Estimating and Visualizing Nonlinear Relations Among Latent Variables: A Semiparametric Approach

Abstract

The graphical presentation of any scientific finding enhances its description, interpretation, and evaluation. Research involving latent variables is no exception, especially when potential nonlinear effects are suspect. This article has multiple aims. First, it provides a nontechnical overview of a semiparametric approach to modeling nonlinear relationships among latent variables using mixtures of linear structural equations. Second, it provides several examples showing how the method works and how it is implemented and interpreted in practical applications. In particular, this article examines the potentially nonlinear relationships between positive and negative affect and cognitive processing. Third, a recommended display format for illustrating latent bivariate relationships is demonstrated. Finally, the article describes an R package and an online utility that generate these displays automatically.

Notes

¹Note that this use of the SEMM involves no assumption that the component distributions, often referred to as latent classes, reflect true groups within the population. The mixture is estimated only as a statistical expedience to obtain an estimate of the global regression function. This type of mixture application has been referred to as “indirect” by Titterington, Smith, and Makov (1985), in contrast to “direct” applications aimed at modeling population heterogeneity.

²A second analogy may be drawn with nonparametric regression splines for observed data where a global curve is obtained by joining knot points together with a piecewise polynomial function (Smith, 1979; Wold, 1974). The number of knot points functions similar to the number of components in the SEMM approach. Spline functions, however, partition the data between user-defined knots into nonoverlapping sets to fit pieces of the function, whereas in SEMM the components overlap and their locations (means) are estimated rather than fixed. The overlap permits smoothing from one within-component linear function to another, unlike, for instance, a piecewise linear model.

³The assumption of strict measurement invariance over classes is consistent with the standard single-group SEM, which assumes strict measurement invariance for all individuals. Because the classes are not believed to represent distinct groups of individuals in this type of SEMM application, this assumption is retained for parsimony.

⁴Although there are no other obvious ways for Equation 5 to reduce to a strictly linear function, we have observed in practice that the aggregate function implied by Equation 5 can sometimes appear roughly linear even when conditions A and B are rejected. For instance, one component may be so small that the aggregate function is minimally influenced by the presence of this component.