  Longitudinal imaging studies have moved to the forefront of medical research
due to their ability to characterize spatio-temporal features of biological
structures across the lifespan. Credible models of the correlations in
longitudinal imaging require two or more pattern components. Valid inference
requires enough flexibility of the correlation model to allow reasonable
fidelity to the true pattern. On the other hand, the existence of computable
estimates demands a parsimonious parameterization of the correlation structure.
For many one-dimensional spatial or temporal arrays, the linear exponent
autoregressive (LEAR) correlation structure meets these two opposing goals in
one model. The LEAR structure is a flexible two-parameter correlation model
that applies in situations in which the within-subject correlation decreases
exponentially in time or space. It allows for an attenuation or acceleration of
the exponential decay rate imposed by the commonly used continuous-time AR(1)
structure. Here we propose the Kronecker product LEAR correlation structure for
multivariate repeated measures data in which the correlation between
measurements for a given subject is induced by two factors. We also provide a
scientifically informed approach to assessing the adequacy of a Kronecker
product LEAR model and a general unstructured Kronecker product model. The
approach provides useful guidance for high dimension, low sample size data that
preclude using standard likelihood based tests. Longitudinal medical imaging
data of caudate morphology in schizophrenia illustrates the appeal of the
Kronecker product LEAR correlation structure.
