Effect size measures for longitudinal growth analyses: Extending a framework of multilevel model R-squareds to accommodate heteroscedasticity, autocorrelation, nonlinearity, and alternative centering strategies.

Abstract

Developmental researchers commonly utilize multilevel models (MLMs) to describe and predict individual differences in change over time. In such growth model applications, researchers have been widely encouraged to supplement reporting of statistical significance with measures of effect size, such as R-squareds (R² ) that convey variance explained by terms in the model. An integrative framework for computing R-squareds in MLMs with random intercepts and/or slopes was recently introduced by Rights and Sterba and it subsumed pre-existing MLM R-squareds as special cases. However, this work focused on cross-sectional applications, and hence did not address how the computation and interpretation of MLM R-squareds are affected by modeling considerations typically arising in longitudinal settings: (a) alternative centering choices for time (e.g., centering-at-a-constant vs. person-mean-centering), (b) nonlinear effects of predictors such as time, (c) heteroscedastic level-1 errors and/or (d) autocorrelated level-1 errors. This paper addresses these gaps by extending the Rights and Sterba R-squared framework to longitudinal contexts. We: (a) provide a full framework of total and level-specific R-squared measures for MLMs that utilize any type of centering, and contrast these with Rights and Sterba's measures assuming cluster-mean-centering, (b) explain and derive which measures are applicable for MLMs with nonlinear terms, and extend the R-squared computation to accommodate (c) heteroscedastic and/or (d) autocorrelated errors. Additionally, we show how to use differences in R-squared (ΔR² ) measures between growth models (adding, for instance, time-varying covariates as level-1 predictors or time-invariant covariates as level-2 predictors) to obtain effects sizes for individual terms. We provide R software (r2MLMlong) and a running pedagogical example analyzing growth in adolescent self-efficacy to illustrate these methodological developments. With these developments, researchers will have greater ability to consider effect size when analyzing and predicting change using MLMs.