Odds Ratios

For a recording of the "5 Minutes of EBM" presentation associated with this summary, click here.

Odds ratios are a commonly reported metric, particularly in retrospective (e.g. case-control) studies.  The OR gives one information about the strength of a proposed association between an exposure and an outcome.  It's fairly easy to remember how it's calculated if you understand exactly what the terms mean.

The "odds" of something happening are subtly different from the percent probability of the same event.  To calculate percent probability from a data set, you simply divide the number of people who experienced the event by the total number of people - so if 12 out of 100 people being observed for acute MI do, in fact, have an MI, then the probability of having an MI was 20%.  The odds of having an MI, on the other hand, are the probability of having the event divided by the probability of not having the event.  So in the same example, the odds of having an MI are 20/80, i.e. one in four (which could also be expressed as "four to one against").

The odds ratio, then, is just a ratio of odds; the odds of the outcome of interest occurring in the group with the exposure divided by the same outcome in the group who lack the exposure. To expand on the previous example, consider a case-control study evaluating the relationship between of watching the Republican primary debates and swearing under one's breath.  Suppose that in the group who watched, say, Santorum speaking, the incidence of cursing was 50%, whereas in the group who were reading a good book at the time only 10% were heard to utter any foul language.  Try to calculate the odds ratio associated with watching Rick Santorum for the outcome of sotto voce profanity (answer below).











In this example, the odds in the exposed group would be 50/50 (i.e. 1/1), and the odds in the unexposed group would be 10/90 (i.e. 1/9).  The odds of the event in the exposed group divided by the odds of the event in the unexposed group is equal to 9, which is the odds ratio.  For a good 2X2 table diagram showing how to calculate the odds ratio, click here.  

The main thing to notice about odds ratios is that they are relative measures of the strength of association, which tell you nothing about its absolute size.  Sometimes they're necessary, but they can also be used (like relative risk) to make trivial effects appear larger than they actually are - so whenever you see an OR, you should try to get some sense of how important the association under consideration is in absolute terms.

Relative Risk

By the end of this post, you should be able to describe in words what "relative risk" is, understand its significance, and possibly even calculate it given the appropriate data.

Whenever you're evaluating a study of treatment effect, try to think in terms of outcomes and exposures.  (The term "exposure" should be understood broadly to include therapeutics and risk factors.)  The important questions usually revolve around the relationship between the incidence of an outcome (e.g. myocardial infarction) to an exposure (e.g. aspirin).  If the study you're reading doesn't give you the actual incidence of the outcome you're interested in, you can calculate it (and most measures of treatment effect) by creating a simple 4x4 table like this one:

(This table should remind you of something).  It's very similar to the one you use to calculate the characteristics of diagnostic tests.)

It's easy to see that the incidence of an outcome in either the exposed group or the unexposed group is simply the number of people with the outcome divided by the total number of people in either group.  So, for instance, the incidence of the outcome in those exposed is A/(A+B).  As described in a previous post, the absolute risk difference is arithmetic difference between the incidence in the group exposed and the incidence in the group not exposed - in mathematical terms, that's [A/(A+B) - C/(C+D)].  The relative risk, on the other hand, is the ratio between the two incidences - or, in mathematical terms, [A/(A+B) / C/(C+D)].

The crucial question with relative risk is always "relative to what?"  It gives you no information at all about the actual incidence of the problem in question.  Imagine two treatments, one of which reduces the incidence of a common condition from 30% per year to 20% per year, and another which reduces the incidence of a rare condition from 0.03% per year to 0.02% per year.  The first corresponds to a number needed to treat (NNT) of 10, the latter to a NNT of 10,000, but the relative risk difference is the same.

The reason this is important to understand is that a lot of therapeutic effects are small, and therefore the relative risk change is usually a bigger number than the absolute risk change.  Because of this, it's frequently reported by people who want their treatments to look successful; so when you see a study that only reports a relative risk change, you should smell a rat and try to figure out what the absolute risk change is.  Most of the time, you will find that it makes the relative risk change look significantly less impressive.