A Second-Order Conditionally Linear Mixed Effects Model With Observed and Latent Variable Covariates

Abstract

A conditionally linear mixed effects model is an appropriate framework for investigating nonlinear change in a continuous latent variable that is repeatedly measured over time. The efficacy of the model is that it allows parameters that enter the specified nonlinear time-response function to be stochastic, whereas those parameters that enter in a nonlinear manner are common to all subjects. In this article we describe how a variant of the Michaelis–Menten (M–M) function can be fit within this modeling framework using Mplus 6.0. We demonstrate how observed and latent covariates can be incorporated to help explain individual differences in growth characteristics. Features of the model including an explication of key analytic decision points are illustrated using longitudinal reading data. To aid in making this class of models accessible, annotated Mplus code is provided.

Keywords:

• conditionally linear models
• latent variables
• nonlinear mixed effects models
• second-order latent growth models
• structured latent curve models

Notes

¹Frequently, this is accomplished by plotting random samples of individuals via a spaghetti plot and fitting candidate functions to individuals using nonlinear least squares estimation (see e.g., Cudeck & Harring, 2010).

²Two alternative candidate functions were an increasing decelerating exponential function and a Gompertz function (Browne, 1993). When fit to the empirical means, the residual sum of squares for the Michaelis–Menten model was smallest among the three alternatives and was parameterized to the satisfaction of the reading scientists.

³ Grimm and Ram (2009) demonstrated how to specify nonlinear latent growth curve models for observed variables using phantom variables. This was needed prior to the nonlinear constraints functionality in Mplus.

⁴These labels are completely arbitrary and could be set using any notation that was deemed acceptable.

⁵The standardized values are not shown in Table 4 but were obtained from the standardized output in Mplus.

⁶Computation of individual coefficients, β _i is based on the joint normal distribution of β _i and y _i (see Fitzmaurice, Laird, & Ware, 2011). Evaluated at the maximum likelihood estimates, the estimator has an empirical Bayes interpretation (see, e.g., du Toit & Cudeck, 2009, for more details).