Mammography problem from 'Intro to Bayes' in my own words/picture

As far as the take-home practical message goes, on my reading it was never about how well doctors could "diagnose cancer" per se based on mammogram results--rather, the reason we ask about P(cancer | positive) is because it ought to inform our decision about whether a biopsy is really warranted. If a healthy young woman from a population with an exceedingly low base rate for breast cancer has a positive mammogram, the prior probability of her having cancer may still be low enough that there might actually be negative expected value in following up with a biopsy; after all, let's not forgot that a biopsy is not a trivial procedure and things do sometimes go wrong.

So I think this actually does have some implication for real-world clinical care: we ought to question whether it is wise to automatically follow up all positive mammograms with biopsies. Maybe it is, and maybe it isn't, but I don't think we should take the question for granted as appears to be the case.

Now that's some interesting back story. I could see that if one knew that the route from positive test through to establishing conclusively if cancer is present was a fluid path, one might not be overly concerned with the test result itself.

As to this specific problem, I just used what EY used; perhaps there are more applicable/pertinent statistics problems that could be used.

I've been trying to think of other visualization tools that might be more universal or intuitive. I get the sliding java applets, but think if one can tie what they're showing to a real world tool or process of some sort, it will help. What these are doing is no different than his. The "top bar" are the two original spheres. The "bottom bar" is the reduced amount of each sphere (0.8 x 0.01 and 0.096 x 0.99) that remains after sifting.

Just a different way to look at it.

Absolutely, and thanks for the link. That was fantastic. Using a Venn diagram occurred to me, but I hadn't worked out exactly how to use it.

The one thing I will say about this is that I like the idea that using something (e.g. a test) tangible on the individuals or objects contributes to the idea that the new evidence is adding something to the pre-existing knowledge. The Venn diagram shows quite clearly what the final result was, but I like the idea that you're gathering more information, which contributes to the body of knowledge, and that this is what alone allows you to update/slide your estimate.

In the Venn diagram, they just start drawing circles and that's that.

In this one, it's a little more clear (at least to me) that the test is doing different things depending on who is being tested and that this is what creates the end inequality between cancer and non-cancer positive tests.

Does that make sense?

Thanks again for sharing. Honestly, I don't see these visualizations as one size fits all; the more who contribute or popularize the various tools out there, the better everyone is!

Check the update. I'm not quite sure how to describe visually what's "left" in the sieve. I don't want to show both test+ and false+ as outputs, exactly, because a sieve is suppose to keep some stuff back while letting other stuff through. But, I think the idea of the circles above makes things more clear than the bars in terms of what's going on as well as proportionality.

I do cover the equivalent of fasle+ verbally, but it would be nice to make it visually. I'll keep thinking about this. Part of it is that I was trying to set things up onto just one page. If I ditch that (it's already on two now), I could maybe spread things out even more and show what's left in the sieves for each group.

Thanks for the suggestion.

Yeah, I can definitely see that. I thought about two different sieves and more of a "splitting' of the groups, but tried to illustrate that the same test is being used on everyone, not different tests depending on whether you have cancer or not.

Maybe "l'll try that version and see if it's clearer.

Thanks. In the end, I agree that the Venn diagram is much "cleaner." I actually thought of using one at one point, I guess it's been done!