Estimation of marginal odds ratios

Coefficients from logistic regression are affected by noncollapsibility, which means that the comparison of coefficients across models may be misleading. Several strategies have been proposed in the literature to respond to these difficulties, the most popular of which is to report average marginal effects (on the probability scale) rather than odds ratios. Average marginal effects (AMEs) have many desirable properties but at least in part they throw the baby out with the bathwater. The size of an AME strongly depends on the marginal distribution of the dependent variable; for events that are very likely or very unlikely the AME necessarily has to be small because the probability space is bounded. Logistic regression, in contrast, estimates odds ratios which are free from such flooring and ceiling effects. Hence, odds ratios may be more appropriate than AMEs for comparison of effect sizes in many applications. Yet, logistic regression estimates conditional odds ratios, which are not comparable across different specifications. In this paper, we aim to remedy the declining popularity of the odds ratio by introducing an estimand that we term the "marginal odds ratio"; that is, logit coefficients that have properties similar to AMEs, but which retain the odds ratio interpretation. We define the marginal odds ratio theoretically in terms of potential outcomes, both for binary and continuous treatments, we develop estimation methods using three different approaches (G-computation, inverse probability weighting, RIF regression), and we present an example that illustrates the usefulness and interpretation of the marginal odds ratio.