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

Abstract

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. First, in the LMCB commentary the authors 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. Second, 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. Third, 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.

Cross-classified Multilevel Analysis of Individual Heterogeneity and Discriminatory Accuracy (MAIHDA) to evaluate hospital performance: the case of hospital differences in patient survival after acute myocardial infarction

“…The second reason is that the equivalence between traditional and multilevel regression results only occurs when the hospital GCE (ie, the clustering) is low and the number of patients at the hospitals is very high (ie, reliable estimation of hospital averages). 26 In other words, traditional non-multilevel analyses give similar results to the multilevel analysis only when the hospital differences are not relevant (ie, low VPC) and the patient load is very large in every hospital (which is rarely the case). In addition, hospital level variables appear paradoxically more statistically ‘significant’ when the hospital level is less relevant (ie, low VPC).…”

An Intersectional Analysis Providing More Precise Information on Inequities in Self-Rated Health

“…Our study could have been performed using multilevel AIHDA, i.e., MAIHDA, as described [11,53] and implemented [17,54,55] elsewhere. While both AIHDA and MAIHDA conceptually consider the intersectional strata as contexts, the MAIHDA has the advantage of providing a statistical technique that ts better with this concept.…”

“…In addition, MAIHDA provides precision-weighted estimates of risk in strata with few individuals, as the strata estimations are shrunken towards the grand mean. Besides this, the multilevel approach uses the grand mean rather than a speci c stratum as the reference in the intersectional comparisons [53]. A practical limitation of the MAIHDA is that being a multilevel random effects approach it is more suitable for implementation in large databases and particularly when analyzing many strata.…”

Non-response in a national health survey in Germany: An intersectionality-informed multilevel analysis of individual heterogeneity and discriminatory accuracy

“…However, in presence of interaction between fixed effects, precision-weighting performs less well than in regular applications of multilevel modelling [26]. Furthermore, estimates of fixed effects in MAIHDA are different from fixed effects in single level models as they are derived from precision-weighted grand means instead of population means [25,27]. A grand mean in MAIHDA is the mean of means of intersectional strata [25].…”

“…Finally, intersectional Multilevel Analysis of Individual Heterogeneity and Discriminatory Accuracy (MAIHDA) has recently been developed to inform quantitative data analysis by an intercategorical intersectional framework [21][22][23]. Main goals of MAIHDA are to estimate measures of discriminatory accuracy, measures of disease frequency, and stratum-specific total interaction effects (so-called intersectional effects) when using multiple dimensions of social location to build intersectional strata [21,24,25]. Evaluations of this method regarding complementarity with intersectional theory as well as statistical properties have been published recently [24][25][26][27][28].…”