Estimating a Piecewise Growth Curve Model With Longitudinal Data That Contains Individual Mobility Across Clusters

• In Event: Cross-Classified and Multiple Membership Random Effects Models: Demonstrations, Evaluations, and Extensions

Abstract

Theoretical Framework and Objectives

A piecewise growth curve model (PGCM) is beneficial for potentially nonlinear data, because it breaks up curvilinear growth trajectories into separate linear components. This modeling approach is useful, for example, when wanting to compare growth rates during two or more different time periods, with longitudinal data before treatment as well as during treatment, or with longitudinal data during treatment as well as follow-up data after treatment.

However to date, no study has proposed a PGCM to handle mobile students who change schools (clusters) during the period of data collection. The proposed cross-classified multiple membership PGCM (CCMM-PGCM) handles individual mobility across clusters frequently encountered in longitudinal studies (Grady & Beretvas, 2010; Leroux & Beretvas, 2015; Luo & Kwok, 2012).

Methods

ECLS-K data (Tourangeau, Nord, Lê, Sorongon, & Najarian, 2009) were used with time nested within students nested within schools and a multiple membership structure due to some students’ switching elementary schools across the course of data collection. Time-points included springs of kindergarten, 1st, 3rd, and 5th grade. Reading IRT-scaled scores were the outcome, and exploratory analyses suggested a two-piece GCM because growth rates from kindergarten through 3rd grade appeared faster than those from 3rd through 5th grade (see Table 1). Gender (coded 1 for female and 0 for male) and school type (coded 1 for public and 0 for private) were used as the level-2 and level-3 predictors, respectively. The sample included 10,914 students (26.9% were mobile) from 1,036 schools with no missing school identifiers nor predictors.

The unconditional and conditional versions of the three-level PGCM ignoring mobility and the CCMM-PGCM were estimated. Two Time variables coded with and were used to capture the piecewise growth. For the CCMM-PGCM, the intercept was modeled as varying across first schools while the slopes were modeled as varying across both first and subsequent schools (see Grady & Beretvas, 2010). The weights were based on the proportion of time-points a student was associated with a school for each slope (see Table 2). Models were fit using R with MCMC estimation using R2jags. Noninformative normal priors were used for fixed effects parameters and inverse-Wishart distributions for the covariance matrices, with a burn-in period of 5,000 iterations and an additional 25,000 iterations.

Results

The two models were compared using both a baseline unconditional and conditional model. Differences were found for estimates of the level-3 predictor as well as the between-schools variance component estimates (see Tables 3 through 6). In addition, the CCMM-PGCM fit to the data better based on the deviance information criterion value (Spiegelhalter, Best, Carlin, & van der Linde, 2002).

Scholarly Significance

Differences were found in the cluster-level predictors’ effects on the two slopes and their SEs as well as in cluster-level variances. Researchers using a PGCM ignoring multiple membership data should be careful when interpreting cluster-level predictors’ effects on slopes, and cluster-level slope variances may be inaccurate under the three-level PGCM. Further guidelines and suggestions for future research will be provided along with many details missing due to word restrictions.

Author

• Audrey J. Leroux, Georgia State University