STATISTICAL ANALYSIS OF CORTICAL MORPHOMETRICS USING POOLED DISTANCES BASED ON LABELED CORTICAL DISTANCE MAPS

2.1 Data Description and Acquisition

A cohort of 34 right-handed young female twin pairs between the ages of 15 and 24 years old were obtained from the Missouri Twin Registry in order to study cortical changes in the VMPFC associated with MDD. The inclusion criteria for affected twin pairs were onset prior to age 16 and the DSM-IV criteria for MDD being greater than duration of 4 weeks. Control twin pairs had no personal or first degree of family history of MDD. Both monozygotic and dizygotic twin pairs were included, of which 14 pairs were controls (Ctrl) and 20 pairs had one twin affected with MDD, their co-twins were designated as the HR group. Three high resolution T1-weighted MPRAGE magnetic resonance scans of each subject in this population were acquired using a Siemens scanner with 1 mm³ isotropic resolution. Images were then averaged, corrected for intensity inhomogeneity and interpolated to 0.5×0.5×0.5 mm³ isotropic voxels. Following [23], a region of interest (ROI) comprising the VMPFC stripped of the basal ganglia, eyes, sinus, cavity, was defined manually and segmented into gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF) by Bayesian segmentation using the expectation maximization algorithm [12]. A triangulated representation of the cortex at the GM/WM boundary was generated using isocontouring algorithms [12].

Bayesian segmentation [12] automatically segments the tissue via the Expectation-Maximization minimization of Gaussians for the three tissue classes at each voxel. Partial volume i.e. voxels that share mixtures were resolved via a Neymann-Pearson recalibration of the segmentation based on a training set [23]. The threshold between GM and WM was used to generate a triangulated isosurface via the marching tetrahedra algorithm i.e. the mesh is dense. Validation with several VMPFC subvolumes yielded misclassification errors of 0.05-0.10 (n=5) for the segmentation and sub-voxel accuracy of the isosurface with 50 percent of the vertices within 0.12-0.28 mm (n=14) from semi-automated contours [23].

LCDM is generated as follows: first, the ROI subvolume is partitioned by a regular lattice of voxels of specific size h , denoted V(h). Every voxel is labeled by tissue type as gray matter (GM), white matter (WM), or cerebrospinal fluid (CSF) (see, e.g., [12, 17]). For every GM voxel in the ROI, the distance from the centroid of the voxel to the closest point on GM/WM surface is computed. Let S(Δ) be the triangulated graph representing the GM/WM surface. An LCDM distance is a set distance function d : ν_i ∈ V(h) → d (ν_i , S(Δ)), the distance between the centroid of voxel ν_i and the set S(Δ) ; that is, it is the distance from the center of the voxel to the closest vertex on the surface. More precisely,

where C_M (·) stands for center of mass (or centroid), and ∥·∥₂ is the usual L₂ – norm . We use a signed (or labeled) distance to indicate the location of each voxel with respect to the GM/WM surface. Figure 1 illustrates the computation of distances between labeled voxels and the cortical surface; also shown is the corresponding non-normalized histograms of LCDM distances. Observe that GM tissue comprises most of the cortex, and by construction, while most of GM distances are positive, most of WM distances are negative, and all of CSF distances are positive. Negative distances for some GM close to the GM/WM boundary are possible by construction, because the surface is constructed in such a way that a surface is always intersecting voxels, i.e., partial volume. So some appropriately labeled GM voxels may fall on a side of surface that they should not belong to. However, these mislabeled voxels constitute a small proportion of all voxels and do not affect the overall analysis. Reliability of LCDMs is dependent on GM segmentation and reconstruction of GM/WM surfaces which has been validated for several cortical structures including VMPFC [23], cingulate cortex [24, 25] and planum temporale [22]. Condensing to a single distance value for each vertex on the surface is the next logical step in extending LCDM. This is called Local LCDM or LLCDM and is useful in comparing thickness across multiple subjects for a cortical structure (see [20, 21]).

For the left ROI, let D^L be the set of LCDM distances, $D_{ijk}^L$ be the distance calculated as in Equation (1) and associated with k^th voxel in subject j in group i for k = 1, 2,…,K_ij , j = 1, 2,…,n_i and i = 1, 2,3 (here group 1 is for MDD, 2 is for HR, and 3 is for Ctrl). Thus, n₁ = 20, n₂ = 20, and n₃ = 28. Right distances D^R are denoted similarly as $D_{ijk}^R$ . Based on prior anatomical knowledge (e.g., [14]), cortical thickness of the VMPFC is roughly 6 mm, so we can safely retain distances between −0.5 mm and 5.5 mm so that (potentially) mislabeled GM is excluded from the data. In this particular case for the left and right VMPFC, only 0.16% and 0.14% of distances were below −0.5 mm respectively; similarly, only 0.22% and 0.07% of distances were above 5.5 mm , respectively.

2.2 Pooling LCDM Distances by Group

Although the descriptive measures such as mean, median, and variance of LCDM distances are global measures regarding the morphometry of VMPFC, they are summary statistics (such as volume or median), so they tend to oversimplify the data since instead of a large number of LCDM distances per subject, we will have two (e.g., one mean value for left, one for right VMPFC) measures for each subject [5]. Hence we lose most of the information conveyed by the LCDM distances. A solution to this problem is using all the LCDM distances in our analysis. So we pool LCDM distances of subjects from the same group and thereby obtain yet another global measure of morphometry. That is, we pool the LCDM distances for all left MDD VMPFCs, those for all left HR VMPFCs, and those for all left Ctrl VMPFCs. Likewise, we pool the right VMPFC LCDM distances. Thus, for left VMPFCs

where $D_{i\ell }^L$ is the ℓ^th distance in group i and $N_i={\sum }_{j=1}^{n_i}{K}_{ij}$ is the number of distances (i.e., GM voxels) in group i for i = 1, 2,3 (group 1 is for MDD, group 2 for HR, and group 3 for Ctrl). Similarly, we denote the right pooled distances as $D_i^R$ . Furthermore, we denote the overall (i.e., groups combined) pooled left and right distances as $D^L={\bigcup }_{i=1}^3{D}_i^L$ and $D^R={\bigcup }_{i=1}^3{D}_i^R$ , respectively. See Table 1 for the corresponding sample sizes, means, and standard deviations of the pooled LCDM distances, overall and for each group. For pooling the LCDM distances, the most crucial assumption is that the subjects with the same diagnosis have similar VMPFC in morphometry, which is reasonable in practice. By pooling, the most common characteristics of the VMPFC specific to a diagnostic group are emphasized, while the differences at the individual (i.e., subject) level are downplayed. Furthermore, the pooled distances will be more powerful in detecting the differences between LCDM distances (hence differences in morphometry).

2.3 Statistical Tests

There is an inherent dependence between LCDM distances of voxels to the gray matter/white matter boundary due to spatial correlation at the level of individual subjects. When we pool the LCDM distances by group, this spatial dependence is not removed. That is, pooling neither creates nor removes the inherent dependence of the distances, as it only ignores the subject information. We compare the distributions and central measures (e.g., means) of the LCDM distances using various statistical tests. In particular, we consider Kruskal-Wallis (K-W) test for omnibus multi-group comparison of the LCDM distributions and ANOVA F-tests for omnibus multi-group comparison of the LCDM means. For k groups the null hypothesis for K-W test is H₀ : F₁ = F₂ =…=F_k where F_i is the distribution function of group i and the null hypothesis for ANOVA F-test is H₀ : μ₁ = μ₂ =…=μ_k where μ_i is the mean of group i, for i = 1, 2,…,k. For comparison of distributions of LCDM distances of pairs of groups, we apply Wilcoxon rank sum test and Kolmogorov-Smirnov (K-S) tests; and for comparisons of means of pairs of LCDM distance groups, we apply Welch’s t-test (see [8] for more detail on these tests). For pairwise comparisons, Wilcoxon rank sum test is done as a post hoc test after a significant K-W test, because Wilcoxon rank sum and K-W tests are both variants of the same test for multiple or two group comparison. K-S test is performed to determine the stochastic ordering. Wilcoxon rank sum test (also called the Mann–Whitney U test) is a non-parametric test for assessing whether two independent samples of observations have similar values. It is based on the sum of the ranks of the two independent samples, when the samples are pooled together. Under the null hypothesis, it is assumed that the distributions of both groups are equal, i.e., H₀ : F₁ = F₂ . In other words, the probability of an observation from the first population being larger than the one from the second population is the same as the probability of an observation from the second population being larger than the first. For two groups, the K-S test is a nonparametric test based on the estimated maximum difference between the cumulative distributions of the two groups. Under the null hypothesis, it is assumed that the distributions are equal, i.e., H₀ : F₁ = F₂ . Welch’s t-test is an extension of the usual Student’s t-test and is intended for use with two samples having (possibly) unequal variances. The null hypothesis for Welch’s t-test is H₀ : μ₁ = μ₂ .

Wilcoxon and t-tests imply an ordering in a location parameter such as mean or median. Stochastic ordering, if present, can be deduced from the direction of the alternative, together with the graph of the cumulative distribution functions (cdfs). However, we can also use Kolmogorov-Smirnov (K-S) tests for H₀ : F₁ = F₂ . Although Wilcoxon rank sum and K-S tests have the same null hypothesis, Wilcoxon test gives an overall distribution comparison based on the rankings of the observations, while K-S test compares the cdfs of the observations at values where the maximum differences between cdfs occur. Hence Wilcoxon test can be significant for only one of the one-sided alternatives, while K-S test yields p-values that are not complementary for the one-sided alternatives (i.e., they don’t add up to 1). Hence, p-values can be significant for both or none of the directional alternatives. This results from the fact that, the order of the cdfs F₁ and F₂ can be different at different distance values (plotted on the horizontal axis). Moreover, if p-value based on K-S test is significant for only one-sided alternative, then we can also deduce stochastic ordering. The p-values being insignificant or significant for both one-sided alternatives imply lack of stochastic ordering. But the first case implies that equality of the distributions is retained, while the latter implies that the distributions are different. Although K-S test does not provide the actual values where the significant differences between cdfs occur, it is more informative and suggestive of distributional differences compared to Wilcoxon rank sum test. Furthermore, different cdf orderings at different values can be masked by the Wilcoxon rank sum test. Hence K-S test is more informative compared to Wilcoxon rank sum test.

We perform the omnibus multi-group tests before the pairwise comparison tests, because if a multi-group test is not significant, there is no need to perform the pairwise tests. For example, if K-W test is not significant, then the distributions of the LCDM distances of groups are not significantly different, hence Wilcoxon rank sum test on each pair of groups is redundant. On the other hand, if a multi-group test yields a significant result, it only means that there are some significant differences between the groups, but does not indicate which groups are different. To determine the pairs that have significant difference, we have to perform the pairwise comparison methods. Among the tests we consider, K-W and ANOVA F-tests are omnibus tests, and Wilcoxon rank sum and Welch’s t-tests are for commonly used multiple comparison procedures after obtaining a significant omnibus test result. Rejecting an omnibus test for k groups suggest that there are differences between some pair(s) of the groups, and to determine which pair(s) exhibit significant differences, k(k – 1)/2 pairwise comparisons are needed. Hence, for large k values, an omnibus test might save a great deal of time and energy since after an insignificant omnibus test, there is no need for the pairwise tests. For small k values, one might do an omnibus test followed by pairwise tests, or just the pairwise tests directly. However, for even k=4, we need 6 pairwise tests, and this might still be too many pairwise tests, if omnibus test were insignificant.

For the nonparametric tests (K-W, Wilcoxon rank sum, and K-S tests) only within sample independence is violated, but for the parametric tests (ANOVA F-tests and t-test), the assumptions of normality (i.e., Gaussianity) and within sample independence are violated. See [4] for a complete list of assumptions for each of these tests. However, we investigate the influence of assumption violations on both nonparametric and parametric tests in Section 4 by an extensive Monte Carlo simulation study where we find the effect of assumption violations is negligible and we conjecture that this is due to the fact that the correlation structure is similar for each person (hence for each group). Moreover, our analysis does not concern inference for single populations but comparison of multiple populations. Given the difficulty to develop a method that accounts for spatial correlations, we ignore this type of spatial dependence henceforth.

In the analysis of the pooled distances, we apply classical parametric and nonparametric tests to detect the differences in LCDM distances due to diagnostic group factors. Such differences will imply morphometric changes (if any) due to the particular disease in question. K-W test provides an overall test of distributional equality for multiple groups. That is, if K-W test yields a significant p-value, then we conclude that LCDM distances are different in distribution for at least two groups, but it does not indicate which pair or pairs of groups exhibit differences. To find out which pairs exhibit significant distributional differences, we apply Wilcoxon rank sum test for each pair of LCDM groups. On the other hand, K-S test is only applicable to compare the distributions of two LCDM groups. Similarly, if an ANOVA F-test yields a significant p-value, it implies that the mean LCDM distances are different for at least two LCDM distance groups. To find out which pairs exhibit significant mean differences, we apply Welch’s t-test for each pair of LCDM groups. The p-values for the t-test and Wilcoxon rank sum test are complementary, in the sense that p-values for the one-sided alternatives add up to 1 and can be significant for only one of the one-sided alternatives. Hence, Wilcoxon test provides an overall distributional comparison for two LCDM groups. On the other hand, p-values for K-S test are not complementary, as they do not add up to 1 for the one-sided alternatives. For example, one might have significant p-values for both of the one-sided alternatives, which implies that at a particular distance value, a group’s empirical cumulative distribution function (ecdf) is significantly larger, while at another distance value the other group’s ecdf is larger. Wilcoxon test (together with the ecdf plots) and K-S tests (either with p-values for both one-sided tests or with the ecdf plots) might provide the stochastic ordering (if present) of pooled distances.

4.1 Simulation of Distances that Resemble LCDM Distances

In our Monte Carlo study, we generate three samples $\mathcal{X}$ , $\mathcal{Y}$ , and $\mathcal{Z}$ with sizes n_x , n_y , and n_z , respectively, and set n_x = n_y = n_z =10000. Each sample is generated similar to the procedure described above. For example, sample $\mathcal{X}$ is generated as follows: First we generate

where $P_X(J=i)={\nu }_i^x∕{\sum }_{i=2}^{12}{\nu }_i^x$ with ${\nu }_i^x$ is the i^th entry in ${\nu }_x=({\nu }_0^x,{\nu }_1^x,\dots ,{\nu }_{12}^x)$ and is also the i^th value after the entries |ν_i-η_x | are sorted in descending order for i = 0,1,2,…,11 and ${\nu }_{12}^x=11659-{\sum }_{i=0}^{11}\mid {\nu }_i-{\eta }_x\mid$. Let $n_i^x$ be the frequency of i among the n_x generated numbers from P_X . Then we generate $U_{ik}~\mathcal{U}(0,r_x)$ for $k=1,\dots ,n_i^x$ for each i . Hence the set of simulated distances for set $\mathcal{X}$ is

Samples $\mathcal{Y}$ and $\mathcal{Z}$ are generated similarly with parameter subscripts in (7) and (8) are modified accordingly.

4.2 Empirical Size Estimates for the Multi-Sample Case

For the null hypothesis of multi-sample case which states the equality of the distributions of LCDM distances, we generate three samples $\mathcal{X}$ , $\mathcal{Y}$ , and $\mathcal{Z}$ with the below parameters:

Notice that each sample is generated so as to resemble those of the left VMPFC of HR subject 1 up to scale. This is done without loss of generality, since any other VMPFC can either be obtained by a rescaling of the generated distances or by modifying the parameters. So for example, for sample $\mathcal{X}$ , P_X(X_j = i) = ν_p,i with ν_p,i being the i^th entry in v_p in Equation (5) and the set of simulated distances for set $\mathcal{X}$ is as in Equation (8) with r_x = 1.0 and η_x = 0.

We repeat this sample generation procedure N_mc = 10000 times. We count the number of times the null hypothesis is rejected at α = 0.05 level for K-W test of distributional equality and ANOVA F-tests (with and without HOV) of equality of mean distances. The ratio of the number of significant results by each test to N_mc yields the estimated significance levels under H_o. The estimated significance levels for various values of n_x , n_y , and n_z are provided in Table 5, where ${\widehat{\alpha }}_{BF}$ is the empirical size estimate for K-W test, ${\widehat{\alpha }}_{F_1}$ is for ANOVA F-test with HOV, and ${\widehat{\alpha }}_{F_2}$ is for ANOVA F-test without HOV. Furthermore, ${\widehat{\alpha }}_{KW,F_1}$ is the proportion of agreement between K-W and ANOVA F-test with HOV, i.e., the number of times out of 10000 Monte Carlo replicates both KW and ANOVA F-test with HOV reject the null hypothesis. Similarly, ${\widehat{\alpha }}_{KW,F_2}$ is the proportion of agreement between K-W and ANOVA F-test without HOV, and ${\widehat{\alpha }}_{F_1,F_2}$ is the proportion of agreement between ANOVA F-test with HOV and ANOVA F-test without HOV. Using the asymptotic normality of the proportions, we test the equality of the empirical size estimates with 0.05, and compare the empirical sizes pairwise. We observe that the K-W test is at the desired significance level, while ANOVA F-tests with and without HOV are at the desired level or slightly conservative. Notice also that under H_o, the tests tend to be more conservative as the sample sizes increase. Hence, if the distances are not that different; i.e., the frequency of distances for each bin and the distances for each bin are identically distributed for each group, the inherent spatial correlation does not seem to influence the significance levels. Moreover, we observe that for LCDM distances K-W and ANOVA with HOV tests have significantly different rejection (hence acceptance) regions, because the proportion of agreement for these tests, ${\widehat{\alpha }}_{KW,F_1}$ is significantly smaller than the minimum of ${\widehat{\alpha }}_{KW}$ and ${\widehat{\alpha }}_{F_1}$ , $\text{min}({\widehat{\alpha }}_{KW},{\widehat{\alpha }}_{F_1})$ . Similarly, K-W and ANOVA without HOV tests have significantly different rejection (hence acceptance) regions because, the proportion of agreement for these tests, ${\widehat{\alpha }}_{KW,F_2}$ is significantly smaller than $\text{min}({\widehat{\alpha }}_{KW},{\widehat{\alpha }}_{F_2})$ . However, ANOVA with and without HOV tests have about the same rejection (hence acceptance) regions because, the proportion of agreement for these tests, ${\widehat{\alpha }}_{F_1,F_2}$ is not significantly different from $\text{min}({\widehat{\alpha }}_{F_1},{\widehat{\alpha }}_{F_2})$ . This mainly results from the fact that K-W and ANOVA with HOV tests test different hypotheses, and so do the K-W and ANOVA without HOV tests. But, ANOVA with and without HOV tests basically test the same hypotheses.

4.3 Empirical Power Estimates for Multi-Sample Case

For the alternative hypothesis, we generate sample $\mathcal{X}$ as in the null case, so $D_{mc}^{\mathcal{X}}$ is as in Equation (8). We consider various r_y and η_y values for sample $\mathcal{Y}$ and various r_z and η_z values for sample $\mathcal{Z}$ . The five alternative cases we consider are

See Figure 4 for the kernel density estimates of sample distances under the null case and various alternatives.

We repeat the sample generation N_mc = 10000 times under each alternative case. We count the number of times the null hypothesis is rejected at α = 0.05 for K-W test of distributional equality, and ANOVA F-tests (with and without HOV) of equality of mean distances, and find the ratio of number of significant results by each test to N_mc . Thus we obtain the empirical power estimates under H_a which are provided in Table 6, where ${\widehat{\beta }}_{KW}$ is the empirical power estimate for K-W test, ${\widehat{\beta }}_{F_1}$ is for ANOVA F-test with HOV, and ${\widehat{\beta }}_{F_2}$ is for ANOVA F-test without HOV. Using the asymptotic normality of the empirical power estimates, we observe that under each of H_a cases with (r_y,r_z,η_y,η_z)∈{(1.1,1.0,0,0),(1.1,1.2,0,0)} the distributions are different, so the larger the r_y and r_z from 1.0, the higher the power estimates for K-W and ANOVA F-tests. Furthermore, as the sample size n increases, the power estimates for K-W and ANOVA F-tests also increase. Notice that under these alternatives, the K-W test tends to be more powerful than ANOVA F-tests, since such alternatives influence the ranking (hence the distribution) of the distances, more than the mean of the distances. Furthermore, under these alternatives, it is not the size or scale that is really different; it is the difference in shape that is more emphasized. The size component is distance with respect to the GM/WM surface; i.e., if the GM voxels from the GM/WM surface are at about the same distance, the K-W test is more sensitive to the differences in the distributions of the LCDM distances. We also note that ANOVA F-tests with and without HOV have about the same power estimates.

Under each of alternative cases with

as η_y and η_z deviate more from 0, the power estimates for K-W and ANOVA F-tests increase. Note that as n increases, the power estimates also increase under these alternative cases. Under these second type of alternatives, ANOVA F-tests tend to be more powerful, since the right skewness (tail) of distances are more emphasized, which in turn implies that the differences in the mean distances are emphasized more. Under these alternatives, both the size or scale and shape are different. If the GM voxels from the GM/WM surface are at different distances, ANOVA F-tests are more sensitive to the differences in LCDM distances. We also note that both ANOVA F-tests (with and without HOV) have about the same power estimates.

4.4 Empirical Size Estimates for the Two-Sample Case

For the null hypothesis for the two-sample case, we generate two samples $\mathcal{X}$ and $\mathcal{Y}$ each of size n_x and n_y , respectively. Each sample is generated as described in Section 4.2. We repeat the sample generation N_mc = 10000 times.

We count the number of times the null hypothesis is rejected at α = 0.05 for Lilliefor’s test of normality, Wilcoxon rank sum test of distributional equality, t-test of equality of mean distances, and K-S test of equality of cdfs, and find the ratio of the number of significant results by each test to N_mc , thereby obtain the estimated significance levels. Unlike the multi-sample case, for the two-sample case, except for Lilliefor’s test there are three types of alternative hypotheses possible: two-sided, left, and right-sided alternatives. The estimated significance levels are provided in Table 7, where ${\widehat{\alpha }}_W$ is the empirical size estimate for Wilcoxon rank sum test, ${\widehat{\alpha }}_t$ is for t-test, ${\widehat{\alpha }}_{KS}$ is for K-S test. Furthermore, ${\widehat{\alpha }}_{W,t}$ is the proportion of agreement between Wilcoxon rank sum and t-tests, ${\widehat{\alpha }}_{W,KS}$ is the proportion of agreement between Wilcoxon rank sum and K-S tests, and ${\widehat{\alpha }}_{t,KS}$ is the proportion of agreement between t-test and K-S test. We omit the Lilliefor’s test, since by construction, our samples are severely non-normal, so normality is rejected for virtually all samples generated. Observe that under H_o, the empirical significance levels are about the desired level for all three types of alternatives, although Wilcoxon tests are slightly liberal, while K-S test is slightly conservative. Hence, if the distances are not that different; i.e., the frequency of distances for each bin and the distances for each bin are identically distributed for each group, the inherent spatial correlation does not influence the significance levels. However, Wilcoxon rank sum, t-test, and K-S methods test different hypotheses, so their acceptance and rejection regions are significantly different for LCDM distances, since the proportion of agreement for each pair is significantly smaller than the minimum of the empirical size estimates for each pair of tests.

4.5 Empirical Power Estimates for the Two-Sample Case

For the alternative hypotheses, we generate samples $\mathcal{X}$ and $\mathcal{Y}$ as in Section 4.3 also. Note that when r_y = 1 and η_y = 0, we obtain the null case. The five alternative cases we consider are (r_y,η_y )∈{(1.1,0),(1.2,0),(1.0,10),(1.0,30),(1.0,50)}. We count the number of times the null hypothesis is rejected for Lilliefor’s test of normality, Wilcoxon rank sum test of distributional equality, t-test of equality of mean distances, and K-S test of equality of cdfs, thereby obtain the estimated significance levels as before. The power estimates are provided in Table 8, where ${\widehat{\beta }}_W$ is the power estimate for Wilcoxon rank sum test, ${\widehat{\beta }}_t$ is for t-test, ${\widehat{\beta }}_{KS}$ is for K-S test.

Under the alternative cases with (r_y,η_y)∈{(1.1, 0),(1.2,0)}, we see that the distributions start to differ. As r_y deviates further away from 1.0, then the power estimates for Wilcoxon rank sum, t-test, and K-S tests increase. Furthermore, as the sample size n increases, the power estimates for Wilcoxon test, t-test, and K-S test also increase. Observe that as in the multi-sample case, under these alternatives, Wilcoxon test is more powerful than t-test, since the ranking of the distances are affected more than the mean distances under these alternatives. But K-S test has the highest power estimates for sample sizes larger than 1000. Thus, for differences in shape rather than the distance from the GM/WM surface, K-S test and Wilcoxon rank sum test are more sensitive (i.e., powerful) than t-test. Furthermore, as the sample sizes increase, the left-sided tests become more powerful than their two-sided counterparts. Notice that we omit the power estimates for the right-sided alternatives, since by construction (i.e., due to our parameter choices in our simulations) $\mathcal{X}$ values tend to be smaller than $\mathcal{Y}$ values for these alternatives; hence the tests virtually have no power for the right-sided alternatives.

Under the H_a cases with (r_y,η_y)∈{(1.0,10),(1.0,30),(1.0,50)}, as η_y deviates further away from 0, the power estimates for Wilcoxon rank sum, t-test, and K-S tests increase. Note that as n increases, the power estimates also increase under each alternative case. Under these alternatives, t-test is more powerful than Wilcoxon test, since mean distances are more affected than the rankings under such alternatives. However, K-S test has higher power estimates for larger deviations from the null case. These alternatives imply that the distances of the GM voxels are at different scales, t-test has the best performance for small differences, while for large differences, K-S has the best performance. Furthermore, as the sample sizes increase, the left-sided tests become more powerful than their two-sided counterparts. Again, we omit the power estimates for the right-sided alternatives, because, by construction, $\mathcal{X}$ values tend to be smaller than $\mathcal{Y}$ values for these alternatives.

We do not report the power estimates for Lilliefor’s test of normality, since by construction our data is severely non-normal, and we get power estimates of 1.000 under both null and alternative cases.