Abstract
The primary goal of this article is to demonstrate the close relationship between 2 classes of dynamic models in psychological research: latent change score models and continuous time models. The secondary goal is to point out some differences. We begin with a brief review of both approaches, before demonstrating how the 2 methods are mathematically and conceptually related. It will be shown that most commonly used latent change score models are related to continuous time models by the difference equation approximation to the differential equation. One way in which the 2 approaches differ is the treatment of time. Whereas there are theoretical and practical restrictions regarding observation time points and intervals in latent change score models, no such limitations exist in continuous time models. We illustrate our arguments with three simulated data sets using a univariate and bivariate model with equal and unequal time intervals. As a by-product of this comparison, we discuss the use of phantom and definition variables to account for varying time intervals in latent change score models. We end with a reanalysis of the Bradway–McArdle longitudinal study on intellectual abilities (used before by McArdle & Hamagami, 2004) by means of the proportional change score model and the dual change score model in discrete and continuous time.
Highlights
The primary goal of this article is to demonstrate the close relationship between 2 classes of dynamic models in psychological research: latent change score models and continuous time models
This first example demonstrates that latent change score (LCS) models can be reformulated as ARCL models and both yield identical results
For j,i = 1 the model fit (–2LogL) and discrete time parameters of the LCS, the reformulated LCS, and the continuous time (CT) model are identical. When it comes to identifying the CT parameters used to generate the data (a = –0.4 and q = 1), with estimates of a = –0.41 and q = 1.00, not surprisingly the CT model gets closer to the true parameters than the LCS models (a∗ = –0.34 and q∗ = 0.68)
Summary
The primary goal of this article is to demonstrate the close relationship between 2 classes of dynamic models in psychological research: latent change score models and continuous time models. This, for example, is the case in latent growth curve models (Bollen & Curran, 2006; Duncan, Duncan, & Strycker, 2006) or multilevel/mixed-effects models (Hox, 2010; Pinheiro & Bates, 2000; Singer & Willett, 2003) Common examples include autoregressive, crosslagged, or latent change score (LCS) models (e.g., Du Toit & Browne, 2001; Jöreskog, 1970; Jöreskog & Sörbom, 1977; McArdle, 2001, 2009; McArdle & Hamagami, 2004) In these models, time is considered implicitly by the order of the measurement occasions, but is not explicitly used as a predictor. Even though growth curve models remain very popular, with the development and advancement of (dual) change score models during the last 10 years, researchers are increasingly considering dynamic modeling approaches as well
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