Marginal odds ratios: What they are, how to compute them, and why applied researchers might want to use them

Logistic response models form the backbone of much applied quantitative research in epidemiology and the social sciences. However, recent methodological research highlights difficulties in interpreting odds ratios, particularly in a multivariate modeling setting. These difficulties arise from the fact that coefficients from nonlinear probability models such as the logistic response model (for example, log odds-ratios) depend on model specification in ways that differ from the linear model. Applied researchers have responded to this situation by reporting marginal effects on the probability scale implied by the nonlinear probability model or obtained by the linear probability model. Although marginal effects on the probability scale have many desirable properties, they do not align well with research in which relative inequality is a key concept. We argue that, in many cases, the odds ratio is preferable because it is a relative measure that does not depend on the marginal distribution of the dependent variable. In our presentation, we aim to remedy the declining popularity of the odds ratio by introducing what we term the "marginal odds ratio", that is, logit coefficients that have similar properties as marginal effects on the probability scale but that 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 examples that illustrate the usefulness and interpretation of the marginal odds ratio.