Structural After Measurement Estimation of higher order and (non)linear in n-level Structural Equation Models

Abstract Information: Conventional applications and developments of MLSEM in education have largely focused simple (fully) nested two- or three-level structures such as students within classrooms/schools. However, more recent work has widely demonstrated how common many level or n-level structures are in education and has extended the MLSEM framework to address, for example, partially nested, cross-classified and many-level SEMs (e.g., Mehta & Petscher, 2016). Despite these developments, a notable practical limitation is how to consistently estimate model parameters when sample sizes are in the common but small-to-moderate range ( < 100 classes/schools) with multiple disparate nesting structures.
Consider a prototypical example of a professional development study that probes the theory of action for teacher development (Figure 1). Teachers are assigned to a control or professional development condition (treatment; T) designed to improve teaching (latent mediator; ) in ways that advance student achievement (latent outcome; ) proportional to teacher self-efficacy (latent moderator; η_W). A diagram of the (latent) moderated mediation model is outlined in Figure 1.
Estimation of the moderated indirect effect requires the connection of two disparate nesting structures—a teacher side that uses multiple lessons-nested-within-each-teacher and a student side that uses multiple students-nested-within-each-teacher. Conventional MLSEM applications cannot appropriately capture such disparate structures because their purview is restricted to models that contain the same hierarchical or cross-classified (sub-)structure across levels.
Recent advances in n-level SEM provide a theoretical bridge for designs with multiple disparate structures (e.g., Pritikin et al., 2017). However, stable estimation of such models requires 100 clusters/teachers (e.g., Kline, 2011) with 10-20 individuals each (e.g., Wolf et al., 2013) as a minimum. In this study, we build on recent developments of limited information estimators and introduce an alternative estimator for estimating small-to-moderate scale MLSEMs with multiple disparate nesting structures.
Despite the flexibility of multilevel structural equation modeling (MLSEM), a practical limitation many researchers encounter is how to effectively estimate model parameters and latent interactions with typical sample sizes when there are many levels of (potentially disparate) nesting. We develop a method-of-moments corrected maximum likelihood (ML) estimator for n-level SEMs that is well-suited to the types of small-to-moderate sample sizes typically seen in education research. We probe the consistency, variability and convergence of the estimator with small-to-moderate n-level samples. The estimator emerges as a practical alternative or complement to conventional ML because it often outperforms ML in small-to-moderate n-level samples in terms of convergence, bias, and variance. The proposed estimators are implemented in the [BLINDED] R package and are illustrated through an n-level teacher development example.

Relevance Statement: Evaluation literature has recognized that well-executed small/moderate scale empirical studies can offer critical contributions to theory and practice. Yet the vast majority of evaluation studies draw on sophisticated theories but only what is statistically considered to be a small to moderate sample size. The combination of small/moderate sample sizes coupled with sophisticated n-level SEMs poses challenges for estimation methods because stable and unbiased estimates typically demand a large sample-to-parameter ratio. The results of this study coupled with those of other studies suggest that the proposed SAM estimator serves as an important alternative/complement when samples contain fewer than about 100 clusters/organizations/schools (depending on model complexity).