Estimating Latent Variable Interactions With Nonnormal Observed Data: A Comparison of Four Approaches

Abstract

A Monte Carlo simulation was conducted to investigate the robustness of four latent variable interaction modeling approaches (Constrained Product Indicator [CPI], Generalized Appended Product Indicator [GAPI], Unconstrained Product Indicator [UPI], and Latent Moderated Structural Equations [LMS]) under high degrees of non-normality of the observed exogenous variables. Results showed that the CPI and LMS approaches yielded biased estimates of the interaction effect when the exogenous variables were highly non-normal. When the violation of non-normality was not severe (normal; symmetric with excess kurtosis < 1), the LMS approach yielded the most efficient estimates of the latent interaction effect with the highest statistical power. In highly non-normal conditions, the GAPI and UPI approaches with ML estimation yielded unbiased latent interaction effect estimates, with acceptable actual Type-I error rates for both the Wald and likelihood ratio tests of interaction effect at N ≥ 500. An empirical example illustrated the use of the four approaches in testing a latent variable interaction between academic self-efficacy and positive family role models in the prediction of academic performance.

Full Information Maximum Likelihood Estimation for Latent Variable Interactions With Incomplete Indicators

“…The first issue is the selection of product indicators of ξ 1 ξ 2 . Consider a widely studied situation where ξ 1 and ξ 2 have three indicators each ( X 1 , X 2 , X 3 and X 4 , X 5 , X 6 ) (e.g., Cham, West, Ma, & Aiken, 2012; Kelava et al, 2011; Marsh et al, 2004; Wall & Amemiya, 2001), so there are nine possible product indicators ( X 1 X 4 , X 1 X 5 , X 1 X 6 , X 2 X 4 , X 2 X 5 , X 2 X 6 , X 3 X 4 , X 3 X 5 , X 3 X 6 ). Simulation studies suggest creating three product indicators using the three most reliable indicators of ξ 1 and ξ 2 , respectively (Jackman, Leite, & Cochrane, 2011; Marsh et al, 2004; Wu, Wen, Marsh, & Hau, 2013).…”

“…The three variants of PI add model constraints related to ξ 1 ξ 2 , which provide more information to describe the distribution of ξ 1 ξ 2 (Aroian, 1944; Bohrnstedt & Goldberger, 1969; Jöreskog & Yang, 1996). Table 1 summarizes the constraints for each PI variant (see also Cham et al, 2012; Kelava et al, 2011). CPI has the most constraints, followed by GAPI and UPI.…”

Bayesian Factor Analysis as a Variable-Selection Problem: Alternative Priors and Consequences

“…Third, the spike and slab prior is only used in factor analysis models. In SEMs, selection of the structural model and SEMs with nonlinear and interaction effects of latent variables are also important issues (Cham, West, Ma, & Aiken, 2012; Kenny & Judd, 1984). In Muthén and Asparouhov (2012), the ridge prior was also used to select the correlation elements in Ψ , and also to identify substantial nonlinear interaction effects between latent variables in nonlinear SEMs.…”

A Heterogeneous Growth Curve Model for Nonnormal Data

“…As a consequence, most of the approaches for nonlinear structural equation modeling are not applicable. Approaches that are robust against a violation of distributional assumptions Cham, West, Ma, & Aiken, 2012;Marsh et al, 2004;Marsh, Wen, & Hau, 2006), for example, product indicator approaches or the 2SMM estimator by Amemiya (2000, 2003), cannot be specified for the HGM-R. For Bayesian approaches, the necessary prior knowledge about the parameters of the model is not always available (e.g., Kelava & Nagengast, 2012;Song, Li, Cai, & Ip, 2013). Particularly for the HGM-R, there is little information available about the effect size of a heteroscedastic variance component: the semi-parametric information about the heterogeneity that may be retrieved from GMMs do not provide insight in the actual effect size.…”