A multilevel structured latent curve model for disaggregating student and school contributions to learning

Educational researchers continue to debate the relative contribution of individual and environmental factors to learning. Concomitant with the proliferation of longitudinal educational testing following students and schools over time, recent research has shown that nonlinear mixed effect models can be parameterized to directly estimate quantities meaningful to learning processes and are situated to address questions about whether learning is driven by the individuals or the context. However, three-level nonlinear models pose estimation challenges because the likelihood does not have a closed-form solution and integral approximations are intractable when there are multiple random effects at multiple levels of the model. Multivariate reparameterization to a structured latent curve model has been suggested as a method to circumvent similar issues in two-level models, but the approach has not yet to be extended to the context of three-level models. We extend the idea of structured latent curve models to accommodate data with a three-level hierarchy. We apply the model to six years of mathematics and reading scores from 6346 students in 68 schools to partition the variance of learning parameters into school- and student-level components. The results show that—compared to reading—learning in mathematics is more heavily influenced by school-level factors and that there is evidence for stronger Matthew effects (“the rich get richer”) in mathematics than in reading.