Bayesian Hierarchical Multivariate Formulation with Factor Analysis for Nested Ordinal Data

This article devises a Bayesian multivariate formulation for analysis of ordinal data that records teacher classroom performance along multiple dimensions to assess aspects characterizing good instruction. Study designs for scoring teachers seek to measure instructional performance over multiple classroom measurement event sessions at varied occasions using disjoint intervals within each session and employment of multiple ratings on intervals scored by different raters; a design which instantiates a nesting structure with each level contributing a source of variation in recorded scores. We generally possess little a priori knowledge of the existence or form of a sparse generating structure for the multivariate dimensions at any level in the nesting that would permit collapsing over dimensions as is done under univariate modeling. Our approach composes a Bayesian data augmentation scheme that introduces a latent continuous multivariate response linked to the observed ordinal scores with the latent response mean constructed as an additive multivariate decomposition of nested level means that permits the extraction of de-noised continuous teacher-level scores and the associated correlation matrix. A semi-parametric extension facilitates inference for teacher-level dependence among the dimensions of classroom performance under multi-modality induced by sub-groupings of rater perspectives. We next replace an inverse Wishart prior specified for the teacher covariance matrix over dimensions of instruction with a factor analytic structure to allow the simultaneous assessment of an underlying sparse generating structure. Our formulation for Bayesian factor analysis employs parameter expansion with an accompanying post-processing sign re-labeling step of factor loadings that together reduce posterior correlations among sampled parameters to improve parameter mixing in our Markov chain Monte Carlo (MCMC) scheme. We evaluate the performance of our formulation on simulated data and make an application for the assessment of the teacher covariance structure with a dataset derived from a study of middle and high school algebra teachers.