  Human mortality data sets 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.
