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 illustrates the approaches appeal.

Selecting a separable parametric spatiotemporal covariance structure for longitudinal imaging data

“…It is an extremely useful assumption that we feel is reasonable, but still one that must be checked. Simpson et al [26] recently proposed a likelihood ratio test for whether a separable parametric structure should be rejected when compared to a fully unstructured covariance matrix. Such an unstructured matrix was estimated by looking at subsets of temporal and spatial observations to fill out the area around the main diagonal, with the assertion that correlation quickly decays to zero with distant observations.…”

“…Using matrix algebra, one can create a full correlation matrix from two independent sources of correlation by taking the direct/Kronecker product (denoted ⊗) of the two correlation matrices [35, 36]. Separable covariance structures for repeated measures imaging data have been used previously by Simpson et al [25], who have also investigated a test for separability [26]. …”

“…Here, five temporal observations made fitting an unstructured model simple while the 16 spatial locations meant that a spatial unstructured model could not be estimated. The values in Figure 8 were calculated by taking pairs of segments from every time point and calculating the correlation with an unstructured model in space and time; this strategy was inspired by the method used in Simpson et al [26]. …”

Score test for a separable covariance structure with the first component as compound symmetric correlation matrix

“…So, even if the methods are available for modeling data using separable structure when n ≤ pq, the testing is not, which is the limiting factor of any statistical analysis for two-level data. However, Simpson et al [33] recently provide a method in the small sample sized separability test where n < pq, however their method only covers a large fraction of the covariance space, and is especially useful in situations with most of the information contained along diagonal blocks (say c diagonal blocks). They have conducted c marginal LRTs using subsets of the data corresponding to c diagonal blocks and then applied a false discovery rate correction to control the multiple testings.…”

Tests for separability in nonparametric covariance operators of random surfaces

“…These methods, however, rely on the estimation of the full multidimensional covariance structure, which can be troublesome. It is sometimes possible to circumvent this problem by considering a parametric model for the full covariance structure (Simpson 2010, Simpson et al 2014, Liu et al 2014). On the contrary, when the covariance is being non-parametrically specified, as will be the case in this paper, estimation of the full covariance is at best computationally complex with large estimation errors, and in many cases simply computationally infeasible.…”