A Hierarchical Linear Model With Factor Analysis Structure at Level 2

Abstract

In this article the authors develop a model that employs a factor analysis structure at Level 2 of a two-level hierarchical linear model (HLM). The model (HLM2F) imposes a structure on a deficient rank Level 2 covariance matrix (τ), and facilitates estimation of a relatively large τ matrix. Maximum likelihood estimators are derived via the Fisher scoring algorithm. An application of HLM2F to early vocabulary growth data illustrates the utility of the methodology. The methodology shown in this article has several advantages over the current two-level HLM (HLM2). Those include: (a) a parsimonious model that facilitates interplay between the theory and empirical results, (b) estimation of a deficient rank Level 2 covariance matrix, (c) calculation of change in deviance to evaluate the HLM2F model compared to HLM2, (d) construction of the distribution of a set of random effects in terms of an underlying latent factor, and (e) more efficient estimates for the variance component parameters.