Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 41 / 1 / e / Solution

Let $Y_i=1$ indicate a depressed outcome and $C_i=1$ indicate the control arm. A logistic regression could specifywith $x_i$ containing prespecified baseline characteristics, including the stratification factor. Fit the coefficients by maximizing the likelihood for independent Bernoulli random variables ${\prod }_i{p}_i^{Y_i}(1-p_i{)}^{1-Y_i}$. For equal covariate values the control-to-intervention odds ratio is $e^{{\beta }_C}$. Thus an estimate ${\hat{\beta }}_C=\log (2.1)$ produces the quoted adjusted ratio. An ordinary Wald confidence interval is obtained by exponentiating the endpoints of the coefficient's confidence interval.

The raw control-to-intervention odds ratio, using the actual counts, isUsing only the quoted percentages gives about $2.05$, so both versions are approximately $2.1$. The printed $14\%$ is not the usual whole-percentage rounding of $40/297\simeq 13.47\%$; using the counts avoids that small numerical inconsistency. The reversed intervention-to-control ratio is about $0.475$.

The apparent surprise is a coding issue: with depression as the adverse outcome and intervention relative to control, protection would give an odds ratio below one. A ratio above one is favourable if the comparison is control relative to intervention, as above, or if the outcome has been recoded as absence of depression and intervention is compared with control. Both comparison and outcome must be specified. The adjusted ratio need not equal the crude one, even in a randomized trial: covariate adjustment and noncollapsibility of the odds ratio can change an odds ratio. Their close numerical values here do not prove that the regression adjustment was unnecessary.