Defining R-squared measures for mixed-effects location scale models

Abstract

Ecological momentary assessment and other modern data collection technologies facilitate research on both within-subject and between-subject variability of health outcomes and behaviors. For such intensively measured longitudinal data, Hedeker et al extended the usual two-level mixed-effects model to a two-level mixed-effects location scale (MELS) model to accommodate covariates' influence as well as random subject effects on both mean (location) and variability (scale) of the outcome. However, there is a lack of existing standardized effect size measures for the MELS model. To fill this gap, our study extends Rights and Sterba's framework of $R^2$ measures for multilevel models, which is based on model-implied variances, to MELS models. Our proposed framework applies to two different specifications of the random location effects, namely, through covariate-influenced random intercepts and through random intercepts combined with random slopes of observation-level covariates. We also provide an R function, R2MELS, that outputs summary tables and visualization for values of our $R^2$ measures. This framework is validated through a simulation study, and data from a health behaviors study and a depression study are used as examples to demonstrate this framework. These $R^2$ measures can help researchers provide greater interpretation of their findings using MELS models.

Keywords: EMA; R-squared; mixed-effects location scale model; standardized effect size.