Flexible Bias-Corrected Multilevel Factor Score Path Analysis

• In Event: Composite/Factor Scores Analyses With Bias Correction: Good Alternatives to Full Structural Equation Modeling?

Abstract

The purpose of this study was to derive and assess multilevel extensions to Croon’s method of moments corrected maximum likelihood estimator. Full information maximum likelihood estimation of multilevel structural equation models (MLSEMs) is a preferred approach to operationalize sophisticated theories involving multiple latent variables in clustered or multilevel education settings. Although full information estimators have many desirable properties including consistency, a major limitation is that they often incur significant bias for structural parameters when implemented in studies with a small to moderate number of groups (e.g., 100 or fewer schools). More generally, prior research has shown that the quality of full information estimates of parameters in MLSEMs can be sensitive to the balance between model complexity and sample size because its core properties (e.g., consistency) lean heavily on large sample theory that may not apply to complex models in finite samples (e.g., Li & Beretvas, 2013). For instance, samples of 80 to 100 schools are often considered a minimum in order to produce stable unbiased estimates with moderately complex models while samples of less than 80 schools are often classified as precarious (e.g., Kline, 2011; Li & Beretvas, 2013). More carefully, sample size recommendations in past research have often indicated that at least 10 to 20 cases per parameter will be needed to provide a minimal basis for unbiased estimation and inference and even these guidelines can be contingent upon, for example, the strength of the measurement models (e.g., Wolf, Harrington, Clark & Miller, 2013). Yet, many studies in education research draw on sophisticated theories with samples of 100 or few schools (e.g., Spybrook, Shi, & Kelcey, 2016).

To address similar limitations within the context of single-level studies, recent literature has developed a two-stage limited information estimator that draws on a method of moments type correction that has been shown to yield unbiased estimates in single-level settings, even with small to moderate sample sizes. In this study, we derive extensions to this estimator for MLSEMs and probe the degree to which the estimator can produce unbiased estimates in small to moderate multilevel samples (e.g., 100 or fewer schools). We consider a wide range of MLSEMs including two- and three-level structures and designs with multiple disparate nesting structures.

The results suggest the promise of the estimator—the proposed estimator outperforms full information estimation in terms of bias, error variance, and power under a wide range of models when samples include 100 or fewer schools and converges with full information estimation in samples larger than 100 schools. The proposed estimators are implemented in R and illustrated through several different types of multilevel mediation examples.

Authors

• Benjamin Kelcey, University of Cincinnati

• Fangxing Bai, University of Cincinnati

• Kyle T. Cox, University of North Carolina - Charlotte