Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 34 / 1 / a / Solution

The odds ratio compares event odds, not event probabilities. Let group 1 denote the elective group and group 0 the reference group. Their fitted event probabilities are ${\hat{p}}_1=9/1515\simeq 0.00594$ and ${\hat{p}}_0=1215/75057\simeq 0.01619$. ThusThe risk ratio is similarly about $0.367$. The unadjusted observations show a lower adverse-event frequency in the elective group, rather than an increase. This is an association, not evidence that choosing the procedure prevents adverse outcomes. The direction is opposite to the reported adjusted odds ratio above one; the two analyses compare different populations or different conditional risks.