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Binary Outcomes part 2

Relative Risk

The statistic usually used in RCTs and cohort studies to express these results is the relative risk (RR). This is a very straightforward comparison of the proportions in each group, and is calculated as the proportion experiencing the event on the "test" or treatment arm divided by the proportion on the control arm. This convention means that the if the event is undesirable and the test treatment more effective at preventing it, the RR will be less than 1.0.

  proportion recovering on MiracleCureTM  20/100   0.2  
RR= _____________________________________________ = __________ = _____ = 0.33
  proportion recovering on OldBoring   60/100   0.6    


As we discovered in the last exercise, the RR for the trial of MiracleCure was 0.33 (or 1/3). In this case, the outcome (recovery) is desirable and so an RR of less than 1.0 suggests that the control treatment was more effective. A cursory glance at the raw data reassures us that this is the case.

Odds Ratio

In a case control study, the RR cannot be used as there is no way to estimate the risk associated with treatment or exposure. This is because the numbers recruited to the study are fixed by the research design. However, when the proportion experiencing the event is fairly low, the odds ratio (OR) provides a reasonable estimate of the RR. When the event rate is large, it will be substantially higher than the RR but is still a useful measure, if somewhat less straightforward to interpret.

Odds are commonly encountered at the bookmakers in everyday life, and are defined as the chance of the event happening divided by the chance that it does not.

  odds of recovering on MiracleCureTM   20/80   0.25  
RR= __________________________________________ = _________ = ______ = 0.17
  odds of recovering on OldBoring   60/40   1.5    

In this trial, the event rate is very high and so the OR is substantially different from the RR, suggesting a treatment effect twice as large if the OR is interpreted as a direct estimate of the RR. As this was an RCT, the RR would usually be preferred in any case, although the OR is often considered more convenient and elegant for mathematical reasons. The most common use in RCTs and cohort studies is in logistic regression, a regression model developed to analyse binary (0,1) outcome data.

Click the "quiz" link below to take a short quiz on Binary Outcomes.