Odds Ratio and Relative Risk

Odds Ratio (OR) and Relative Risk (RR) are two ratios often used in the epidemiology studies and the clinical trials. They are related, but the calculation and the interpretation are quite different. Notice that the Relative Risk (RR) may also be called Risk Ratio with the same abbreviation of RR.

Relative Risk

P0=The probability of events (eg., responder, cardiovascular event) when a covaraite is at a give level (eg, treatment = Placebo, gender=female)
P1=The probability of events (eg., responder, cardiovascular event) when a covariate is at one unit higher than the previous level (eg., treatment = New Drug) or simply another level (gender=male)
RR = P1 / P0

Odds Ratio
Odds = The probability of events / the probablity of non-events
OR = Odds in one group / odds in another group = P1/((1-P1) divided by P0/(1-P0)
For example, OR = Odds in treated group / Odds in Placebo group

When P1 and P0 are small, OR can be used to estimate RR. However, then P1 and P0 are close to 0.5, the OR is typically much larger than RR.

Steve Simon wrote an excellent web page about the comparison of Odds Ratio versus Relative Risk.

In epidemiology class, we are typically advised to use OR for case-control study and use RR for cohort study. For cross-section studies, both OR and RR may be used.

In clinical trial setting, there is no consensus of using OR or RR. In practice, both of them are used. Sometimes, using OR or RR could be manipulated to serve one purpose or another. Brian Attig and Alison Clabaugh criticized the misuse of statistical interpretation of the odds ratio in APPLIED CLINICAL TRIALS.

Both OR and RR can be easily obtained from SAS Proc Freq with RISKDIFF option. An example by Mimi Chong from the following website illustrates this. We just need to be careful when we read the results from the SAS outputs. Odds ratio is clearly labelled and the Risk ratio is the the numbers corresponding to 'Col1 Risk' or 'col 2 risk' depending on which column is defined as 'event'.

If additional covariates (or in epimiology term, confounding factors) need to be considered, SAS Proc Freq with CMH option or Proc Logistic regression or Proc Genmod can be used.