# Tensor Statistics

The following functions are designed for performing advanced tensor statistics in the context of voxelwise group comparisons. These functions assume that the DTI images have been normalized to the same coordinate frame (e.g. MNI coordinates) so that each voxel coresponds to the same anatomical structure in all subjects.

## Contents

###  Testing for group differences in the principal diffusion direction (PDD)

Schwartzmann, Dougherty and Taylor developed a statistical model for the principal diffusion direction. This is published in MRM (see below).

The statistic they used is the Watson distribution. This distribution is useful for modeling unit vectors with undefined sign, such as the principal diffusion direction (first eigenvector of the diffusion tensor). The following functions are related to this model (the particle Dir means direction).

There are a set of scripts that illustrate how to use this statistic for the PDD. These are briefly described here.

• dtiDirMean.m: Computes voxel-wise mean directions and dispersions from an array of DTI directions. It computes the summary statistics required for dtiDirTest.m.
• dtiDirTest.m: Computes a voxel-wise 2-sample test statistic for testing equality between group-mean principal diffusion directions.
• dtiDirSample.m: Random vectors from the Watson distribution.
• dtiDirConcentration.m: Computes concentration parameter of Watson distribution from the corresponding dispersion value.
• dtiDirTestStatDistr.m: Montecarlo estimate of distribution of 2-sample test statistic (useful for computing statistical power).

For a longer description of what the scripts do, see the headers of the m-files.

For details on the mathematics of the method see Schwartzman, Dougherty & Taylor, MRM 2005 (PDF or pubmed).

###  Lognormal model for the full diffusion tensor

The log-normal distribution for positive definite matrices provides a mathematically valid model for the full diffusion tensor. This model can be understood in two steps:

1. A matrix log transformation is applied to the tensors. The resulting log-tensors are 3-by-3 symmetric matrices with no positive definite constraints.
2. The log-tensors are modeled with a normal distribution for symmetric matrices.

Based on this distribution, three types of 2-sample tests have been derived: an omnibus test for the full diffusion tensor, a test for tensor eigenvalues and a test for tensor eigenvectors. The list below also includes a standard t-test for comparison with scalar quantities (e.g. FA).

• dtiTTest.m: Computes a voxel-wise 2-sample T-test statistic for testing equality between two scalar quantities (e.g. FA).
• dtiLogTensorMean.m: Computes voxel-wise log-tensor mean and variance. It computes the summary statistics required for dtiLogTensorTest.m.
• dtiLogTensorTest.m: Computes a voxel-wise 2-sample test statistic for testing equality between group-mean log-tensors. Three types of tests are available:
• Hotelling T2 test of equality between group-mean log-tensors (omnibus log-tensor test).
• Test of equality between sets of group-mean eigenvalues of log-tensors (eigenvalue test).
• Test of equality between frames of group-mean eigenvectors of log-tensors (eigenvector test).

Both functions can also be used to compare a single subject to a control population.

For details, see Schwartzman, Dougherty & Taylor, MRM (submitted) 2006 (Link coming soon!).

For an example analysis script, see BROKEN LINK BOB TO FIX and the associated dataset.

###  False-Discovery Rate

Voxelwise testing leads to a multiple testing problem. The following functions implement control of the false discovery rate, or the expected fraction of falsely selected voxels among those selected at a particular threshold. This is a different criterion from the familywise error rate (probability of getting at least one falsely selected voxel) implemented by Bonferroni and random field corrections. This particular implementation is designed to allow for an empirical null distribution, i.e. an empirical Bayes adjustment to the theoretical null distribution based on the observed data.

• fdrEmpNull.m: Empirical FDR function. Computes data histograms and empirical null parameters (if desired).
• fdrHist.m: Constructs a histogram structure of the test statistics. Used within fdrEmpNull.m.
• fdrEmpNull0.m: Adjusts the theoretical null distribution of the test statistics to the actual distribution of the test statistics. Used within fdrEmpNull.m.
• fdrCurve.m: Based on the true histogram and the empiricall null found by fdrEmpNull.m, this function computes an FDR curve, i.e. estimates of FDR at all possible thresholds.
• fdrThresh.m: Based on an FDR curve, finds a threshold that results on the desired FDR level.

For details, see (Link coming soon!).

###  Example Analyses

To compare a single subject to a group template, see dtiAnalyzeSubjectAL.m in mrDiffusion/analysisScripts.

To compare two groups of subjects, see dtiAnalysisExample1.m and the associated dataset.

There are several other scripts within VISTASOFT/mrDiffusion/analysisScripts that might be of interest. For example, the most general would be dtiAnalyzeEigVecs, which can do a Watson test (principal eigenvector difference) or it can do a logNormal tensor eigenvector or eigenvalue test. This latter test takes into account all three eigenvectors/values and does a voxel-by-voxel analysis. The one-subject version of this test is dtiAnalyzeSubjectAL, which was written specifically for subject AL but should apply to any single subject.

Another useful script might be dtiAnalyzeFA, which produced a correlation map of FA and basic reading score in normalized brains. It is also voxel-based.