Highly adaptive tests for group differences in brain functional connectivity

Abstract

Resting-state functional magnetic resonance imaging (rs-fMRI) and other technologies have been offering evidence and insights showing that altered brain functional networks are associated with neurological illnesses such as Alzheimer's disease. Exploring brain networks of clinical populations compared to those of controls would be a key inquiry to reveal underlying neurological processes related to such illnesses. For such a purpose, group-level inference is a necessary first step in order to establish whether there are any genuinely disrupted brain subnetworks. Such an analysis is also challenging due to the high dimensionality of the parameters in a network model and high noise levels in neuroimaging data. We are still in the early stage of method development as highlighted by Varoquaux and Craddock (2013) that "there is currently no unique solution, but a spectrum of related methods and analytical strategies" to learn and compare brain connectivity. In practice the important issue of how to choose several critical parameters in estimating a network, such as what association measure to use and what is the sparsity of the estimated network, has not been carefully addressed, largely because the answers are unknown yet. For example, even though the choice of tuning parameters in model estimation has been extensively discussed in the literature, as to be shown here, an optimal choice of a parameter for network estimation may not be optimal in the current context of hypothesis testing. Arbitrarily choosing or mis-specifying such parameters may lead to extremely low-powered tests. Here we develop highly adaptive tests to detect group differences in brain connectivity while accounting for unknown optimal choices of some tuning parameters. The proposed tests combine statistical evidence against a null hypothesis from multiple sources across a range of plausible tuning parameter values reflecting uncertainty with the unknown truth. These highly adaptive tests are not only easy to use, but also high-powered robustly across various scenarios. The usage and advantages of these novel tests are demonstrated on an Alzheimer's disease dataset and simulated data.

Keywords: Covariance matrix; Graphical lasso; NBS; Precision matrix; SPU tests; Sparse estimation; Statistical power; rs-fMRI.

Fig. 1

p-Values for testing brain network differences between the AD group and CN group in the ADNI data. The left column (a) shows the p-values using correlations. The middle column (b) shows the p-values with using partial correlations. The last column (c) taSPU and taNBS shows the p-values by combining two inferences evaluated with correlations or partial correlations.

Fig. 2

Altered network edges selected by the most significant NBS tests and the top 50 ranked by univariate score components.

Fig. 3

Altered brain connectivity for Alzheimer's disease: Left side shows the brain in sagittal view and right side is for axial view; brain areas are color coded; the node size is proportional to the degree of the node size; abbreviations for brain regions are in Table 2.

Fig. 4

Simulation set-up 1: power for testing network differences with true sparse precision matrices when ϕ = 0.009; the left panel (a) shows the power using correlations; the middle panel (b) for the power with using partial correlations; in the last panel (c), is the power of taSPU and taNBS.

Fig. 5

Simulation set-up 1: power of SPUs for testing network differences with true sparse precision matrices; in the top row, ϕ = 0.01, and in the bottom ϕ = 0.008. The left panel (a) shows the power using correlations; the middle panel (b) for the power with using partial correlations; in the last panel (c), is the power of taSPU.

Fig. 6

Simulation set-up 1: CV-selected regularization for testing network differences and estimation errors with true sparse precision matrices and ϕ = 0.01.

Fig. 7

Simulation set-up 2: power for testing network differences with true sparse covariance matrices; the left panel (a) shows the power using correlations; the middle panel (b) for the power with using partial correlations; in the last panel (c), is the power of taSPU and taNBS.