Developing a Bayesian Cross-Classified Growth Mixture Model to Account for Student Mobility

• In Event: Advancing Mixture Modeling Methodology: Model Specification, Class Enumeration, and Measurement Invariance for Complex Data Structures

Abstract

Growth mixture modeling (GMM) is an increasingly popular longitudinal data analytic approach to identify subpopulations of students that differ in their growth trajectories of academic outcomes. Among applications of GMM, and longitudinal data analysis in general, the prevalence of student mobility has been well documented (Rumberger, 2003; Welsh, 2017). Formally defined, student or school mobility refers to non-promotional school changes initiated by families and students or school and district officials; however often mobility is caused by forces outside of school control such as housing instability and economic insecurity (Rumberger, 2015). Despite the prevalence and significance of student mobility, the current methodological techniques of GMM do not allow researchers to account for student mobility (Bowers & Sprott, 2012; Lu, 2016; McCoach et al., 2006).
To address this methodological limitation, this study proposes cross-classified GMM (CC-GMM) to incorporate student mobility while identifying distinct academic trajectories among students. Specifically, CC-GMM models repeated measures as cross-classified by students and schools which are viewed as separate higher levels of analysis (rather than students nested within schools), thus allowing for changing school memberships over time (e.g., Cafri et al., 2015; Chen & Leroux, 2018; Grady & Beretvas, 2010; Luo & Kwok, 2012; Ortega et al., 2018). Random effects in growth parameters due to students and schools can be decomposed and estimated, and latent classes are estimated at the student level. Additionally, we propose a conditional version of CC-GMM which allows covariates to explain latent class membership.
We demonstrated CC-GMM using a subset of data from the Early Childhood Longitudinal Study, Kindergarten Class of 1998-1999 (Tourangeau et al., 2009). Five repeated measures of student math achievement were used, including fall and spring of kindergarten, fall and spring of first grade, and spring of third grade. A total of 404 students were included in the analysis, among which 20% of them changed schools over time. We estimated unconditional models with 1- to 4-classes using JAGS (Plummer, 2017) through the runjags package (Denwood, 2016) in R (R Core team, 2025). Diffuse priors were used for the Bayesian estimation with convergence assessed through the potential scale reduction factor (Gelman & Rubin, 1992). Model fit was compared using the deviance information criterion (Spiegelhalter et al., 2002), Watanabe-Akaike information criterion (Watanabe, 2010), and leave-one-out cross-validation (Vehtari et al., 2017). After the optimal model was selected, age, sex, and SES were entered as covariates into the conditional model.
Preliminary results showed that the three-class model had the optimal fit. Class 1 (43%) was characterized by a relatively low baseline and low growth rate; Class 2 (52%) had a moderate baseline and high growth rate; and Class 3 (5%) had a high baseline and moderate growth rate. Using Class 1 as the reference, females were less likely to be in Class 2 than males and higher SES was associated with a higher likelihood of being in Class 3. This study will provide educational researchers with a cutting-edge statistical tool to conduct research that accounts for student mobility. Additional implications will be further discussed.

Authors

• Audrey J. Leroux, Georgia Institute of Technology

• Tonghui Xu, University of Houston

• Yan Wang, University of Massachusetts Lowell