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

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. Valid inference in longitudinal imaging requires enough flexibility of the covariance model to allow reasonable fidelity to the true pattern. On the other hand, the existence of computable estimates demands a parsimonious parameterization of the covariance structure. Separable (Kronecker product) covariance models provide one such parameterization in which the spatial and temporal covariances are modeled separately. However, evaluating the validity of this parameterization in high dimensions remains a challenge. Here we provide a scientifically informed approach to assessing the adequacy of separable (Kronecker product) covariance models when the number of observations is large relative to the number of independent sampling units (sample size). We address both the general case, in which unstructured matrices are considered for each covariance model, and the structured case, which assumes a particular structure for each model. For the structured case, we focus on the situation where the within-subject correlation is believed to decrease exponentially in time and space as is common in longitudinal imaging studies. However, the provided framework equally applies to all covariance patterns used within the more general multivariate repeated measures context. Our 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 illustrate the approaches appeal.

Keywords:

• Kronecker product
• separable covariance
• multivariate repeated measures
• likelihood ratio test
• spatio-temporal data
• linear exponent autoregressive model

Acknowledgements

The authors thank the editor and referees for their comments that considerably improved the paper. We especially thank one of the referees for the observation that the KP LEAR model can be reparameterized as an exponential model (or continuous-time AR(1) model) with a multiplicative nugget effect. An earlier version of this manuscript can be found at arxiv.org (Simpson et al., arXiv:1010.4471v2 [stat.AP]). Sean Simpson's support includes the Translational Scholar Award from the Translational Science Institute of Wake Forest School of Medicine and NIBIB K25 EB012236-01A1. Lloyd Edwards was supported by the National Center for Research Resources and the National Center for Advancing Translational Sciences, National Institutes of Health, through Grant Award Number UL1TR000083. Keith Muller's support includes NIDCR R01 DE020832, R01 DE020832-01A1S1, U54 DE019261, NICHD P01 HD065647, NCRR 3UL1RR029890-03S1, NHLBI R01 HL091005, NIAAA R01 AA016549, NIDA R01 DA031017, and NIDDK R01 DK072398.