I have a problem, which is rooted in a very discouraging observation. I have the task of explaining Bayes' theorem to some regular folks who aren't necessarily mathematically minded. I tried to explain it to readers here once a while back, and I thought I'd done okay at the time but maybe not?

It turns out that a very large proportion of the people really don't get math at all. Bayes' theorem isn't actually very sophisticated mathematics -- all you need is high school algebra. There are no exponents, just multiplication, division and addition. But people tell me my explanation is just incomprehensible gobbledygook. I can't for the life of me imagine why -- it's straightforward, perfectly simple, and crystal clear as far as I'm concerned. The problem is that it has numbers in it, and an equation, and that seems to be enough to make people run screaming for the exits.

We really need to do something about this. I think that the shrieking mobs (metaphorically) who attacked the new mammography screening recommendations were largely a function of innumeracy, as is the paranoia about health care "rationing" and death panels. If people could just get into their heads the basic ideas about probability and absolute and relative risk, and the magnitude of risks and their consequences, we could have much more fruitful conversations about public health and health care policy. But it seems they just can't.

I did a mini probability and statistics course a while back. I can re-run it and try to improve it if that seems like a good idea. (Any advice will be appreciated.) And if anyone cares to check out the post on Bayes' theorem and tell me why it's absolutely impossible to understand, I'll take that too.

## Tuesday, March 30, 2010

### Math is hard!

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## 10 comments:

You lost me at ".9/10.8"

Look, I bailed out of math as early as possible in high school. I never "got" long division; neither did most of the rest of the class, so I blame it on poor teaching. The teacher did not go back over it, but just moved on. Most of the rest of what I studied I understood and could do, but I've never had to use it, so it's all forgotten. Never studied statistics.

And I went to a supposedly "good" public school. I'm always astounded by how vastly much more kids who went to private school were taught at the high school level.

Well, .9 is a number, and 10.8 is a number. So I divided .9 by 10.8

You don't have to be able to do the calculation to grok the concept, right? It's the same, BTW, as 9/108, if that seems less confusing.

I'm thinking that you are not kidding here, so I have two words for you...Willful. Ignorance. I believe the political dialogue is so polarized that people only want to hear information, facts and opinions which support their own mind set. It's the McD's of citizenship these days and the media is only too happy to oblige by tailoring newscasts(and I use that term loosely since some are simply big on opinion and short on facts.I'll look at the math on your past posting once I finish my coffee and walk the dog.

Since math confuses folks, what not try using less of it? How about something like this (a revamped version of your original post):

A virus infects 1% of the population. In other words, 1 out of 100 people have it, so in a town of 1,000 people, 10 have the virus and 990 do not.

There's a diagnostic test for the virus which comes back positive 90% of the time when people actually have the virus. (That's called the "sensitivity" of the test.) If you don't have the virus, it comes back negative 90% of the time. (That's called the "specificity.") Sounds like a pretty good test, huh? Let’s see.

All 1,000 people in the town take the test.

Of the 10 who have the virus, 9 test positive but 1 tests negative (because the sensitivity of the test is 90%, and 90% of 10 is 9.)

Of the 990 who do not have the virus, 891 test negative but 99 test positive (because the specificity of the test is 90%, and 90% of 990 is 891.)

So in total 1,000 people took the test and 99 + 9 = 108 of them tested positive. But only 9 of those 108 actually have the virus, 99 of them do not! 9 is about 8% of 108, so if you tested positive there is only an 8% chance that you actually have the virus.

Thanks Eric, yes I think the illustration is a good way for people to get the basic idea. I like to show the precise calculation and the general formula as well in case people want to know it. But does that tend to intimidate people who are math-phobic? I'm saying, you don't need to memorize the formula, I'm just giving it to you in case you're interested. But is that counterproductive with some folks?

Got it, C. Typical of me to read an obscure mathematical formulation where something straight forward was meant. I seem to recall actually understanding what you wrote at the time, but it's next to impossible for me to retain stuff like that. Math's always been a struggle for me. I wouldn't call myself phobic, but neither did I like it enough to continue to pursue it when I saw an escape route.

I also have a tendency to swap numbers between the time I read them and the time I write them down. I'll see and say to myself "forty-three" but write 34. Sucks really, especially when I'm balancing my checkbook. I try to do math on paper whenever possible, rather than rely on a calculator, just to force myself to keep those brain cells functioning at some level.

Weren't you the one who pointed out that the media doesn't hesitate to throw around sports stats, but won't touch stats in the context of policy debates?

i think the take-home message -- most people who test "positive" probably don't have it -- is the important one. that gets lost in the numbers, at a quick read and for those of us who primarily use numbers to figure out how much is in the bank account, or [advanced] how much to tip.

Ok, back again today. Quite honestly, don't consider myself math phobic, but know myself well enough to know I need to translate math problems into something that makes sense to me, which means I use multiple pages of paper until the "A HA" moment occurs. Sounds a bit moronic of me, but there it is. Eric's approach helped with the "translation," but your explanation helped with the math equations.

me: eric is spot on with the concrete ex. kathy is right, the consequences have to be brought home close and personal:

lecture: what happens to the 9 falsely diagnosed positive ppl? how would you feel if you were told you had a deathly virus/breast cancer/whatever when you don’t? what medical procedures will you undergo that may cost you money, time, anguish? might you even be operated on, or quarantined, in error? or would you be lucky and be offered a second test, which would then most likely be negative? then what? are you afflicted or not? maybe yes (one test = pos), maybe no (one test = neg)...what will you doctor do, order a third test? maybe you know someone who has been thru something like this?

what about the societal cost, not just the cost to you, but to the community? Many tests are quite expensive, and ppl must pay for them in one way or another. e.g. if you have private health insurance, or if you pay taxes to fund a ‘single payer’ system, your payments include paying for that test to be administered to /pop. that is suspected of/being at risk/having disease/etc.

me aside: why do medical costs rise and rise? until they can’t be paid for - working that in brings up a slew of other points, not for now.

any exposition of this type must, to get through, not just give equations or ‘do the math’ but propose pathways forward. this is of course bloody difficult.

one thing to do is to show first of all that the math is all very well but not the only consideration, “reality”, or “patient input” counts for a lot.

lecture: cold numbers are *necessary*, but don’t unequivocally tell the medical community what to do. ex: a false negative (use other words) in the case of a deadly virus might kill the whole town, everyone about; with the false positives leading to anguish and maybe ‘disruption’, financial costs for sure, but worth it in the long run, to save X ppl, the whole town, the children in the schools, the babies in their cradles, etc. :)

for other /afflictions/ it is the other way about; false positives create a lot of misery and the false negative is unimportant (prostrate cancer or other ex.)

so who decides? - here is the argument for central planning, hazardous as it may be.

going forward, ppl differ in their attitudes to their own health, and should be given choices..a GP should advise, etc. etc. Yet, personal considerations can’t override what is mandatory for the community...so some balance must be found...

me: C can surely work that out

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apologies for the quick typing, maybe C could write a white paper outlining the main issues ...

Ana

All good points Ana. I will take your assignment.

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