"Exposure" and "Outcome" can be switched without changing the odds ratio? I'm enrolled in an introductory course in statistics. In one of my lecture notes, the instructor explains why Odds Ratio is popular and he mentions:
"Exposure" and "Outcome" can be switched without changing the odds ratio. Could anyone explain what that means?
I appreciate your help.
 A: To answer your question consider a binary outcome Y (0 or 1) and a binary exposure variable X (0 or 1). In prospective studies we are interested in $P[Y = 1 | X = 0]$ v. $P[Y = 1 | X = 1]$ whereas in retrospective studies we are interested in $P[X = 1 | Y = 0]$ v. $P[X = 1 | Y = 1]$.
First let's consider a prospective study with the setting below:
$P_0 = P(Y = 1 | X = 0)$ , $P_1 = P(Y = 1 | X = 1)$, $OR_1 = \frac{p_1/(1 - p_1)}{p_0/(1 - p_0)}$
Now consider a retrospective study and define:
$P_0' = P(X = 1 | Y = 0)$ , $P_1' = P(X = 1 | Y = 1)$
Note that seems that our retrospective study is the same as prospective study where the outcome and the exposure variables are exchanged:
Now let's calculate the odds ratio for our new setting:
$OR_2 = \frac{p_1'/(1 - p_1')}{p_0'/(1 - p_0')}$
Using Bayes theorem, we know that $P_0' = P(X = 1 | Y = 0) = \frac{P(X = 1)*P(Y = 0 | X = 1)}{P(Y = 0)}$
Do the same for $P_1'$ and substitute them in the $OR_2$ formula, then you will see that you get $OR_1 = OR_2$
Hope it helps.
A: Here is a common example:  we want to investigate the relationship between smoking and lung cancer, so smoking (yes/no) is the "exposure" variable and lung cancer (yes/no) is the "outcome" variable.  We could do a prospective cohort study by finding a large group of smokers and a large group of non-smokers and following them for a period of time and computing the proportion in each group that develope lung cancer.  From those results we can calculate the relative risk of developing cancer given smoking status.
More commonly (due to time, expense, etc.) we are more likely to do a case control study where we find groups of patients with cancer and without cancer then investigate whether they were smokers or not.  In this case we can calculate the risk ratio of being a smoker given cancer status (which is opposite of what we usually want).  Without extra information we cannot calculate the probability (or risk ratio) of getting cancer based on smoking status.
The odds ratio on the other hand is valid in both studies (and approximates the risk ratio of interest in the later), switiching which is exposure and which is outcome makes no difference to the odds ratio (but can make a huge difference in the risk ratio).
