# Separable Factor Analysis with Applications to Mortality Data

## Bailey Kathryn Fosdick and Peter D. Hoff

November 2012 CSSS Working Paper #124

### Abstract

Human mortality datasets can be expressed as multiway data arrays, the dimensions of which correspond to categories by which mortality rates are reported, such as age, sex, country and year. Regression models for such data typically assume an independent error distribution, or an error model that allows for dependence along at most one or two dimensions of the data array. However, failing to account for other dependencies can lead to inefficient estimates of regression parameters, inaccurate standard errors and poor predictions. An alternative to assuming independent errors is to allow for dependence along each dimension of the array using a separable covariance model. However, the number of parameters in this model increases rapidly with the dimensions of the array, and for many arrays, maximum likelihood estimates of the covariance parameters do not exist. In this paper, we propose a submodel of the separable covariance model that estimates the covariance matrix for each dimension as having factor analytic structure. 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 dimension of the array. We discuss properties of this model as they relate to ordinary factor analysis, describe maximum likelihood and Bayesian estimation methods, and provide a likelihood ratio testing procedure for selecting the factor model ranks. We apply this methodology to the analysis of data from the Human Mortality Database, and show in a cross-validation experiment how it outperforms simpler methods. Additionally, we use this model to impute mortality rates for countries that have no mortality data for several years. Unlike other approaches, our methodology is able to estimate similarities between the mortality rates of countries, time periods, and sexes and use this information to assist with the imputations.

Keywords: array normal, Kronecker product, multiway data, Bayesian estimation, imputation