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Human mortality data sets can be expressed as multiway data arrays, thedimensions of which correspond to categories by which mortality rates arereported, such as age, sex, country and year. Regression models for such datatypically assume an independent error distribution or an error model thatallows for dependence along at most one or two dimensions of the data array.However, failing to account for other dependencies can lead to inefficientestimates of regression parameters, inaccurate standard errors and poorpredictions. An alternative to assuming independent errors is to allow fordependence along each dimension of the array using a separable covariancemodel. However, the number of parameters in this model increases rapidly withthe dimensions of the array and, for many arrays, maximum likelihood estimatesof the covariance parameters do not exist. In this paper, we propose a submodelof the separable covariance model that estimates the covariance matrix for eachdimension as having factor analytic structure. This model can be viewed as anextension of factor analysis to array-valued data, as it uses a factor model toestimate the covariance along each dimension of the array. We discussproperties of this model as they relate to ordinary factor analysis, describemaximum likelihood and Bayesian estimation methods, and provide a likelihoodratio testing procedure for selecting the factor model ranks. We apply thismethodology to the analysis of data from the Human Mortality Database, and showin a cross-validation experiment how it outperforms simpler methods.Additionally, we use this model to impute mortality rates for countries thathave no mortality data for several years. Unlike other approaches, ourmethodology is able to estimate similarities between the mortality rates ofcountries, time periods and sexes, and use this information to assist with theimputations.

Separable factor analysis with applications to mortality data