Factor analysis has been one of the most powerful and flexible tools for assessment of multivariate dependence and codependence. Loosely speaking, it could be argued that the origin of its success rests in its very exploratory nature, where various kinds of data-relationships amongst the variables at study can be iteratively verified and/or refuted. Bayesian inference in factor analytic models has received renewed attention in recent years, partly due to computational advances but also partly to applied focuses generating factor structures as exemplified by recent work in financial time series modeling. The focus of our current work is on exploring questions of uncertainty about the number of latent factors in a multivariate factor model, combined with methodological and computational issues of model specification and model fitting. We explore reversible jump MCMC methods that build on sets of parallel Gibbs sampling-based analyses to generate suitable empirical proposal distributions and that address the challenging problem of finding e#cient proposals in high-dimensional models. Alternative MCMC methods based on bridge sampling are discussed, and these fully Bayesian MCMC approaches are compared with a collection of popular model selection methods in empirical studies.

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, e.g., output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we illustrate on a simple errors-in-variables model.

Abstract not found

We propose a new method for analyzing factor analysis models using a Bayesian approach. Normal theory is used for the sampling distribution, and we adopt a model with a full disturbance covariance matrix. Using vague and natural conjugate priors for the parameters, we find that the marginal posterior distribution of the factor scores is approximately a matrix T-distribution, in large samples. This explicit result permits simple interval estimation and hypothesis testing of the factor scores. Explicit point and interval estimators of the factor score elements, in large samples, are obtained as means as means of the respective marginal posterior distributions. Factor loadings are estimated as joint modes (with factor scores), or alternatively as means or modes of the distribution of the factor loadings conditional upon the estimated factor scores. Disturbance variances and covariances are estimated conditional upon the estimated factor scores and factor loadings. This revision includes the correction of some typographical errors and some revised computations, plus an appendix that provides some intermediate results. 1

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Structural Equation Modeling, 15:705–728, 2008 Copyright © Taylor & Francis Group, LLC ISSN: 1070-5511 print/1532-8007 online

Structural equation modeling (SEM) has advanced considerably in the social sciences. The direction of advances has varied by the substantive problems faced by individual disciplines. For example, path analysis developed to model inheritance in population genetics, and later to model status attainment in sociology. Factor analysis developed in psychology to explore the structure of intelligence, and simultaneous equation models developed in economics to examine supply and demand. These largely discipline-specific advances came together in the early 1970s to create a multidisciplinary approach to SEM. Later, during the 1980s, responding to criticisms of SEM for failing to meet assumptions implied by maximum likelihood estimation and testing, SEM proponents responded with estimators for data that departed from multivariate normality, and for modeling categorical, ordinal, and limited dependent variables. More recently, advances in SEM have incorporated additional statistical models (growth models, latent class growth models, generalized linear models, and multi-level models), drawn upon artificial intelligence research to attempt to “discover ” causal structures, and finally, returned to the question of causality with formal methods for specifying assumptions necessary for inferring causality with nonexperimental