Abstract: Adequacy of Semiparametric Approximations for Growth Models with Nonnormal Random Effects

Abstract

Conventional growth models assume that the random effects describing individual trajectories are conditionally normal. In practice, this assumption may often be unrealistic. As an alternative, Nagin (2005) suggested a semiparametric group-based approach (SPGA) which approximates an unknown, continuous distribution of individual trajectories with a mixture of group trajectories.

Prior simulations (Brame, Nagin, & Wasserman, 2006; Nagin, 2005) indicated that SPGA could generate nearly-unbiased estimates of means and variances of a nonnormal distribution of individual trajectories, as functions of group-trajectory estimates. However, these studies used few random effects—usually only a random intercept. Based on the analytical relationship between SPGA and adaptive quadrature, we hypothesized that SPGA's ability to approximate (a) random effect variances/covariances and (b) effects of time-invariant predictors of growth should deteriorate as the dimensionality of the random effects distribution increases. We expected this problem to be mitigated by correlations among the random effects (highly correlated random effects functioning as fewer dimensions) and sample size (larger N supporting more groups).

We tested these hypotheses via simulation, varying the number of random effects (1, 2, or 3), correlation among the random effects (0 or .6), and N (250, 500). Results indicated that, as the number of random effects increased, SPGA approximations remained acceptable for fixed effects, but became increasingly negatively biased for random effect variances. Whereas correlated random effects and larger N reduced this underestimation, correlated random effects sometimes distorted recovery of predictor effects. To illustrate this underestimation, Figure 1 depicts SPGA's approximation of the intercept variance from a three correlated random effect generating model (N < eqid1 > 500). These results suggest SPGA approximations are inadequate for the nonnormal, high-dimensional distributions of individual trajectories often seen in practice.

FIGURE 1 SPGA-approximated intercept variance from a three correlated random effect generating model. Notes. The dashed horizontal lines denote + 10% bias. The solid horizontal line denotes the population-generating parameter value; * denotes the best-BIC selected number of groups. The vertical bars denote 90% confidence intervals.