Understanding Individual-level Change Through the Basis Functions of a Latent Curve Model

Abstract

Latent curve models have become a popular approach to the analysis of longitudinal data. At the individual level, the model expresses an individual's response as a linear combination of what are called ''basis functions'' that are common to all members of a population and weights that may vary among individuals. This article uses differential calculus to define the basis functions of a latent curve model. This provides a meaningful interpretation of the unique and dynamic impact of each basis function on the individual-level response. Examples are provided to illustrate this sensitivity, as well as the sensitivity of the basis functions, to changes in the measure of time.

Differentiating between mixed-effects and latent-curve approaches to growth modeling

“…The estimation and specification of nonlinear models can be vastly different between the ME and LC frameworks, with many different types of models being uniquely estimable in only one framework. We will not be able to fully cover all of the nuances of nonlinear models in a single section, since many full-length articles have been dedicated solely to this topic (e.g., Blozis & Harring, 2016b); however, we will attempt to highlight the most salient of the differences that are likely to arise in empirical research.…”

“…ME software is able to accommodate either of these specifications without changing the interpretation of the model parameters (e.g., SAS Proc Mixed for models nonlinear in their variables, SAS Proc NLMIXED for models nonlinear in their parameters; Blozis & Harring, 2016b). SEM software can accommodate models that are nonlinear in their variables but cannot directly accommodate models that are nonlinear in their parameters (Blozis & Harring, 2016b). A common method to fit models that are nonlinear in their parameters using SEM software is through a structured latent-curve model (SLCM; Blozis, 2004;M.…”

“…W. Browne, 1993), which linearizes the nonlinear portions of the model using Taylor series expansions (for details on this procedure, see Blozis & Harring, 2016a). Although many consider the SLCM to be the LC equivalent of nonlinear ME models, as detailed by Blozis and Harring (2016b), the linearization process changes the interpretation of the model and of the random effects (the interpretation of the factor means takes a population-averaged rather than a subject-specific interpretation, which are not equivalent in nonlinear models; Fitzmaurice, Laird, & Ware, 2004;Zeger et al, 1988). In SLCMs, subject-specific curves are not required to follow the same functional form as the mean curve-the only restriction is that the sum of the subject-specific curves must equal the mean trajectory.…”

“…Although it is common to consider these models interchangeable, the variance estimates are noticeably different even though the quality of the data is rather high (e.g., large sample, normally distributed outcomes based on IRT scores, no missing data). The difference stems from the ME model retaining the subject-specific interpretation of the parameters, whereas the linearization process in the SLCM has a marginal interpretation (Blozis & Harring, 2016b). As we noted previously, the change in the random effects and subjectspecific curves is stark, which can be seen by the vastly different variance-covariance parameters for the three random effects.…”

The (Ir)Responsibility of (Under)Estimating Missing Data

“…Although these designs can cover various objectives, often the interest is in testing the efficacy of a treatment and assessing the evolution of the behavior experienced over a period of time. Sometimes it is intended only to make population inferences, but sometimes we are interested in examining the individual behavior from the responses of each subject (growth curve analysis, see Blozis and Harring, 2015 ; Blozis, 2016 ). The most sensible thing to respond to both types of hypothesis is to analyze the data using the mixed linear model (MLM), or, if we also have latent variables, using Structural Equation Modeling (SEM).…”