Distance-based analysis of variance for brain connectivity

Abstract

The field of neuroimaging dedicated to mapping connections in the brain is increasingly being recognized as key for understanding neurodevelopment and pathology. Networks of these connections are quantitatively represented using complex structures, including matrices, functions, and graphs, which require specialized statistical techniques for estimation and inference about developmental and disorder-related changes. Unfortunately, classical statistical testing procedures are not well suited to high-dimensional testing problems. In the context of global or regional tests for differences in neuroimaging data, traditional analysis of variance (ANOVA) is not directly applicable without first summarizing the data into univariate or low-dimensional features, a process that might mask the salient features of high-dimensional distributions. In this work, we consider a general framework for two-sample testing of complex structures by studying generalized within-group and between-group variances based on distances between complex and potentially high-dimensional observations. We derive an asymptotic approximation to the null distribution of the ANOVA test statistic, and conduct simulation studies with scalar and graph outcomes to study finite sample properties of the test. Finally, we apply our test to our motivating study of structural connectivity in autism spectrum disorder.

Keywords: biostatistics; distance statistics; kernel ANOVA; neuroimaging.

FIGURE 1

Observed data for two selected subjects, consisting of volume-normalized counts of streamline connections between each pair of regions. Regions are represented spatially in sagittal, coronal, and axial views as red dots, and blue lines are connections. Darker and wider blue lines indicate stronger connections between regions

FIGURE 2

Mean networks (first and second rows) for subjects younger than the median age (12.2 years) compared to older subjects. Regions are represented as red dots, and connections are shown as blue lines. Thicker lines indicate stronger connections on average between regions, and the legend indicates the number of streamline connections estimated. In the third row, the differences between the maps are shown with blue lines indicating stronger connections in older subjects, and red lines indicating weaker connections in these groups

FIGURE 3

Q-Q plots comparing the proposed approximation to the empirical null distribution of the squared Euclidean (top) and absolute (bottom) distance-based ANOVA test statistic Q_n in the scalar case. ANOVA, analysis of variance; Q-Q, quantile-quantile

FIGURE 4

Graph-outcome simulation design. On the left, the adjacency matrix for the simulated graphs is shown, and on the right three example subjects are shown with τ₀ =5% for the first population and τ₁ =10% for the second

FIGURE 5

Figures showing the type I error rates and power for various settings with network outcomes in scenario 4. The top row shows type I error rates for several noise levels and power under several alternatives for the case of π₀ = 1/2, and the bottom rows show results for the case of imbalanced group sizes