# 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

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'.