A Heterogeneous Growth Curve Model for Nonnormal Data

Abstract

The heterogeneous growth curve model (HGM; Klein & Muthén, 2006) is a method for modeling heterogeneity of growth rates with a heteroscedastic residual structure for the slope factor. It has been developed as an extension of a conventional growth curve model and a complementary tool to growth curve mixture models. In this article, a robust version of the heterogeneous growth curve model (HGM-R) is presented that extends the original HGM with a mixture model to allow for an unbiased parameter estimation under the condition of nonnormal data. In two simulation studies, the performance of the method is examined under the condition of nonnormality and a misspecified heteroscedastic residual structure. The results of the simulation studies suggest an unbiased estimation of the heterogeneity by the HGM-R when sample size was large enough and a good approximation of the heteroscedastic residual structure even when the functional form of the heteroscedasticity was misspecified. The practical application of the approach is demonstrated for a data set from HIV-infected patients.

Notes

For a minimal growth model that only includes two growth factors, the HGM has two additional parameters compared to the LGM [see Equation (4)]. A minimal 2 class GMM needs at least 2 additional parameters (one class-specific model parameter and a mean for the latent-class variable), but typically a GMM involves more parameters.

The residual ζ_1i is uncorrelated with η_0i, w_i, because E[ζ_1i|η_0i, w_i] = (γ₀ + γ₁η_0i + γ₂w_i)E[ζ_2i|η_0i, w_i] + E[ζ_3i|η_0i, w_i] = 0 = E[ζ_1i] (see, e.g., Robinson, 1987; White, 1980 for regression models).

The simulation results did not depend on the number of repeated measures. A model with eight repeated measures showed only slight differences in comparison to the results reported here.

The reliability of the other indicator variables varied depending on the degree of heterogeneity.