Abstract: Comparing Semiparametric and Parametric Methods for Modeling Interactions Among Latent Variables

Many approaches have been proposed to estimate interactions among latent variables. These methods often assume a specific functional form for the interaction, such as a bilinear interaction. Theory is seldom specific enough to provide a functional form for an interaction, however, so a more exploratory, diagnostic approach may often be required. Bauer (2005) proposed a semiparametric approach that allows for the estimation of interaction effects of unknown functional form among latent variables. A structural equation mixture model (SEMM) is first fit to the data. Then an approximation of the interaction is obtained by aggregating over the mixing components. A simulation study is used to examine the performance of this semiparametric approach to two parametric approaches: the latent moderated structures approach (Klein & Moosbrugger, 2000) and the unconstrained product-indicator approach (Marsh, Wen, & Hau, 2004). Data were generated from four functional forms: main effects only, quadratic trend, bilinear interaction, and exponential interaction. Estimates of bias and root mean squared error of approximation were calculated by comparing the surface used to generate the data and the model-implied surface constructed from each approach. As expected, the parametric approaches were more efficient than the SEMM. For the main effects model, bias was similar for both the SEMM and parametric approaches. For the bilinear interaction, the parametric approaches provided nearly identical results, although the SEMM approach was slightly more biased. When the parametric approaches assumed a bilinear interaction and the data were generated from a quadratic trend or an exponential interaction, the parametric approaches generated biased estimates of the true surface. The SEMM approach approximated the true data generation surface with a similarly low level of bias for all the nonlinear surfaces. For example, Figure 1 shows the true surface for the bilinear interaction along with the SEMM estimated average surface. The results suggest that the SEMM approach can provide a relatively unbiased approximation to variety of nonlinear relationships among latent variables.