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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.

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Objectives Although schools and neighborhoods influence health outcomes, little is known about their relative importance, or the influence of one context after accounting for the other. Our objective was to simultaneously examine the influence of each setting on levels of depressive symptoms among adolescents. Methods Analyzing cross-sectional data from the National Longitudinal Study of Adolescent Health (Add Health), we used cross-classified multilevel modeling (CCMM) to examine between-level variation (random effects) and individual-, school-, and neighborhood-level predictors of adolescent depressive symptoms (fixed effects). We also compared CCMM results to results from a multilevel model (MLM) where either school or neighborhood was ignored. Results In CCMMs examining each context simultaneously, the school-level random effect was statistically significant and more than three times the neighborhood-level random effect, even after accounting for individual-level characteristics. While individual-level indicators (e.g., race/ethnicity, gender, socioeconomic status) were significantly associated with depressive symptoms, neither school- nor neighborhood-level fixed effects were. CCMM results showed that the between-level variance in depressive symptoms was driven largely by the school (ICC=3.0%) and not by the neighborhood (ICC=0.8%), as suggested by the school- (ICC=3.6%) and neighborhood-only (ICC=3.2%) MLM. Conclusions Schools appear more salient than neighborhoods in explaining variation in depressive symptoms. However, the school-level demographic variables examined were not determinants of youth depression. Future work using CCMM is needed to better understand the relative effect of schools and neighborhoods on youth mental health. These findings also underscore the need for CCMM over MLM when youth are nested in more than one context.

Drawing on the traditions of environmental justice, intersectionality, and social determinants of health, and using data from the EPA's NATA 2014 estimates of cancer risk from air toxics, we demonstrate a novel quantitative approach to evaluate intersectional environmental health risks to communities: Eco-Intersectional Multilevel (EIM) modeling. Results from previous case studies were found to generalize to national-level patterns, with multiply marginalized tracts with a high percent of Black and Latinx residents, high percent femaleheaded households, lower educational attainment, and metro location experiencing the highest risk. Overall, environmental health inequalities in cancer risk from air toxics are: (1) experienced intersectionally at the community-level, (2) significant in magnitude, and (3) socially patterned across numerous intersecting axes of marginalization, including axes rarely evaluated such as gendered family structure. EIM provides an innovative approach that will enable explicit consideration of structural/institutional social processes in the social production of intersectional and geospatial inequalities.