A Multivariate Generalized Linear Model Approach to Mediation Analysis and Application of Confidence Ellipses

Mediation analysis evaluates the significance of an intermediate variable on the causal pathway between an exposure and an outcome. One commonly utilized test for mediation involves evaluation of counterfactual effects, estimated from separate regression models, corresponding to a composite null hypothesis. However, the “compositeness” of this null hypothesis is not commonly acknowledged and accounted for in mediation analyses. We describe a generalized multivariate approach in which these separate regression models are fit simultaneously in a single parsimonious model. This multivariate modeling approach can reproduce standard mediation analysis and has notable advantages over separate regression models, including the ability to combine distributions in the exponential family with any link functions and perform likelihood-based tests of some relevant hypotheses using existing software. We propose the use of a novel visual representation of confidence intervals of the two estimates for the indirect path with the use of a confidence ellipse. The calculation of the confidence ellipse is facilitated by the multivariate approach, can test the components of the composite null hypothesis under a single experiment-wise type I error rate, and does not require estimation of the standard error of the product of coefficients from two separate regressions. This method is illustrated using three examples. The first compares results between the multivariate method and separate regression models. The second example illustrates the proposed methods in the presence of an exposure–mediator interaction, missing data and confounding, and the third example utilizes these proposed methods for an outcome and mediator with negative binomial distributions.