Interpretable whole-brain prediction analysis with GraphNet.

Multivariate machine learning methods are increasingly used to analyze neuroimaging data, often replacing more traditional "mass univariate" techniques that fit data one voxel at a time. In the functional magnetic resonance imaging (fMRI) literature, this has led to broad application of "off-the-shelf" classification and regression methods. These generic approaches allow investigators to use ready-made algorithms to accurately decode perceptual, cognitive, or behavioral states from distributed patterns of neural activity. However, when applied to correlated whole-brain fMRI data these methods suffer from coefficient instability, are sensitive to outliers, and yield dense solutions that are hard to interpret without arbitrary thresholding. Here, we develop variants of the Graph-constrained Elastic-Net (GraphNet), a fast, whole-brain regression and classification method developed for spatially and temporally correlated data that automatically yields interpretable coefficient maps (Grosenick et al., 2009b). GraphNet methods yield sparse but structured solutions by combining structured graph constraints (based on knowledge about coefficient smoothness or connectivity) with a global sparsity-inducing prior that automatically selects important variables. Because GraphNet methods can efficiently fit regression or classification models to whole-brain, multiple time-point data sets and enhance classification accuracy relative to volume-of-interest (VOI) approaches, they eliminate the need for inherently biased VOI analyses and allow whole-brain fitting without the multiple comparison problems that plague mass univariate and roaming VOI ("searchlight") methods. As fMRI data are unlikely to be normally distributed, we (1) extend GraphNet to include robust loss functions that confer insensitivity to outliers, (2) equip them with "adaptive" penalties that asymptotically guarantee correct variable selection, and (3) develop a novel sparse structured Support Vector GraphNet classifier (SVGN). When applied to previously published data (Knutson et al., 2007), these efficient whole-brain methods significantly improved classification accuracy over previously reported VOI-based analyses on the same data (Grosenick et al., 2008; Knutson et al., 2007) while discovering task-related regions not documented in the original VOI approach. Critically, GraphNet estimates fit to the Knutson et al. (2007) data generalize well to out-of-sample data collected more than three years later on the same task but with different subjects and stimuli (Karmarkar et al., submitted for publication). By enabling robust and efficient selection of important voxels from whole-brain data taken over multiple time points (>100,000 "features"), these methods enable data-driven selection of brain areas that accurately predict single-trial behavior within and across individuals.

Group Comparison of Eigenvalues and Eigenvectors of Diffusion Tensors.

Diffusion tensor imaging (DTI) data differ from most medical images in that values at each voxel are not scalars, but 3 × 3 symmetric positive definite matrices called diffusion tensors (DTs). The anatomic characteristics of the tissue at each voxel are reflected by the DT eigenvalues and eigenvectors. In this article we consider the problem of testing whether the means of two groups of DT images are equal at each voxel in terms of the DT's eigenvalues, eigenvectors, or both. Because eigendecompositions are highly nonlinear, existing likelihood ratio statistics (LRTs) for testing differences in eigenvalues or eigenvectors of means of Gaussian symmetric matrices assume an orthogonally invariant covariance structure between the matrix entries. While retaining the form of the LRTs, we derive new approximations to their true distributions when the covariance between the DT entries is arbitrary and possibly different between the two groups. The approximate distributions are those of similar LRT statistics computed on the tangent space to the parameter manifold at the true value of the parameter, but plugging in an estimate for the point of application of the tangent space. The resulting distributions, which are weighted sums of chi-squared distributions, are further approximated by scaled chi-squared distributions by matching the first two moments. For validity of the Gaussian model, the positive definite constraints on the DT are removed via a matrix log transformation, although this is not crucial asymptotically. Voxelwise application of the test statistics leads to a multiple-testing problem, which is solved by false discovery rate inference. The foregoing methods are illustrated in a DTI group comparison of boys versus girls.

Empirical null and false discovery rate analysis in neuroimaging.

Current strategies for thresholding statistical parametric maps in neuroimaging include control of the family-wise error rate, control of the false discovery rate (FDR) and thresholding of the posterior probability of a voxel being active given the data, the latter derived from a mixture model of active and inactive voxels. Correct inference using any of these criteria depends crucially on the specification of the null distribution of the test statistics. In this article we show examples from fMRI and DTI data where the theoretical null distribution does not match well the observed distribution of the test statistics. As a solution, we introduce the use of an empirical null, a null distribution empirically estimated from the data itself, allowing for global corrections of theoretical null assumptions. The theoretical null distributions considered are normal, t, chi(2) and F, all commonly encountered in neuroimaging. The empirical null estimate is accompanied by an estimate of the proportion of non-active voxels in the data. Based on the two-class mixture model, we present the equivalence between the strategies of controlling FDR and thresholding posterior probabilities in the context of neuroimaging and show that the FDR estimates derived from the empirical null can be seen as empirical Bayes estimates.

FALSE DISCOVERY RATE ANALYSIS OF BRAIN DIFFUSION DIRECTION MAPS.

Diffusion tensor imaging (DTI) is a novel modality of magnetic resonance imaging that allows noninvasive mapping of the brain's white matter. A particular map derived from DTI measurements is a map of water principal diffusion directions, which are proxies for neural fiber directions. We consider a study in which diffusion direction maps were acquired for two groups of subjects. The objective of the analysis is to find regions of the brain in which the corresponding diffusion directions differ between the groups. This is attained by first computing a test statistic for the difference in direction at every brain location using a Watson model for directional data. Interesting locations are subsequently selected with control of the false discovery rate. More accurate modeling of the null distribution is obtained using an empirical null density based on the empirical distribution of the test statistics across the brain. Further, substantial improvements in power are achieved by local spatial averaging of the test statistic map. Although the focus is on one particular study and imaging technology, the proposed inference methods can be applied to other large scale simultaneous hypothesis testing problems with a continuous underlying spatial structure.

Cross-subject comparison of principal diffusion direction maps.

Diffusion tensor imaging (DTI) data differ fundamentally from most brain imaging data in that values at each voxel are not scalars but 3 x 3 positive definite matrices also called diffusion tensors. Frequently, investigators simplify the data analysis by reducing the tensor to a scalar, such as fractional anisotropy (FA). New statistical methods are needed for analyzing vector and tensor valued imaging data. A statistical model is proposed for the principal eigenvector of the diffusion tensor based on the bipolar Watson distribution. Methods are presented for computing mean direction and dispersion of a sample of directions and for testing whether two samples of directions (e.g., same voxel across two groups of subjects) have the same mean. False discovery rate theory is used to identify voxels for which the two-sample test is significant. These methods are illustrated in a DTI data set collected to study reading ability. It is shown that comparison of directions reveals differences in gross anatomic structure that are invisible to FA.

Unified univariate and multivariate random field theory.

We report new random field theory P values for peaks of canonical correlation SPMs for detecting multiple contrasts in a linear model for multivariate image data. This completes results for all types of univariate and multivariate image data analysis. All other known univariate and multivariate random field theory results are now special cases, so these new results present a true unification of all currently known results. As an illustration, we use these results in a deformation-based morphometry (DBM) analysis to look for regions of the brain where vector deformations of nonmissile trauma patients are related to several verbal memory scores, to detect regions of changes in anatomical effective connectivity between the trauma patients and a group of age- and sex-matched controls, and to look for anatomical connectivity in cortical thickness.