Multilevel versus single-level regression for the analysis of multilevel information: The case of quantitative intersectional analysis.

Intersectional MAIHDA involves applying multilevel models in order to estimate intercategorical inequalities. The approach has been validated thus far using both simulations and empirical applications, and has numerous methodological and theoretical advantages over single-level approaches, including parsimony and reliability for analyzing high-dimensional interactions. In this issue of SSM, Lizotte, Mahendran, Churchill and Bauer (hereafter “LMCB”) assert that there has been insufficient clarity on the interpretation of fixed effects regression coefficients in intersectional MAIHDA, and that stratum-level residuals in intersectional MAIHDA are not interpretable as interaction effects. We disagree with their second assertion; however, the authors are right to call for greater clarity. For this purpose, in this response we have three main objectives. (1) In their commentary, LMCB incorrectly describe model predictions based on MAIHDA fixed effects as estimates of “grand means” (or the mean of means), when they are actually “precision-weighted grand means.” We clarify the differences between average predicted values obtained by different models, and argue that predictions obtained by MAIHDA are more suitable to serve as reference points for residual/interaction effects. This further enables us to clarify the interpretation of residual/interaction effects in MAIHDA and conventional models. Using simple simulations, we demonstrate conditions under which the precision-weighted grand mean resembles a grand mean, and when it resembles a population mean (or the mean of all individual observations) obtained using single-level regression, explaining the results obtained by LMCB and informing future research. (2) We construct a modification to MAIHDA that constrains the fixed effects so that the resulting model predictions provide estimates of population means, which we use to demonstrate the robustness of results reported by Evans et al. (2018). We find that stratum-specific residuals obtained using the two approaches are highly correlated (Pearson corr = 0.98, p < 0.0001) and no substantive conclusions would have been affected if the preference had been for estimating population means. However, we advise researchers to use the original, unconstrained MAIHDA. (3) Finally, we outline the extent to which single-level and MAIHDA approaches address the fundamental goals of quantitative intersectional analyses and conclude that intersectional MAIHDA remains a promising new approach for the examination of inequalities.