# How to interpret odds ratio?

Suppose we have a baseline exposure group and 2 other exposure groups for a case control study. Suppose the odds ratio for the first exposure is $1.5$ and the odds ratio for the second exposure is $1.8$. Does this mean that cases are $1.5$ times as likely to have exposure 1 than the controls? Also does this mean that cases are $1.8$ times as likely to have exposure 2 than the controls?

• For example, you could have $90$ cases out of $100$ members of the control group i.e. $\frac{90}{100}=90\%$ of them with odds of $\frac{90}{10}=9$, and $27$ cases out of $29$ of the exposure 1 group i.e. $\frac{27}{29}=93.1\%$ of them with odds of $\frac{27}{2}=13.5$, and this would give an odds ratio of $\frac{13.5}{9}=1.5$ but would not mean "cases are $1.5$ times as likely to have exposure 1 than the controls". Arguably, more than three-quarters of the $90+27=117$ cases come from the control group – Henry Jun 10 '18 at 11:19

## 3 Answers

It means that the odds of a case having had exposure #1 are 1.5 times the odds of its having the baseline exposure. This is not the same as being 1.5 times as probable: odds are not the same as probability (odds of 2:1 against means a probability of $\frac{1}{3}$). So it comes down to what you mean by 'likely'.

In answer to the second part of your question, what you say is correct (assuming that by "likely" you mean in terms of odds not probability). Both odds ratios should be relative to the controls.

However, you should check carefully the definition of the exposures - it is possible that having the second exposure is only possible if you have already had the first exposure, in which case the second odds ratio is not the relative odds of having an exposure at time point 2, but rather of having two exposures. If you can provide more information on the methods, I can give a clearer answer specific to your case.

Your language, "cases are 1.5 times as likely to have exposure 1 than the controls" is a fine description of the interpretation of an odds ratio. As some have noted "likely" is something of an ambiguous phrase, though I doubt anyone in epidemiology is going to raise an eyebrow at your language.

You might consider either "The odds of having exposure 1 in cases was 1.5 times that of controls" or some such as a slightly more precise wording.