Even though I’ve had a fair amount of training in technical areas of statistics and probability, my work (primarily in the realm of “public policy”) does not generally lend itself to having to understand the intricacies of, say, GMM estimators. (Though sometimes I develop Monte Carlo simulations for sensitivity analysis.) The vast majority of what I do involves fairly straightforward things — writing proposals and plans; reading papers; occasionally generating spreadsheets to help write the proposals and plans. Very little math is required, and most of it is basic algebra. I don’t even use that much statistical knowledge except on a tacit level. And it turns out that a lot of things I run into are things that I’ve never formally studied (for example, ANOVA, odds ratios, experimental design, surveying methods), so I wind up having to teach myself fairly basic concepts.

A fair amount of what I am reading these days involves trying to make sense of medical literature. For example, I am trying to understand the risk of someone with particular characteristics getting a disease, like asthma or congestive heart failure. Or, I would read a set of papers outlining the effectiveness of various treatments (measured in either hospitalizations reduced, or length of time in the hospital reduced). A few concepts crop up over and over, some of which I have an understanding of from other fields (e.g. p-values). One of those concepts that crops up fairly frequently is that of the odds ratio, which I haven’t seen much outside of the medical literature thus far. It took me a while to understand what the odds ratio was, and I’m still not quite sure.

But I’ll try to explain, and if anyone is reading this wants to provide suggestions/corrections, please let me know!

So, we start with two groups of people. Let’s call them A and B. They’re a pool of people who have had heart attacks in the past year. We are running a clinical trial to determine if a new medication, jokeapril, will reduce the probability of a heart attack in the next year. ^{1} People are randomly assigned to group A or group B. Those in group A get the jokeapril, while, those in group B do not. Out of a total pool of 100 people, 35 are in group A, and 65 are in group B. ^{2}

Over the next year, 10 of the people in group A have heart attacks, and 32 of the people in group B have heart attacks. The probability of people in group A having a heart attack in the next year, then, is 10/35 = 0.286, which means that the probability of people in group A not having a heart attack is 25/35=0.714. The odds of having a heart attack while being in group A are

In other words, the odds are 1 in 2.5 that someone in the next year with this treatment will have a heart attack under this treatment.

Similarly, the probability of people in group B getting heart attacks is 32/65=0.492, and that of people in group B not getting heart attacks is 33/65=1-0.492=0.508. The odds of someone in group B getting a heart attack in such a case are:

That is, the odds are about even (1:1) that someone in the next year in the control group will have a heart attack.

So, we have the odds of someone in group A getting a heart attack in the next year, and the odds of someone in group B getting a heart attack. The odds ratio, as the term implies, is the ratio between the two:

Here, since the denominators in both parts of group A are the same, and those in both parts of group B are the same, I just canceled them. The odds ratio in this case is 0.4125, which means that the treatment group (A) is 59% (well, 1-0.4125=0.5875) less likely than the control group (B) to have a heart attack in the next year.

Now, suppose what we were counting was the number of people who *avoided* getting heart attacks in the next year. What would change? Simply, the numbers would be flipped around. The odds for group A would be 25/10 = 2.5, and the odds for group B would be 33/32= 1.03125. Dividing these two would give an odds ratio of 2.424, which means the people in group A would be 142% more likely to avoid a heart attack in a year than group B.

What I’ve just done is an analysis of odds ratios in a simple situation — two groups of people, and two levels of treatment. What I am trying to figure out is to what extent it’s generalizable to multiple levels of treatment and multiple groups. I know there are ways, but I don’t know enough yet to make sense of it.

- As far as I know, this is not an actual medication — I hope the
*jokea*– prefix makes that clear — but the*-pril*suffix denotes an ACE inhibitor, which expands blood vessels and decreases resistance. ↩ - I am making up numbers here — this has no correspondence to any clinical trial that I know of. ↩