Causal Mediation Analysis with a Binary Outcome and Multiple Unordered Continuous/Ordinal Mediators: Extending Muthén's Method Using Probit SEM to Estimate Natural Direct and Indirect Effects

Introduction: We investigate a technique for estimating the combined effects of multiple unordered continuous or ordinal mediators on a binary outcome. It is an extension of Muthén's method to estimate natural direct and indirect effects (NDE and NIE) for a binary outcome and a single mediator, which involves: (i) fitting a probit structural equation model, (ii) using parameter estimates to predict potential outcome probabilities and (iii) using such probabilities to compute NDE and NIE. Step (ii) requires rescaling the latent continuous variable underlying the binary outcome to address the uncertainty due to mediator residual variance. With multiple mediators, we replace the mediator residual variance with the mediator residual covariance matrix.

Methods: Identification assumptions include no unmeasured confounding of treatment-mediator, treatment-outcome and mediator-outcome relationships, and no treatment-induced confounding of mediator-outcome relationships. For simplicity, we also assume no treatment-mediator interaction. We investigate three broad situations, all with three mediators: (1) no confounders, (2) randomized treatment with one confounder of mediator-outcome paths, and (3) non-randomized treatment with one confounder of all paths. Under no confounding, we investigate variations in (a) the signs of the paths and mediator residual correlations, (b) path strengths, and (c) correlation strengths; as well as (d) the small outcome probability case and (e) the special case where the mediator residual correlation matrix is semi-positive-definite. Under randomized treatment, we investigate mediator-outcome confounding that is positive, negative, and mixed but mostly positive/negative. Under non-randomized treatment, we use the same variation in mediator-outcome confounding but add a moderate confounder effect on treatment assignment. We evaluate the estimation of risk difference (RD) and risk ratio (RR) based NDE and NIE using the maximum likelihood (ML), weighted least squares (WLSMV) and Bayes estimators in Mplus.

Results: Across most path/correlation sign combinations and across variations in path/correlation strengths, the method performs well with all estimators, but favors ML and WLSMV for continuous mediators. When the mediators have the same relationships with the treatment and outcome but their residuals are all negatively correlated, model parameters are imprecisely estimated, inducing bias and poor coverage for some RR-based effects; in this case Bayes estimates have less bias and better coverage. Bias is also an issue when outcome probabilities are small, and here Bayes also performs better. With semi-positive-definite residual correlation matrices, ML fails, WLSMV results in large variances, and Bayes, although biased, performs the best. We recommend researchers use ML/WLSMV generally but switch to Bayes in these specific cases.