Using SAS PROC CALIS to fit Level-1 error covariance structures of latent growth models

Abstract

In the present article, we demonstrates the use of SAS PROC CALIS to fit various types of Level-1 error covariance structures of latent growth models (LGM). Advantages of the SEM approach, on which PROC CALIS is based, include the capabilities of modeling the change over time for latent constructs, measured by multiple indicators; embedding LGM into a larger latent variable model; incorporating measurement models for latent predictors; and better assessing model fit and the flexibility in specifying error covariance structures. The strength of PROC CALIS is always accompanied with technical coding work, which needs to be specifically addressed. We provide a tutorial on the SAS syntax for modeling the growth of a manifest variable and the growth of a latent construct, focusing the documentation on the specification of Level-1 error covariance structures. Illustrations are conducted with the data generated from two given latent growth models. The coding provided is helpful when the growth model has been well determined and the Level-1 error covariance structure is to be identified.Keywords Error covariance structure . Latent growth model . Structural equation modeling The latent growth model (LGM) plays an important role in repeated measure analysis over a limited number of occasions in large samples (e.g., Meredith & Tisak, 1990;Muthén & Khoo, 1998; Preacher, Wichman, MacCallum, & Briggs, 2008, p. 12). The model can not only characterize intraindividual (within subjects) change over time, but also examine interindividual (between subjects) difference by means of random growth coefficients, and is a typical application of hierarchical linear modeling (HLM). The within-subjects errors over time and the between-subjects errors are conventionally referred to as "Level-1" and "Level-2" errors, respectively. LGM can also be handled by using structural equation modeling (SEM) (e.g., Bauer, 2003;Bollen & Curran, 2006;Chan, 1998;Curran, 2003;Duncan, Duncan, & Hops, 1996; Mehta & Neal 2005;Meredith & Tisak, 1990; Willet & Sayer, 1994). SEM and HLM stem from different statistical theory, and each has developed its own terminology and standard ways of framing research questions. However, there exists much overlap between the two methodologies under some circumstances. Typically, when a two-level data structure arises from the repeated observations of a variable over time for a set of individuals (so that time is hierarchically nested within each individual), SEM is analytically equivalent

A systematic approach for identifying level-1 error covariance structures in latent growth modeling

“…Therefore, ARH(1) and AR(1), representing a non-stationary structure and a stationary one, respectively, were selected for Θ ∊ , coupled with UN for normalbold Ψ bold-italicζ bold-italicη , in the simulation studies. In each study, 1000 replications were generated for each of the sample sizes (denoted by N ) of 150, 300, and 500 (Ding & Jane, 2012; Ferron et al, 2002) from the population linear growth model. According to Kwok et al (2007), among the multiwave longitudinal studies published in Developmental Psychology in 2002, more than half (52%) of these studies collected data with three or four occasions.…”

“…Because the effects of Θ ∊ and normalbold Ψ bold-italicζ bold-italicη of different magnitudes are not of concern, single values were used for them. By following Ding and Jane (2012), the population parameter values in Study 1 were set as γ 00 = 10, γ 01 = 4, σ ζ italicη α 2 = 15, σ ζ italicη β 1 2 = 10, σ ζ italicη α ζ italicη β 1 = 7 , σ italic∊ 1 2 = 36 , σ italic∊ 2 2 = 25 , σ italic∊ 3 2 = 49 , σ italic∊ 4 2 = 64 , …”

“…Ding and Jane (2012) gave a brief review on this issue. A systematic approach to facilitate identifying level-1 error covariance structures seems still unavailable.…”

“…Since the correct covariance structure is difficult to specify by theory (Kwok et al, 2007, p. 588), the identification needs to rely on a specification search. Ding and Jane (2012) gave a brief review on this issue. A systematic approach to facilitate identifying level-1 error covariance structures seems still unavailable.…”

“…Here a more parsimonious covariance structure is defined as one having fewer parameters. Frequently seen error covariance structures in the LGM literature, summarized in Ding and Jane (2012), are considered in the proposed approach. Preliminary simulation results have indicated that the proposed approach performs well when the sample size is large enough.…”