Data that can be represented in the form of an array are present in many of the social and biological sciences. Examples include relational measurements over time or measurements of several variables over time and space. Regression models for such data often assume an independent error distribution, or an error model that allows for dependence along at most one or two dimensions of the data array. Failing to account for other dependencies can lead to incident estimates of regression parameters, inaccurate standard errors, and poor predictions. Previously developed models for estimating dependence within an array are non-stochastic and difficult to interpret, or require a large number of parameters prohibiting likelihood based inference for some arrays.

In this talk I will introduce a model called Separable Factor Analysis (SFA), which is parameterized by a factor-analytic structured covariance matrix for each dimension of the array. This model can be viewed as an extension of factor analysis to array-valued data, as it uses a factor model to estimate the covariance along each array dimension. I will discuss properties of this model as they relate to ordinary factor analysis, de-scribe maximum likelihood and Bayesian estimation methods, and propose a likelihood ratio testing procedure for selecting the factor model ranks. This methodology will be illustrated in the analysis of data from the Human Mortality Database and will be shown to outperform simpler covariance models in a cross-validation experiment.