Individual-level probabilities and cluster-level proportions: Toward interpretable level 2 estimates in unconflated multilevel models for binary outcomes.

Abstract

Multilevel models allow researchers to test hypotheses at multiple levels of analysis-for example, assessing the effects of both individual-level and school-level predictors on a target outcome. To assess these effects with the greatest clarity, researchers are well-advised to cluster mean center all Level 1 predictors and explicitly incorporate the cluster means into the model at Level 2. When an outcome of interest is continuous, this unconflated model specification serves both to increase model accuracy, by separating the level-specific effects of each predictor, and to increase model interpretability, by reframing the random intercepts as unadjusted cluster means. When an outcome of interest is binary or ordinal, however, only the first of these benefits is fully realized: In these models, the intuitive cluster mean interpretations of Level 2 effects are only available on the metric of the linear predictor (e.g., the logit) or, equivalently, the latent response propensity, y_ij∗. Because the calculations for obtaining predicted probabilities, odds, and ORs operate on the entire combined model equation, the interpretations of these quantities are inextricably tied to individual-level, rather than cluster-level, outcomes. This is unfortunate, given that the probability and odds metrics are often of greatest interest to researchers in practice. To address this issue, I propose a novel rescaling method designed to calculate cluster average success proportions, odds, and ORs in two-level binary and ordinal logistic and probit models. I apply the approach to a real data example and provide supplemental R functions to help users implement the method easily. (PsycInfo Database Record (c) 2024 APA, all rights reserved).