Small Sample Bayesian Factor Analysis

Factor analysis (FA) is a statistical technique to explain the correlation structure among observed variables. It can e.g. be useful in the analysis of Patient Reported Outcomes, which typically use multiple questions to measure a single concept. Maximum likelihood factor analysis (ML-FA) relies on large sample theory, and it is consequently often recommended to use it only in large samples (e.g. N=200 or more). In smaller samples, ML-FA can run into problems (e.g. model non-convergence, negative residual variances). Bayesian statistics typically perform better in small samples, and may therefore be useful in (clinical) studies that rely on smaller sample sizes. This paper provides a gentle introduction to Bayesian FA and shows how it can be implemented in Mplus 7.0. The paper then shows the relative performance of Bayesian FA as compared to ML-FA for two CFA models: a onefactor model with four factor indicators, and a two-factor model with 3 indicators for each factor. Several criteria are taken into account to evaluate the FA results in samples of sizes 200, 100, 50, and 25: problems with model fitting (model convergence, negative residual variances), and statistical power. The influence of priors in Bayesian FA on the parameter estimates is also investigated in terms of bias on factor loadings and factor correlations. The paper explains the results from the Monte Carlo studies in practical terms. It concludes by pointing to some novel capabilities in Bayesian FA such as estimation of near-zero cross-loadings or quasi zero residual covariances, which represent a more flexible approach than can typically be done in ML-FA.