Estimating and contextualizing the attenuation of odds ratios due to non collapsibility

The odds ratio is a measure commonly used for expressing the association between an exposure and a binary outcome. A feature of the odds ratio is that its value depends on the choice of the distribution over which the probabilities in the odds ratio are evaluated. In particular, this means that an odds ratio conditional on a covariate may have a different value from an odds ratio marginal on the covariate, even if the covariate is not associated with the exposure (not a confounder). We define the individual odds ratio (IORs) and population odds ratios (PORs) as the ratio of the odds of the outcome for a unit increase in the exposure, respectively, for an individual in the population and for the whole population, in which case the odds are averaged across the population. The attenuation of conditional odds ratio, marginal odds ratio, and PORs from the IOR is demonstrated in a realistic simulation exercise. The degree of attenuation differs in the whole population and in a case-control sample, and the property of invariance to outcome-dependent sampling is only true for the IOR. The relevance of the non collapsibility of odds ratios in a range of methodological areas is discussed.