deception. </p><p>Sometimes the elephant believes something untrue all by itself without the rider ever getting a look in. The claims of <a href="http://elephantinthebrain.com/">elephant in the brain</a> seem to focus on this type of unconscious self-deception.</p><p>At other times the rider is complicit in endorsing a particular known untrue belief. The elephant analyses a situation, determines what it is beneficial to believe and motivates the rider to believe this. The rider has access to information which indicates that this isn't true. If the rider brings this information to full attention then it is one of those rare occasions where he can override the elephant's desires. However the rider also has the option to push the information to the side and believe a beneficial lie. It is possible to do this well enough that the information is forgotten or completely overridden with new, inaccurate, information. </p><p>In pushing the information to the side, the rider can sometimes just never bring the information to full attention. Failing that, it can drown the information out by presenting other information (which agrees with its favoured interpretation) as loudly as possible in order to doubt/ignore/forget the information which it doesn't like.</p><p>At least, this is something I experience but I don't know whether other people do. I have a few examples where this has happened and have even experimented with allowing myself to start down the route of conscious self-deception to see what it feels like. To me it feels like cognitive dissonance (feeling hot, brain feeling "fuzzy", adrenaline kicking in) whilst the rider works on counteracting the information. I guess this would be followed by the relief of resolving said dissonance when the rider starts to believe the lie but I haven't experimented that far!</p><p>The literature appears to be understandably non-committal on whether the subjects are consciously aware of their self-deception - I guess that would be pretty hard to determine. </p><p>So my question is - do other people recognise this as something which happens to them? How would you describe the experience? Is it something which you've trained yourself to recognise when it starts?</p>buckyoDADigGrmZuXMHHmX2018-12-18T11:10:26.965ZStatus model
https://lw2.issarice.com/posts/v6hh8E8Qq8WCJMGcG/status-model
<p>[Epistemic status: My best guess.]</p><p>Following <a href="https://www.lesswrong.com/posts/BktRFP2jtYLJvFLEg/in-praise-of-heuristics#y7tJsFAgKj3CMwvLZ">a conversation</a> on a previous post, I decided to do some research into the community’s thoughts on status. The community talks about status a lot and I spent a few happy hours sifting through everything I could find.</p><p>I came up with a status model based on the posts and comments which I read:</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/a_270/v1543234803/blog%20posts/Status%20model.jpg" class="draft-image " style="width:98%" /></figure></span><p>Essentially, status-related mental adaptations are executed which lead to certain status behaviours. These behaviours determine our social standing which, at least in the ancestral environment, tended to affect out fitness. </p><p>I don’t think there’s anything ground breaking here (a similar model would probably apply to any adaptation execution vs fitness maximising effect) but I haven’t seen it sketched out specifically for status before. </p><p>The word "status" gets used to refer to items on all 4 levels and this can lead to confusion where two people are referring to different levels.</p><hr class="dividerBlock"/><p>For instance, what is status and how can I measure it? One way is to look directly at row 2 (status rating): who respects whom and how much? Maybe I could give everyone a questionnaire to rate each other’s status. Or I can look at row 3 (status behaviours): who acts like they have high status? Or row 1 (status benefits): who has the most social control etc.? Whoever gets the most benefits probably has the most status.</p><p>Each option has its advantages and disadvantages e.g. accuracy, ease of assessment but it is important to know which level is being referred to.</p><p>Probably the most commonly debated issue on this topic is whether status is zero-sum.</p><p>If we consider status in row 2 then status is probably going to be relatively zero-sum, although you can maybe get around this a bit by splitting into smaller sub-cultures.</p><p>If we consider row 1, insofar as the status rating determines the benefits, they are close to zero-sum. However, the benefits are also controlled by things other than status (how good are we at getting food, how well coordinated are we as a group?) and so are able to be positive sum.</p><p>Row 4 is where it gets really interesting - our adaptations which implement status behaviour are not zero-sum. We can feel more self-esteem without increasing our status rating (see <a href="https://www.lesswrong.com/posts/qjSHfbjmSyMnGR9DS/that-other-kind-of-status">That Other Kind Of Status</a>). These mental adaptations are the things which we care about on a gut level and give plenty of scope for positive sum behaviours (e.g. <a href="https://www.lesswrong.com/posts/fg7pqjiKfqLsErAfq/give-praise">give praise</a>).</p><p>I don't pretend that this answers the zero-sum question completely but I think it does put it in a helpful frame.</p><hr class="dividerBlock"/><p>The model is incomplete in a number of ways.</p><p>The “status rating” row is a massive simplification. In reality there are all of the different ways which humans judge status, how status changes depending on group and circumstances and the effects of social allies. I only listed <a href="http://www.overcomingbias.com/2010/02/two-kinds-of-status.html">two kinds of status</a> to simplify visually.</p><p>The status benefits and status adaptations listed are also only a subset of the actual benefits and adaptations.</p><p>The relationships between the rows are leaky. The status adaptations lead to behaviours which aren’t necessarily related to status and the status benefits can be affected by things other than status rating.</p><p>Despite the model's limitations, I hope it is a useful simplification.</p><hr class="dividerBlock"/><p>EDIT:</p><p>Ben Pace <a href="https://www.lesswrong.com/posts/v6hh8E8Qq8WCJMGcG/status-model#HSNu72BwMcsimkBry">suggested</a> I add a list of the posts which I looked at and how I think they relate to my model. I’ve done this below – apologies to any of the contributors if I’m misinterpreting their work.</p><p><a href="https://www.lesswrong.com/posts/qjSHfbjmSyMnGR9DS/that-other-kind-of-status">That Other Kind Of Status</a>: Link between status rating and status adaptations.</p><p><a href="https://www.lesswrong.com/posts/pvfrJACkq4DzGYjFF/the-many-faces-of-status">The Many Faces of Status</a>: Morendil's own investigation into status. Covers all areas of the model.</p><p><a href="https://www.lesswrong.com/posts/7ZkHyrBFaDwZ3XgLi/the-red-paperclip-theory-of-status">The Red Paperclip Theory of Status</a>: Trading within and between status ratings and status benefits plus some discussion of related behaviours.</p><p><a href="https://www.lesswrong.com/posts/n3QgyPG2Q3RC3AYzg/status-is-it-what-we-think-it-is">Status: Is it what we think it is?</a>: Discussion of prestige and dominance status. <a href="https://www.lesswrong.com/posts/n3QgyPG2Q3RC3AYzg/status-is-it-what-we-think-it-is#pQDF7M9arxXcuXNui">Diegocaleiro</a> gives the academic names of the status types and links to google scholar for more info.</p><p><a href="http://www.overcomingbias.com/2009/08/actors-see-status.html">Actors see status</a>: Quotes from Impro, focusing on status behaviour and status rating.</p><p><a href="http://meltingasphalt.com/the-economics-of-social-status/">The Economics of Social Status</a>: Inspired by Red Paperclip and goes into significant detail.</p><p><a href="https://www.lesswrong.com/posts/BFyCM7fgCsFYF69Hf/status-map-and-territory">Status: Map and Territory</a>: Discussion of how some status adaptations attempt to enforce the integrity of the links between other adaptations and status behaviours.</p><p><a href="https://www.lesswrong.com/posts/zMxrkFrB6ka4Lb7fM/making-yourself-small">Making yourself small</a>: How status rating plays into status benefits, specifically freedom of action. Low status rating can make it hard to make yourself big. High status rating allows you to make yourself big or small. </p><p><a href="https://www.lesswrong.com/posts/GCQvWvYvgRcaC23BP/what-if-status-is-a-terminal-value-for-most-people">What if status is a terminal value for most people</a>: This seems to me to be a different model, where status rating is a direct adaptation. Some commenters (e.g. <a href="https://www.lesswrong.com/posts/GCQvWvYvgRcaC23BP/what-if-status-is-a-terminal-value-for-most-people#E27fLX7C7btZyTuHJ">someonewrongonthenet</a>) suggest that separating status from adaptation works better.</p><p><a href="https://www.lesswrong.com/posts/BktRFP2jtYLJvFLEg/in-praise-of-heuristics#y7tJsFAgKj3CMwvLZ">The conversation which started my research</a> – To predict status behaviours, should we go backwards from status rating (zero-sum) or is there a better way (e.g. working forwards from status adaptations)?</p>buckyv6hh8E8Qq8WCJMGcG2018-11-26T15:05:12.105ZBayes Questions
https://lw2.issarice.com/posts/MbSvKHJNcM5EwPiCF/bayes-questions
<p> For the first time ever I’ve had reason to use formal Bayesian statistics in my work. I feel this is a cause for streamers and confetti.</p><p>However, I’ve got a bit stuck on analysing my confidence levels and I thought what better place than lesswrong to check that I’m making sense. I’m not sure this is strictly what lesswrong is for but the site says that this is my own personal blog so I guess its ok?! I can always be down-voted to hell if not!</p><p>***</p><p>I’m trying to calculate estimated life before failure of a particular component. </p><p>We’ve done a number of tests and the results show a larger than expected variance with some components not failing even after an extended lifetime. I’m trying to analyse the results to see which probabilistic failure distribution best suits the available data. I have three different distributions (Weibull, Log-normal & Birnbaum-Saunders) each of which has a shape parameter and a scale parameter.</p><p>For each distribution I created a grid which samples the possible values of these parameters. I’ve given the parameters a log uniform prior by giving each sampled parameter pair a uniform prior but sampling the parameters geometrically (i.e. each sampled value of the parameter is a fixed multiple of the previous value). I’ve tried other priors and the results seem fairly robust over choice of prior.</p><p>For components which have failed, P(E|H) is the probability density function at the number of hours before failure.</p><p>For components which get to a certain age and do not fail, P(E|H) is 1 – the cumulative probability function at this number of hours.</p><p>This is implemented on a spreadsheet with a tab for each test result. It updates the prior probability into a posterior probability and then uses this as the prior for the next tab. The result is normalised to give total probability of 1.</p><p>Initially I calculate the expected life of the worst component in 1,000,000. For this, I just use the inverse cumulative probability function with p=0.000001 and calculate this for all of the potential probability distributions.</p><p>The results of this calculation are multiplied by the final probabilities of each distribution being the correct one. Then I sum this over the entire hypothesis space to give the expected life of the worst component in a million.</p><p><em>So my first question – is all of the above sound or have I made some silly mistake in my logic?</em></p><p>***</p><p>The part that I’m less confident about is how to analyse my 95% confidence level of the life of the worst component in 1,000,000.</p><p>The obvious way to approach this is that I should just calculate my expected value for the worst component in 20,000,000. Then, for any given million that I select, I have a 5% chance of selecting the worst in 20,000,000. This is treating 95% as my confidence from the overall weighted model. </p><p>Alternatively, I can treat the 95% as referring to my confidence in which is the one correct probability distribution. In this case, after I normalise my probabilities so that the sum of all of the hypotheses is 1, I start deleting the least likely hypotheses. I keep deleting unlikely hypotheses until the sum of all of the remaining hypotheses is <0.95. The last hypothesis which was deleted is the top end of my 95% confidence level.</p><p>Now if I calculate the expected life of the worst component in 1,000,000 for that individual model I think I can argue that this also represents my 95% confidence level of the worst component in 1,000,000 but in a different way.</p><p><em>Is either of these better than the other? Is there an alternative definition of confidence level which I should be using?</em></p><p>The confidence-in-which-distribution version gives much more conservative answers, particularly when the number of tests is small; the confidence-from-overall-model is much more forgiving of having fewer tests. Even after only a couple of tests the latter gives a 95% confidence level relatively close to the expected value, whereas the confidence-in-which-distribution version remains further away from the expected value until a larger number of tests is performed.</p><p>This seems to me to be more realistic but I don’t have a proper argument to actually justify a decision either way.</p><p>Any help warmly welcomed.</p>buckyMbSvKHJNcM5EwPiCF2018-11-07T16:54:38.800ZGood Samaritans in experiments
https://lw2.issarice.com/posts/kmT47aLQmqzcw329Y/good-samaritans-in-experiments
<p>Consider 2 people. Both are seminary students who are taking part in an experiment ostensibly to consider different types of religiosity. One is asked to prepare a short talk on the Good Samaritan, the other on potential future careers for seminar graduates.</p><p>They are both told to go to another room to record their talk. The one who is to be giving a talk on the Good Samaritan is told that he is late and needs to hurry. The other participant is told that he has a time to spare.</p><p>If they, separately, come across someone who appears to be in respiratory distress, which do you think is more likely to stop and help?</p><p>Does being in a hurry determine whether someone helps?</p><p>Does reading the Good Samaritan?</p><p>Which is a bigger effect?</p><hr class="dividerBlock"/><p>I was recently told about an experiment which showed that seminary students who had just prepared to give a talk about the Good Samaritan were no more likely to help someone in need than those who had been preparing a talk about an unrelated topic.</p><p>This seemed unexpected to me – people who had just been reading and thinking about a story which was told specifically by the leader of their faith to instruct them to help other people were no more likely to help than the control? I know humanity is crazy but seemed like a new level of crazy which I wouldn’t have predicted.</p><p>So I thought I’d check out the <a href="https://greatergood.berkeley.edu/images/uploads/Darley-JersualemJericho.pdf">study</a> and – Aaaaaaaaaaaaaaaaaah!!!</p><p>I know getting overly upset about bad experiments (especially those from before the replication crisis) is probably bad for my health but still – Aaaaaaaaaaaaaaaaaaaaaaaah!!!</p><p>I don’t want to be too harsh on the authors as this probably isn’t the worst culprit you’ll see but – Aaaaaaaaaaaaaaah!!!</p><p>The paper has 1811 citations listed on google scholar – Aaaaaaaaaaaaaaah!!!</p><p>I’m tempted to pretend that this post has some purpose other than just as a release of my frustration but that would be dishonest. Please consider this post a form of therapy for me. The working title for this post was “Screaming into the void” - consider yourself warned.</p><p>(If you want a more coherent discussion of common misuse of statistics in research papers I highly recommend putanumonit’s <a href="https://putanumonit.com/category/defense-against-the-dark-arts/">defense against the dark arts</a> series)</p><hr class="dividerBlock"/><h2><strong>The Experiment</strong></h2><p>Ok, so the basic premise of the experiment seems to be sound. We want to know what inputs cause people to be more or less likely to help others:</p><p>1. Planning a talk on the Good Samaritan (GS)</p><p>2. Being in a hurry</p><p>3. Type of religiosity (Religion as quest, means or end)</p><p>The setup is to give people a questionnaire to determine their type of religiosity. Then give them some time to plan a short talk (3-5 mins) on GS or an unrelated topic. They are then asked to go to another room to give the talk (with 3 degrees of urgency – low, medium and high). </p><p>Contrary to the example given in the introduction, the level of hurriedness doesn’t depend on which topic the individual has prepared – there are 6 conditions people are put in: GS low, medium and high urgency and control low, medium and high urgency.</p><p>On the way to the other room, you arrange for them to come across someone slumped in a doorway, with an apparent respiratory condition. </p><p>You monitor the subjects’ responses and analyse the results.</p><p>My first question was whether they would adjust their p-value requirement for the 5 variables they were testing but no, it turns out that p<0.05 was deemed adequate for significance. Ok, could be worse I guess. More on this later.</p><hr class="dividerBlock"/><p>The second place where doubts started to creep in were the rankings of responses:</p><p>0 = failed to notice the victim as possibly in need at all; </p><p>1 = perceived the victim as possibly in need but did not offer aid; </p><p>2 = did not stop but helped indirectly (e.g., by telling Steiner's assistant about the victim); </p><p>3 = stopped and asked if victim needed help; </p><p>4 = after stopping, insisted on taking the victim inside and then left him.</p><p>5 = after stopping, refused to leave the victim (after 3-5 minutes) and/or insisted on taking him somewhere outside experimental context (e.g., for coffee or to the infirmary).</p><p>It seems to me that there are two different things being measured here:</p><p>How likely are they to notice someone in distress</p><p>If they notice, how likely are they to help</p><p>However, these two factors have been put on the same scale. Surely it would be far more informative to separate them – does being in a hurry make you less likely to help someone you see who is in need or does it just make you less observant?</p><p>But this is relatively minor and the victim doesn’t really mind why you’re not helping, just whether you help or not – there could still be some useful results here. From the conclusion, it looks like not noticing the victim at all was rare so this failure mode was maybe not too big a deal.</p><hr class="dividerBlock"/><h2><strong>Drawing the wrong conclusions</strong></h2><p>So, on to the conclusions from the report:</p><p>1. People who had been preparing GS talks were no more likely to help</p><p>2. People who were in a hurry were less likely to help</p><p>3. Religiosity type had an effect on the type of help given</p><p>It was the first which I was most interested in so I looked at the actual results. </p><p>53% of the people who had been preparing GS talks offered some kind of help (10/19). 29% of the people preparing non-GS talks offered some kind of help (6/21).</p><p>Wait, surely that means people who prepared a GS talk were 1.8x more likely to help than those with an alternative topic? Oh no, says the report. The difference was not significant at the p<0.05 level. Therefore, there is no effect. This isn’t specifically stated in that way but “lack of significant effect” is immediately followed by “lack of effect”:</p><blockquote>Although the degree to which a person was in a hurry had a clearly significant effect on his likelihood of offering the victim help, whether he was going to give a sermon on the parable or on possible vocational roles of ministers did not. This lack of effect of sermon topic raises certain difficulties for an explanation of helping behavior involving helping norms and their salience.</blockquote><p>The paper goes some way to redeeming itself by stating:</p><blockquote>The results were in the direction suggested by the norm salience hypothesis, but they were not significant. The most accurate conclusion seems to be that salience of helping norms is a less strong determinant of helping behavior in the present situation than many, including the present authors, would expect.</blockquote><p>It then undoes the good work in the next sentence:</p><blockquote>Thinking about the Good Samaritan did not increase helping behaviour</blockquote><p>Part of me wants to be happy that they at least included a fairly accurate description of the evidence but the repeated stating of the incorrect conclusion throughout the report can only lead readers to the wrong conclusion.</p><p>At one point, the paper seems to go even further and claims that the fact that we can’t reject the null hypothesis is confirmation of the null hypothesis:</p><blockquote>The prediction involved in the first hypothesis concerning the message content was based on the parable. The parable itself seemed to suggest that thinking pious thoughts would not increase helping. Another and conflicting prediction might be produced by a norm salience theory. Thinking about the parable should make norms for helping salient and therefore produce more helping. The data, as hypothesized, are more congruent with the prediction drawn from the parable. A person going to speak on the parable of the Good Samaritan is not significantly more likely to stop to help a person by the side of the road than is a person going to talk about possible occupations for seminary graduates.</blockquote><blockquote>Since both situational hypotheses are confirmed…</blockquote><p>Aaaaaaaaaaaaaaaaaaaaaaaaaaaah!!!</p><p>Somehow, the paper manages to make the “pious thoughts are ineffective” hypothesis into the null hypothesis and the “norm salience” hypothesis into the alternative hypothesis. Then, when the results are not significant to reject the null hypothesis this is treated as confirmation that the null hypothesis is true. This is the equivalent of accepting p<0.95 as evidence for the “pious thoughts are ineffective” hypothesis.</p><p>(Aside: I’m no theologian but I’m not really sure that “pious thoughts are ineffective” is really what the parable implies. Jesus often used the religious leaders as the bad guys in his parables so he may just be repeating that point)</p><hr class="dividerBlock"/><h2><strong>Effect Size, Significance and Experimental Power</strong></h2><p>I think the issue here is confusion between effect size and significance.</p><p>The effect size is actually pretty good (80% increase in helping). In fact, in the condition that the GS participants weren’t rushing they averaged an impressive score of 3.8 (compared to 1.667 for the equivalent non-GS participants).</p><p>The fact that this doesn’t rise to significance has little to do with effect size and everything to do with experimental power.</p><p>The sample size was 40. There were 6 categories relating to the first 2 hypotheses (3 hurry conditions x 2 message conditions). If for each of the 3 religiosity type conditions a participant was just rated as “high” or “low” then this is 8 categories. That makes a total of 48 possible categorisations for each subject to cover the 3 hypotheses. We’ve managed to get more potential categorisations of each subject than we have subjects.</p><p>Aaaaaaaaaaaaaaah!!!</p><p>(Actually, this may not be irretrievable in and of itself – it just threw up a big red flag for me. If all the other parts of the experiment were on the money this could just be efficiently testing as many different hypotheses as possible given the limited data points available. The real problem is that if we adjust for multiple variable testing then the required p-value for significance goes down and power goes down with it.)</p><p>In addition to sample size, experimental power depends on variation in the dependant variable due to other sources (I’m happy to accept that they had low measurement error). My best guess is that there is significant variation due to other sources although I don’t have the data to show this. A number of personality traits had been investigated previously (Machiavellianism, authoritarianism, social desirability, alienation, and social responsibility) and found not to significantly correlate with helping behaviour, so my expectation would be that finding a true effect is difficult and unexplained variation in helping is large.</p><p>If experimental power is low, in order to find significant results, the effect size must be large.</p><p>As the effect size of reading GS was below the effect size required, the result is not statistically significant.</p><p>If an effect size of increasing helping by 80% is not significant, you really should have known before the experiment that you didn’t have enough power.</p><hr class="dividerBlock"/><h2><strong>Further reducing experimental power</strong></h2><p>If you thought that N=40 was questionable, wait until you see what comes next. The paper goes on to see if the input variables correlate with the amount of help given when help was given. Only 16 people gave any help so suddenly N=16.</p><p>Aaaaaaaaaaaaaaaaaaaaaaah!!!</p><p>This seems like bad news for finding significance but we suddenly do have a significant effect. It turns out that scoring higher on seeing religion as a quest makes you likely to offer less help than if your score lower on this metric. This is contrary to the experimenters’ expectations.</p><p>After performing some extra calculations, the experimenters conclude that this is because those who scored lower on this metric were likely to offer over-the-top assistance and score a 5 which skewed the results.</p><p>Allow me to offer an alternative explanation. </p><p>The paper has so far calculated 18 different p-values (3 from ANOVA of message x hurry, 10 from linear regression of full data (5 x help vs no help, 5 x scoring system) and 5 from linear regression of only helpful participants). There were actually another 10 p-values calculated in their stepwise multiple regression analysis but these seem to have been ignored so I’ll gloss over that.</p><p>Now for each p-value which you calculate you have a 5% chance of finding a spurious result. I’ll take off the 3 p-value calculations which yielded true effects and say 15 opportunities to get a spurious p-value. </p><p>0.95 ^ 15 = 0.46</p><p>At this point, you are more likely to have achieved a spurious p-value than not from all the calculated p-values. Some of the p-values calculated are related so that may change the exact value but the probability of a spurious result is uncomfortably high.</p><p>Remember that an increase in helping of 80% didn’t achieve significance when N=40. The effect size must be truly huge in order to achieve significance with N=16 (The actual effect size isn’t given in the report).</p><p>Because their prior for this effect being true is fairly low (it’s huge and in the opposite direction to expectation) it would be reasonable to say that the p-value is probably spurious in the report with a note that this might be worth investigating further in the future.</p><p>Instead, the report ends up with a weird conclusion that low religion-as-a-quest scoring people are more likely to offer over-the-top help. The fact that they achieve an additional significant p-value when the introduce a new categorisation system (over-the-top help vs reasonable help) doesn’t add much to the likelihood of their conclusion – it just shows that they are able to look at their data and see a pattern.</p><hr class="dividerBlock"/><h2><strong>Introducing new variables</strong></h2><p>At this point, another input variable is introduced. The original 3 types of religiosity were made up of scores from 6 different scales which were weighted to create the 3 types. Suddenly one of the 6 original scales is grabbed out (doctrinal orthodoxy) and this correlates even more strongly with giving over-the-top help (p<0.01).</p><p>Aaaaaaaaaaaaaaaaaaaah!!!</p><p>Introducing a new categorisation (over-the-top help) and a new variable (doctrinal orthodoxy) to try to explain a (probably) spurious p-value from multiple hypothesis testing is NOT a good idea. </p><p>We now have 4 different potential categorisations and 11 variables (the original 5 plus the 6 newly introduced scales). This makes 44 different potential p-values to calculate even before we consider the different types of tests that the authors might try (simple linear regression, ANOVA, stepwise multiple linear regression). I don’t think they calculated all of these 44+ p-values but rather looked at the data and decided which ones looked promising.</p><p>0.99 ^ 44 = 0.64</p><p>So now, even in the best case, a p<0.01 would happen in more than a third of similar experiments just by coincidence.</p><p>I don’t think that the effect described is impossible but I think the failure to adjust for multiple variables is a much more likely explanation.</p><hr class="dividerBlock"/><h2><strong>Conclusion</strong></h2><p>So in conclusion, against all expectation, reading and preparing a talk on a parable given by the leader of your religion on how we should help people who are in need does, in fact, increase the likelihood that you will, in the next 5 minutes, help someone who is in need.</p><p>The fact that being in a hurry is a larger effect is the truly interesting finding here but I think not a huge surprise.</p><p>This is why I asked the question the way I did in the introduction – I didn’t get the chance to guess this blind and I’m not sure which way I would have voted if I had.</p><p>I’m confident that I wouldn’t have predicted quite such a big drop of help between GS low hurry and GS high hurry so I’ll have to update accordingly (average score 3.8 down to score 1).</p><p>One final thing:</p><p>Aaaaaaaaaaaaaaaaaaaaaaaaah!!!</p>buckykmT47aLQmqzcw329Y2018-10-30T23:34:27.153ZIn praise of heuristics
https://lw2.issarice.com/posts/BktRFP2jtYLJvFLEg/in-praise-of-heuristics
<p> </p><p>We’ll get there in the end, bear with me.</p><h2>Introduction to ZD strategies in IPD</h2><p><em>Feel free to skip if you’re already familiar with ZD strategies.</em></p><p>In the iterated prisoner’s dilemma (IPD) a zero determinant (ZD) strategy is one which forces your opponent’s winnings to be a linear relation to your own winnings. These strategies can take either generous or extortionate forms.</p><p>Think of it as a modified version of tit-for-tat. </p><p>***</p><p>A Generous ZD strategy still always respond to C with C but will sometimes also respond to D with a C (it sometimes fails to punish). With "standard" PD utilities (T=0, R=-1, P=-2, S=-3) my opponent gains 1 utility by defecting. If I defect back in retaliation, I cost him 2 units of utility. If I defect back with probability 0.7, on average I cost him 1.4 units of utility. This still means that defecting is disadvantageous for my opponent (loss of 0.4 utility) but not quite as disadvantageous as it would be if I was playing pure tit-for-tat (loss of 1 utility). </p><p>This gets slightly more complex when you don't have constant gaps between T, R, P and S but the principle remains the same.</p><p>If he defects at all, my opponent will end up gaining more utility than me, but less than he would have got if he had co-operated throughout.</p><p>Advantages of GZD are:</p><p>1. Total utility isn’t damaged as much by accidental defections as it is in pure tit-for-tat.</p><p>2. It won’t get caught in endless C-D, D-C, C-D, D-C as tit-for-tat can.</p><p>***</p><p>On the other hand, Extortionate ZD always responds to D with D but also sometimes responds to C with D. Provided I don’t respond to C with D too often, it is still advantageous to my opponent to play C (in terms of their total utility). </p><p>If my opponent co-operates at all I'll end up with more utility than him. If he gives in and plays C all the time (to maximise his own utility) I can achieve a better utility than I would with C-C. .</p><p>The main disadvantage of EZD in evolutionary games is that it defects against itself.</p><p>For both EZD and GZD you can vary your probabilities to be more or less generous/extortionate, provided you always ensure your opponent gets the most utility by co-operating.</p><h2>Different opinions on fairness</h2><p>An extortionate ZD strategy is similar to an opponent who has different perceptions of what is fair. Maybe your opponent had to pay to play the game but you got in free so he wants a higher percentage of the winnings. Maybe you think this is just his bad luck and think a 50:50 split is fair.</p><p>If you give in to what seems to you to be an extortionate strategy, your opponent is encouraged to make more extortionate demands in future, or modify his definition of what is fair. At some point, the level of extortion is so high that you barely get any advantage from co-operating.</p><p>This brings us to a <a href="https://www.lesswrong.com/posts/z2YwmzuT7nWx62Kfh/cooperating-with-agents-with-different-ideas-of-fairness">proposal</a> of Eliezer’s. </p><p>When choosing whether to give in to an extortioner you can capitulate to some extent, provided that you ensure your opponent gains less utility than he would if he agreed to your favoured position (ideally you should let your opponent know that this is what you're doing).</p><p>This removes any motivation to your opponent to extort and encourages him to give his true estimation of what is fair.</p><h2>Two experiments in ZD strategies</h2><p><a href="https://www.nature.com/articles/ncomms4976">Hilbe et al.</a> performed an experiment on humans playing against computerised ZD strategies. Four different strategies were tried – strong extortion through to strong generosity. Regrettably, pure tit-for-tat wasn’t included as I would have liked to see a comparison with this.</p><p>The two generous strategies achieved higher average utility for the ZD programme than the 2 extortionate strategies. If the human players had acted purely in self-interest (they were paid according to the points gained) the extortionate strategists would have won. So what happened?</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1540381500/blog%20posts/ZD1.png" class="draft-image center" style="width:69%" /></figure></span><p>Firstly, a bit of detail about the experimental setup. The participants were not told that they were playing against a computer programme – the impression given was that they were playing against one of the other experimental subjects (although this wasn’t explicitly stated).</p><p>Looking at the results from each individual player it is clear that none of the human participants allowed the extortionate ZD strategists to beat the score that is achievable from co-operating (R=0.3).</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1540381505/blog%20posts/ZD2.png" class="draft-image center" style="width:54%" /></figure></span><p>It seems that the subjects automatically used a strategy similar to the one suggest by Eliezer (or this represented something of a limit to co-operating) when dealing with a player who seemed to be extortionate.</p><p>In <a href="https://www.nature.com/articles/ncomms11125">another experiment</a> some players did allow the extortionate ZD strategy to achieve higher utility than it would have got by co-operating but over the two experiments there is a strong tendency not to let the unfair strategy get away with it.</p><p>***</p><p>The second experiment tested the effects of:</p><p>1. More rounds of IPD (500 vs 60)</p><p>2. Being told that your opponent is a computer (Aware (A), Unaware (U))</p><p>3. Extortionate (E) / Generous (G) ZD strategies</p><p>Interestingly, human players in this second experiment were, over a long game, much more willing to let their EZD opponent “get away with it” when they were told that their opponent was a computer (see the grouping of red dots in figure a below). For the final 60 rounds of a 500 round IPD the extortionate ZD strategist was achieving 3.127 average utility – significantly (p=0.021) more than the R=3 gained from both players co-operating. </p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1540381509/blog%20posts/ZD3.png" class="draft-image center" style="width:57%" /></figure></span><p>***</p><p>So why are we more willing to let a computer “get away with it”? </p><p>Maybe we view a computer as being insusceptible to change so are therefore more likely to give in. </p><p>Alternatively, if you think you are playing against a human, even after 500 rounds you will probably be annoyed enough with him for not co-operating properly that you won’t be co-operating the whole time. You’re less likely to get annoyed at a computer for beating you at a game. As soon as you realise you can’t beat the computer you can just try to do the best you can for yourself. This doesn't dent your pride as much as it would against a human opponent.</p><p>(There was one person who just defected pretty much throughout the whole 500AE experiment despite knowing he was playing against a computer, maybe he just decided to tit-for-tat or maybe it's just a <a href="http://slatestarcodex.com/2013/04/12/noisy-poll-results-and-reptilian-muslim-climatologists-from-mars/">lizardman constant</a>. Without him the 500AE ZD strategy would have had even more impressive results.)</p><p>***</p><p>To me this looks like humans having a heuristic to deal with extortionate opponents/people who have a different opinion on what a fair split is. People seem to apply this heuristic naturally through emotions such as pride, annoyance and anger.</p><p>The heuristic works out roughly similarly to Eliezer’s suggestion of regulating your opponent’s winnings to less than he would achieve if he played fair/by your rules.</p><p>Being told that your opponent is a computer effectively turns off this heuristic (if you have long enough to get a rough idea of the computer’s strategy). This motivates your opponent to become more extortionate, something which the original heuristic was protecting you against.</p><h2>In praise of heuristics</h2><p>All of that is a very long introduction/example of my main point.</p><p>Heuristics are good.</p><p>Heuristics are very good.</p><p>You don’t even know how many times your heuristics have saved you.</p><p>You possibly have no idea what they are saving you from.</p><p><a href="https://www.lesswrong.com/posts/AdYdLP2sRqPMoe8fb/knowing-about-biases-can-hurt-people">Knowing about biases can hurt people</a>. Getting rid of heuristics without understanding them properly is potentially even more dangerous.</p><p>***</p><p>A recent discussion made me aware of <a href="https://www.lesswrong.com/posts/9thqSN8HDLM3LTxK5/offense-versus-harm-minimization">this post</a> by Scott where he tried to come up with a way of dealing with people who claim you have caused them offense. </p><p>One of the motivations for the post was Everybody Draw Mohammed Day. EDMD seems like it is a natural outworking of the heuristic described above.</p><p>People see the terrorists increasing their utility unfairly by attacking people who draw pictures of Mohammed. To ensure they don’t get an advantage by defecting, people want to decrease their utility back to below where they started – hence Everybody Draw Mohammed Day.</p><p>The terrorists were also following a similar heuristic – the original cartoon decreased their utility, they are trying to decrease the utility of those who created it to demotivate further defection.</p><p>The heuristic isn’t there to improve the world – it is just there so that the person performing it doesn’t encourage increased defection against themselves and increased demands from others.</p><p>Scott’s post was an attempt to turn off the heuristic and replace it with a principled position:</p><blockquote>The offender, for eir part, should stop offending as soon as ey realizes that the amount of pain eir actions cause is greater than the amount of annoyance it would take to avoid the offending action, even if ey can't understand why it would cause any pain at all. If ey wishes, ey may choose to apologize even though no apology was demanded </blockquote><p>In this case, his proposal was criticised by others and Scott ended up <a href="https://www.lesswrong.com/posts/9thqSN8HDLM3LTxK5/offense-versus-harm-minimization#9jRbrJmnkcMMeEEnF">rejecting his own proposal</a>.</p><p>Had Scott applied his policy universally he would likely have ended up losing out if those he dealt with had modified to become more demanding of him. </p><p>It’s likely that our heuristic doesn’t lead us to an optimal result, it just prevents some bad results which Scott’s proposal may have led to. </p><p>Possibly a principled application of Eliezer’s proposal would help optimise the result better than the heuristic. In the experiments there was no standard amount that people chose to penalise the EZD strategist - the results were fairly spread out over the region between full defection and Eliezer's defined maximum co-operation. Sometimes the heuristic doesn't stop there and co-operates more than Eliezer would suggest.</p><p>***</p><p>This all sounds a bit harsh on Scott. Actually, putting an idea out there, engaging with criticism and admitting when you were wrong is exactly the right thing to do.</p><p>I’ll give an example where I didn’t do this and it did, in fact, end up biting me in the butt.</p><p>***</p><p>A while back, I was thinking about status. Status is, within a fixed group, a zero-sum game. People in the workplace are constantly attempting to improve their position on the ladder at the expense of others. This doesn’t just apply to promotions, it applies to pretty much everything. Alex wants to feel like he’s important and will get massively offended if he feels that Bob is trying to take status which should be Alex’s. This probably accounts for ~95% of disagreements in my workplace.</p><p>Zero-sum games are, usually, for suckers. If you can get out of the game and into a positive sum game, you probably should. This is doubly true if you’re competing for a thing you’re not really interested in.</p><p>Status very much matches my definition of a zero-sum game which I don’t want to play. The problem is, status also allows access to things which I do want – it is a very useful instrumental value. It is a game which everyone else plays so it’s hard to unilaterally leave.</p><p>Instead, I made the decision not to play status games unless I really have a need of the status (e.g. I will attempt to achieve status in the eyes of the person who will decide on a potential promotion but not others). Essentially I was trying to turn off the heuristic of “always attempt to gain status with everyone” and replace it with a trimmed down version “attempt to gain status only with those people who make a decision about your pay/promotions etc.”</p><p>Now if you have any experience in how status games work, you may realise that this was a naïve approach. If it isn’t obvious to you, have a think about what might go wrong.</p><p>*</p><p>*</p><p>*</p><p>If you don’t fight for your status with your colleagues, it’s like blood in the water. If they can push themselves up at your expense they will, not always maliciously, it’s just “the thing to do” in a workplace. If there are no consequences then it will happen again and again. In the end, this will mean that the people who you care about impressing will see the status that others treat you as having and start to modify their own opinion of your status.</p><p>It took me a while to realise just how harmful this was. When I did, I had to do a lot of firefighting to re-establish a sense of normality.</p><p>All is fine again now but the experience did teach me that my heuristics are there for a reason and that I shouldn’t get rid of them entirely without properly understanding the consequences. </p><p>***</p><p>I haven’t decided exactly how I should deal with tackling heuristics in future but I have a few initial thoughts.</p><p>1. Don’t be overconfident that you have really understood why the heuristic is there</p><p>2. When comparing potential pros and cons, remember the cons are likely to be worse than you think</p><p>3. Discuss ideas with others</p><p>4. Where possible, make small changes first and monitor progress</p>buckyBktRFP2jtYLJvFLEg2018-10-24T15:44:47.771ZThe tails coming apart as a strategy for success
https://lw2.issarice.com/posts/8jwAvGtrwvKHYgoC7/the-tails-coming-apart-as-a-strategy-for-success
<p>[Epistemic status: A proposed causal mechanism for something I’ve been using for a while which seems to work empirically (n=1!)]</p><p>So, <a href="https://www.lesswrong.com/posts/dC7mP5nSwvpL65Qu5/why-the-tails-come-apart">the tails come apart</a>. I really like the elegance of this theory and how it seems to touch every part of life. It seems highly applicable.</p><p>For example, say you wanted to become the world’s most successful author. There is no single talent for you to learn which makes you the most successful. You must be a good writer but also a good publicist and business manager. Your writing skill can be broken down into various skills such as humour, prose, research ability etc. You need to get good at hiring the right people.</p><p>You don’t necessarily need to be the best at any individual skill. The tails come apart so the person who writes the world’s most beautiful prose might not be able to find themselves an audience if they’re lacking in other skills.</p><p>This is very similar to Scott Adams’ concept of a <a href="http://blog.dilbert.com/2016/11/28/the-trump-talent-stack/">talent stack</a>. He claims that having many good but unspectacular skills, which support each other, is a more efficient strategy than working on a single amazing talent.</p><p>The tails coming apart is a good causal mechanism to explain how the talent stack might work and significantly increases my estimate of how useful the concept is. I’ve been thinking in talent stack terms for a while and it seems to work for me.</p><p>***</p><p>I work in mechanical engineering. The talent which sets me apart the most is maths (I studied maths and <a href="https://xkcd.com/1052/">physics</a>, not engineering). Relatively few people I work with can follow in detail the maths I do, and probably none could generate the work themselves. Within the less wrong community I’m sure I would stand out a lot less but, in my day-to-day work, I could improve my maths overnight +1σ and no-one would notice the difference.</p><p>So in order to improve my overall engineering value it is a better use of my time to focus on some adjacent skills which either:</p><p>1. Fill in gaps in my knowledge from not studying engineering (low time commitment for big relative gains)</p><p>2. Move me into the top 10% within the company in a skill which is useful alongside maths</p><p>Sometimes these are completely new skills, sometimes they may be things I’m already good at which can be modified to be more useful at work.</p><p>***</p><p>Part of me thinks this should be obvious. I suspect some people have a vague model for something similar in their head. </p><p>What surprises me is how few people act on this.</p><p>Most people seem to just improve the skills related to the tasks that they’re carrying out. It’s important to maintain skills in this area but the fact that you’re being paid to work on them suggests that you’re already towards the tail of that particular skill. Getting +1σ better in a skill which you’re already highly competent at is difficult and improves your relative skill over a smaller proportion of the population. </p><p>It is rare to see people work on skills that they’re not familiar with and that don’t come up in their normal work, in anticipation of it being useful later on. With proper planning it seems that this is a better use of time.</p><p>***</p><p>Toy model time. </p><p>Consider two uncorrelated skills which you might learn. You are at the mean skill level (µ) for skill 1 and at µ+2σ for skill 2. Assume a normal distribution for population skill level for both. You have the option of increasing either by 1σ. You only care about getting ahead of as many people as possible with as little effort as possible.</p><p>Going from µ to µ+σ increases your position by 34% of the population, going from µ+2σ to µ+3σ only overtakes 2.1% of the population. Even if the skill 2 is more important, it needs to be 16x more important to give the same impact.</p><p>Of all the people who are above µ, 32% are able to get above µ+σ. Of all the people above µ+2σ, only 8.7% are able to get above µ+3σ. Training skill 2 by σ is 3.7x harder (rarer) than training skill 1.</p><p>Taking into account both impact and difficulty, training skill 1 is 60x more efficient than training skill 2.</p><p>Let's move on before I get arrested for crimes against maths.</p><p>***</p><p>By way of example, here are some of the things I’ve deliberately learnt which aren’t directly related to my work but are adjacent enough to be helpful.</p><p><u>Electronics</u>. Most mechanical engineers have little knowledge of this so getting into the top 10% is relatively easy. I bought a <a href="https://www.raspberrypi.org/">raspberry pi</a> and did a few mini projects.</p><p><u>Programming</u>. I was already a fair programmer (by mechanical engineer standards). The most important additional thing I did was teach myself visual basic so that I can create macros for MS office programs. I also did a bit of Python and SQL as they're used at my work, even though it isn't for something I work on.</p><p><u>Customer understanding</u>. I noticed in my company that a lot of people didn’t really know what factors cause our potential customers to choose a supplier and how they weight each factor. Hanging around with sales-y people and asking questions gave huge relative gains very quickly</p><p><u>Data science/machine learning</u>. This one was mostly for fun and my knowledge is <u>very</u> basic but, because no one else on site has much knowledge in the subjects, I can do things that no one else can.</p><p><u>Mental maths approximations</u>. I was never amazing at mental arithmetic. Getting better at this sometimes helps with work but it is most useful for impressing to people. Other people can’t relate to “real” maths as they don’t really understand what’s going on but they can relate to mental arithmetic and anyone doing it fast looks amazing.</p><p><u>Psychology.</u> My current topic, I’m hoping it will be useful for career building. First book is Influence by Robert Cialdini as everyone seems to recommend it!</p><p>I’d like to claim that I have a mathematical model to determine which skill to work on next but in truth the decision is probably made mostly based on which seems like most fun. This may actually be a valid method of choosing as the more fun something is, the closer it is to a <a href="https://www.lesswrong.com/posts/qjLb9ESxx7Mfp2iX8/neutral-hours-a-tool-for-valuing-time">neutral hour</a>. In truth I didn’t pick the method for any such principled reason!</p><p>***</p><p>What adjacent skills could you learn to maximise your impact?</p>bucky8jwAvGtrwvKHYgoC72018-10-01T15:18:50.228ZDefining by opposites
https://lw2.issarice.com/posts/YkMTXJW5NgcJ3EKYp/defining-by-opposites
<p> In <a href="https://www.lesswrong.com/s/SGB7Y5WERh4skwtnb/p/WBdvyyHLdxZSAMmoz">Taboo your words</a> Eliezer talks about how confusion occurs when people are using the same word to mean something different. Refusing to use the word in question can help with the discussion. He mainly focusses on philosophical debate but it’s a technique which works in everyday life too.</p><p>I came across an example of this. Alex claims that he makes decisions logically. Bob disagrees.</p><p>I’m fairly sure they were talking about two different things but I couldn’t think of a way of explaining this quickly enough. I realised afterwards I should have thought about defining their alternative perceptions of “logically” with their opposites. </p><p>I think for Alex the opposite would have been “haphazardly” and for Bob the opposite would have been “emotionally”.</p><p>“Logically” is loaded up with good karma and I think that this was the main cause of the argument. Alex feels Bob is unfairly robbing him of all the good karma. Bob thinks Alex is unfairly claiming all the good karma. In reality Bob is just trying to stop Alex claiming one bit of good karma (being able to make decisions unemotionally).</p><p>Now had Alex and Bob realised this I’m not sure it would have dissolved the disagreement but they would have been arguing about the same thing. If they had figured this out (or I had explained it to them) the argument may have got worse - use the technique with caution.</p><p>***</p><p>Tabooing your words takes effort. I submit a sub-tool to help if you’re in a hurry – try to think of the opposite of what you mean.</p><p>This won’t always help. In the classic tree falling in the woods argument I can’t think of opposites for the concept of “sound” which would help. However, in more day-to-day experiences I suspect the richness of language would be more likely to bear fruit.</p><p>It is probably most helpful when you have to think fast in conversations where the participants are less likely to be willing to take the time to use the full taboo your words technique.</p>buckyYkMTXJW5NgcJ3EKYp2018-09-18T09:26:38.579ZBirth order effect found in Nobel Laureates in Physics
https://lw2.issarice.com/posts/QTLTic5nZ2DaBtoCv/birth-order-effect-found-in-nobel-laureates-in-physics
<p><em>[Epistemic status: Three different data sets pointing to something similar is at least interesting, make your own mind up as to how interesting!]</em></p><p><strong>Follow-up to:</strong> <a href="http://slatestarcodex.com/2018/01/08/fight-me-psychologists-birth-order-effects-exist-and-are-very-strong/">Fight Me, Psychologists, Birth Order Effects are Real and Very Strong</a>, <a href="https://www.lesswrong.com/posts/x9FNKTEt68Rz6wQ6P/2012-survey-results#KuAdBqWYveHtc6wtC">2012 Survey Results</a>, <a href="https://www.lesswrong.com/posts/tj8QP2EFdP8p54z6i/historical-mathematicians-exhibit-a-birth-order-effect-too">Historical mathematicians exhibit a birth order effect too</a></p><p>In <a href="https://www.lesswrong.com/posts/tj8QP2EFdP8p54z6i/historical-mathematicians-exhibit-a-birth-order-effect-too">Eli Tyre’s analysis</a> of birth order in historical mathematicians, he mentioned analysing other STEM subjects for similar effects. In the comments I <a href="https://www.lesswrong.com/posts/tj8QP2EFdP8p54z6i/historical-mathematicians-exhibit-a-birth-order-effect-too#zb2mKteaB4hxLQBmS">kinda–sorta</a> preregistered a study into this. Following his comments I dropped the age requirement I mentioned as it no longer seemed necessary.</p><p>I found that Nobel Laureates in Physics are more likely to be firstborn than would be expected by chance. This effect (10 percentage points) is smaller than the effect found in the rationalist community or historical mathematicians (22 and 16.7 percentage points respectively) but is significant (p=0.044).</p><p>More brothers were found in the study then sisters (125:92 (58%)). After correcting for the correct expected ratio (~52%) this was found to not be significant (p=0.11).</p><p>I was unable to find sufficient data on Fields medal, Abel prize and Turing award winners.</p><p>My data and analysis is documented <a href="https://drive.google.com/open?id=1egxApi8XQDyclzwgeCXLWrJek26sSrJ7FMgTOwYuiyc">here</a>. With Eli's kind permission I used his spreadsheet as a template. I have kept Eli’s data on the same Table – rows 4-153 are his.</p><h2>Methodology</h2><p>My methods matched Eli’s closely except for the data sets I looked at, see his post for more information.</p><p>Initially I attempted to replicate Eli’s results in other mathematicians by analysing Fields medal and Abel prize winners. Unfortunately I was unable to gather sufficient additional data. This is partly due to crossover in names between these mathematicians and the list from which Eli was working.</p><p>It also seems to be the case that less biographical information is available for people born after ~1950. This might be partly due to these people and their siblings being more likely to be still alive so data protection rules prevent e.g. geni from listing their full details (siblings’ details are often set to “private”) but there could be other reasons. For Fields medals awarded before 1986 I found data on 12/30 recipients, after that only 3/30.</p><p>I had a brief look at Turing award winners, as this would have seemed a relevant field to compare to the results from the rationalist community that inspired the studies, but came across the same problem.</p><p>Finally, I looked at Nobel laureates in Physics. A massive help in data collection here was the fact that since the 1970s Nobel laureates have been asked to supply an autobiography, which is published on the <a href="https://www.nobelprize.org/prizes/uncategorized/all-nobel-prizes-in-physics/">Nobel website</a>. Even before then there are biographies of each laureate although these seldom mention birth order.</p><p>Between the Nobel site, Wikipedia and <a href="http://www.geni.com/">geni</a> I was able to find useful data on 100/207 Physics laureates. The other 107 either had no siblings or I couldn’t find sufficient data on them – either way they weren’t included in the analysis.</p><p>As a comment on data sources, I found geni to be somewhat unreliable. It contradicted the autobiographies or sometimes even contradicted itself. At other times, the list of siblings was incomplete or missing completely.</p><h2>Results</h2><p>Categorising by family size shows that for all family sizes with ≥10 data points there are more firstborns than would be expected by chance. </p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1536052112/blog%20posts/Table1.png" class="draft-image " style="" /></figure></span><p></p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1536058570/blog%20posts/Chart1.png" class="draft-image " style="" /></figure></span><p>Due to small sample size I have grouped all families of 6+ siblings into a single bucket and even then n=14. Expected birth order then varies with higher birth order as there are fewer families in the sample with at least that many children.</p><span><figure><img src="https://res.cloudinary.com/deszvp5h9/image/upload/v1536058429/blog%20posts/chart2.png" class="draft-image " style="" /></figure></span><p>Analysing the data as a whole gives a 10 percentage point effect (0.2 to 19.8 percentage points, 95% confidence). This is less than both the SSC / Less Wrong surveys and Eli’s historical mathematicians analysis (22 and 17 percentage point respectively). I haven’t got a number for overall confidence level for the SSC data but due to the large data set and very low p quoted for the 2 sibling example, it is unlikely that the 95% confidence interval overlaps with this new data, suggesting that the effect is truly a different size and not due to chance.</p><h2>Discussion</h2><p><strong>Autobiographies as source material</strong></p><p>Using autobiographies as the source for a significant number of the data points should have helped with the reliability of the data. It is possible that when writing an autobiography one would be more likely to mention siblings and birth order if one was the eldest but this doesn't seem likely.</p><p><strong>Gender imbalance</strong></p><p>Eli discussed under reporting of females as a potential source of bias. However, he found that the brothers:sisters ratio in his data was not unreasonable.</p><p>Running the same analysis on the physics Nobel laureate data I get a ratio of 125:92 brothers:sisters. This makes the siblings 58% male, with p=0.03 (binomial distribution, two tailed). This effect is actually more significant than the birth order effect.</p><p>Looking at the SSC data and Eli’s data and found that there were 52% brothers in both. I did a little <a href="https://en.wikipedia.org/wiki/Human_sex_ratio">research</a> and found that actually 51-52% is roughly the expected brother:sister ratio. I feel like this is something I should have already known but didn’t. </p><p><a href="https://link.springer.com/article/10.1007/s11692-008-9046-3">Another effect</a> which might increase the proportion of Nobel laureates brothers is that men can have a disposition to have boys or a disposition to have girls. As almost all of the laureates are male it would be reasonable to think more of their Dads were predisposed to having boys. However as this isn’t seen in SSC or historical mathematicians data (both also male dominated) this doesn’t really get us much further.</p><p>Using 52% as the expected ratio (instead of 50%) means that the 58% result from Nobel laureates no longer rises to significance (p=0.11) and should instead be labelled as <a href="https://xkcd.com/1478/">“hey, look at this interesting subgroup analysis”</a> or possibly “slightly odd but not implausible”.</p><p>As I mentioned previously, most of the data since the 1970s Nobels is based on autobiographies. Looking at only data since then, the brother:sister ratio is 51:35 (59%). It seems unlikely that Nobel laureates forgot about some of their sisters, making it less likely that the gender imbalance is due to incorrect data.</p><p>One potential source of error in the gender balance may be in the siblings whose gender I was unable to determine. There were 50 of these. Most (41) of these came from families where I had no data except the number of siblings and the position of the laureate within the family (e.g. “I was the fourth of five children.”). It is possible that some of the missing sisters are in this category. </p><p>However, this would imply that if someone has more brothers they are more likely to list the genders of their siblings than if they have more sisters. Perhaps as most of the laureates were male they might have had more in common with brothers and spend more time with them, making them statistically more likely to mention their brothers’ gender. This seems plausible but unlikely to cause a big effect even if it were true.</p><p>For the moment, I am working with the assumption that the sample is accurate and that the gender imbalance is just an outlier. Any other thoughts on causes of bias are welcome. These would have to explain how this effect was seen both in data from both geni and the laureates’ autobiographies.</p><h2>Conclusion</h2><p>Nobel laureates in physics exhibit a birth order effect such that they are 10 percentage points more likely to be the eldest child than would be expected (p=0.044). This effect is less than data from both SSC readers and historical mathematicians (22 and 17 percentage points respectively). </p><p>There was a gender imbalance between brothers and sisters (58% brothers) but, taking into account the expected ratio of 52%, this was not significant (p=0.11). This effect is not seen in SSC readers or historical mathematicians (52% in both)</p><p>I would recommend that anyone who wishes to collate additional historical data consider Nobel laureates in other awards due to the availability of accurate data from the autobiographies. My analysis took perhaps 12 hours but a lot of that was spent on wild goose chases in looking for data on Fields medal and Abel prize recipients. I saved a lot of time by reusing Eli’s spreadsheet (thanks for the permission). I would estimate getting data on the entire history of another Nobel prize category and analysing it would take ~6-8 hours so it shouldn’t be too daunting for someone to take on.</p>buckyQTLTic5nZ2DaBtoCv2018-09-04T12:17:53.269Z