Kronecker Product Linear Exponent AR(1) Correlation Structures for Multivariate Repeated Measures

Abstract

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 to 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. 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 (e.g., spatial and temporal dependence). Excellent analytic and numerical properties make the Kronecker product LEAR model a valuable addition to the suite of parsimonious correlation structures for multivariate repeated measures data. Longitudinal medical imaging data of caudate morphology in schizophrenia illustrates the appeal of the Kronecker product LEAR correlation structure.

Separability tests for high-dimensional, low-sample size multivariate repeated measures data

“…As evidenced by information criteria and observed vs. predicted correlation plots, Simpson et al (2013) showed that the Kronecker product LEAR model appears to provide a good fit to the caudate morphology data. However, the validity of the separable assumption should be assessed as there may be space × time interactions which cannot be modeled with the Kronecker structure.…”

“…We consider the following structured likelihood ratio test of separability for the Kronecker product linear exponent autoregressive (KP LEAR) model which has been shown to work well for situations in which the within subject correlation is believed to decrease exponentially in time and space (Simpson et al, 2010, 2013). Suppose y i is a t i s i × 1 vector of t i s i observations (e.g., t i temporal measurements and s i spatial measurements) on the i th subject i ∈ {1, …, N }.…”

“…One could also consider tuning the constants if necessary to address, for example, convergence issues. Simpson et al (2010, 2013) contain further details of the model. Note that each factor specific LEAR model can also be reparameterized as an exponential model (or continuous-time AR(1) model) with a multiplicative nugget effect (see Schabenberger and Gotway, 2005 for details on nugget effects).…”

“…Our simulations were aimed to mimic the schizophrenia and caudate morphology data discussed in Simpson et al (2013) and in the next section. To do this, we again generated data under H 0 with ρ = [ ρ γ ρ ω ]′ = [0.8 0.8]′, δ = [ δ γ δ ω ]′ = [( d t ;max − d t ;min )/4 ( d s ;max − d s ;min )/4]′, and σ 2 = 1, and under H 1 based on Theorem 10.13 of Muller and Stewart (2006).…”

“…We model the schizophrenia and caudate morphology data discussed in the Introduction with an intercept-only general linear model for repeated measures (Simpson et al, 2013). That is, ${\mathit{y}}_i~N_{t_{i}s}false({\mathit{\mu }}_i={\mathbf{1}}_{t_{i}s}^{\prime }{\mathit{\beta }}_0,{\mathbf{\sum }}_{i}false)$, where the log 2 (radius) values for each of the s i = s = 21 locations (spatial factor) and t i images (temporal factor) for the i th subject ( i ∈ {1, …, 296}) are contained in y i ( t i · 21 × 1).…”