Catching Up on Multilevel Modeling

Abstract

This review focuses on the use of multilevel models in psychology and other social sciences. We target readers who are catching up on current best practices and sources of controversy in the specification of multilevel models. We first describe common use cases for clustered, longitudinal, and cross-classified designs, as well as their combinations. Using examples from both clustered and longitudinal designs, we then address issues of centering for observed predictor variables: its use in creating interpretable fixed and random effects of predictors, its relationship to endogeneity problems (correlations between predictors and model error terms), and its translation into multivariate multilevel models (using latent-centering within multilevel structural equation models). Finally, we describe novel extensions—mixed-effects location–scale models—designed for predicting differential amounts of variability.

Satisfying singlehood as a function of age and cohort: Satisfaction with being single increases with age after midlife.

“…With grand-mean centered Level-1 variable (time-varying age) included in the model, slope of the Level-2 variable (person mean of age) is interpreted as the effect of person mean of age controlling for aging (contextual-level-2 effect; Hoffman & Walters, 2022). Note that the person mean of age would perfectly correlate with birth year (cohort) had all individuals started at the same time and contributed data at all time points (i.e., those who were born earlier would be older on average when completing the surveys).…”

Resuming social contact after months of contact restrictions: Social traits moderate associations between changes in social contact and well-being

“…All models were estimated using maximum likelihood with random intercepts. We included random slopes of the Level-1 predictors of interest in those instances where likelihood ratio tests indicated that the addition of the random slope significantly improved model fit (Hoffman & Walters, 2022). If this was the case, we report the results of the random slope model herein and of the fixed slope model in the Supplemental Material (Tables S2-S6), and vice versa.…”

Cross-level covariance approach to the disaggregation of between-person effect and within-person effect.

“…Also, because the latent residual contains the within-person variation, its coefficient γ t can be seen as a within-person effect at time t . The model in Figure 1A is called latent centering model (Asparouhov & Muthén, 2019; Hamaker & Muthén, 2020; Hoffman, 2019; Hoffman & Walters, 2022) as centering is performed with the latent intercept…”

“…Thus, if these variables are not properly addressed, for example, if a time-varying outcome is naively regressed on a time-varying predictor, the resulting regression coefficient would be an “uninterpretable blend” of the between-person and within-person effects of the predictor (Cronbach, 1976; Raudenbush & Bryk, 2002). It is also called conflated effect (Preacher et al, 2010; Rights et al, 2020) and smushed effect (Hoffman, 2015, 2019; Hoffman & Walters, 2022) in the multilevel modeling literature, and is mathematically a weighted average of the between-person and within-person effects (Raudenbush & Bryk, 2002).…”