Visualizing Confidence Bands for Semiparametrically Estimated Nonlinear Relations Among Latent Variables

Abstract

Structural equation mixture models (SEMMs), when applied as a semiparametric model (SPM), can adequately recover potentially nonlinear latent relationships without their specification. This SPM is useful for exploratory analysis when the form of the latent regression is unknown. The purpose of this article is to help users familiar with structural equation models to add SEMM to their toolkit of exploratory analytic options. We describe how the SEMM captures potential nonlinearity between latent variables, and how confidence bands (CBs; point wise and simultaneous) for the recovered latent function are constructed and interpreted. We then illustrate the usefulness of CBs for inference with an empirical example on the effect of emotions on cognitive processing. We also introduce a visualization tool that automatically generates plots of the latent regression and their CBs to promote user accessibility. Finally, we conclude with a discussion on the use of this SPM for exploratory research.Structural equation models (SEMs) are widely used in the educational, social, and behavioral sciences, as many phenomena of interest such as intelligence, emotions, and personality are latent in nature. Latent variable models have the advantage of accounting for measurement error and have the ability to model

A Note on the Use of Mixture Models for Individual Prediction

“…By contrast, the goal of indirect applications is to estimate as many latent classes as necessary to adequately represent the range of individual differences, without concern for the existence or recovery of natural groups. The latent classes are then interpreted to reflect local conditions (Nagin, 2005; Bauer and Shanahan, 2007) or reaggregated to glean insights about the population as a whole (e.g., Kelava, Nagengast, and Brandt, 2014; Pek, Chalmers, Kok, and Losardo, 2015; Gottfredson, Bauer, Baldwin, and Okiishi, 2014). Thus, depending on the nature of the application, inferences may be drawn with respect to the characteristics of the latent classes, the total population, or both.…”