Latent variable interaction in structural equation models

This study investigated moderated regression analysis using latent variables. Interactive psychological theories, the stress-buffering hypothesis, and interaction in regression and structural equation models, were reviewed. Moderated regression was recast using latent variables to accommodate measurement error. Three methods of fitting the latent variable interaction (LVI) model were studied: (1) Factor Score Regression (FSR) treats factor score estimates for latent variables, and their product, as predictors. (2) Kenny and Judd's (1984) Product Indicators Approach (PIA) defines an interaction latent variable whose indicators are cross-products of indicators of the additive predictors. (3) McDonald's (1993) Nonlinear Factor Analysis (NOFA) treats the latent predictors as fixed factors, and estimates model parameters and factor scores simultaneously. FSR was shown to be scale invariant, but not origin invariant (indicators with zero and nonzero means yielded different solutions). PIA was neither scale nor origin invariant. NOFA was scale and origin invariant. The methods were compared in a simulation study, controlling for Sample Size (SS), Communality Ratio (CR) or reliability, and Interaction Coefficient (IC) magnitude. The performance ordering, in terms of parameter recovery, was NOFA, PIA, FSR. All methods tended to recover $\rm\beta\sb{43(int)}$, the interaction coefficient, better as SS and CR increased, but worse as IC increased. For FSR, SS interacted with IC (as SS increased, recovery improved only at low IC) and CR (recovery improved only at high CR). Also, IC and CR interacted (recovery declined as IC increased, with steeper decline as CR decreased). For PIA and NOFA, SS and IC interacted (recovery improved as SS increased, with stronger improvement as IC increased). R$\rm\sb{cha}\sp2$ was shown to be unsuitable for testing LVI. For FSR and NOFA, recovery declined as CR increased. PIA and NOFA yielded negative R$\rm\sb{cha}\sp2$ estimates. To demonstrate hypothesis testing, jackknifed standard errors were obtained for data from a study of stress-buffering. No FSR estimates were significant. It was shown that FSR under-estimated coefficients, and that PIA or NOFA might have demonstrated significant effects. However, PIA did not converge, and NOFA estimates converged to starting values, suggesting a problem with the NOFA software. Findings were summarized in terms of their relevance for applied researchers.U of I OnlyETDs are only available to UIUC Users without author permissio