Studying the strength of prediction using indirect mixture modeling: nonlinear latent regression with heteroskedastic residuals

We present a latent regression model in which the regression function is possibly nonlinear, and not necessarily smooth (e.g., a step function), and in which the residual variances are not necessarily homoskedastic. Heteroskedasticity is modeled by making the conditional (on the predictor) residual variance a (user-specified) function of the predictor. We use indirect mixture modeling to estimate the parameters by marginal maximum likelihood estimation, as proposed by Bock and Aitken (1981) in the context of item-response theory modeling and Klein and Moosbrugger (2000) in the context of structural equation modeling. We present a small simulation study to evaluate power and the consequences of model misspecification, and an illustration concerning neuroticism and extroversion. The model can be used to evaluate changes in the strength of the prediction as a function of the predictor.