Random fields of multivariate test statistics, with applications to shape analysis

TL;DR

This paper introduces a simple method for calculating p-values for the maximum of multivariate Gaussian random fields, with applications to shape analysis and detecting brain damage.

Contribution

It extends Euler characteristic-based approximations to a variety of multivariate test statistics, unifying previous results and enabling new applications.

Findings

Unified method for Euler characteristic calculations

Applied to shape analysis for brain damage detection

Validated approach for multiple multivariate test fields

Abstract

Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point. The problem is to find the $P$-value of the maximum of such a random field of test statistics. We approximate this by the expected Euler characteristic of the excursion set. Our main result is a very simple method for calculating this, which not only gives us the previous result of Cao and Worsley [Ann. Statist. 27 (1999) 925--942] for Hotelling's $T^2$, but also random fields of Roy's maximum root, maximum canonical correlations [Ann. Appl. Probab. 9 (1999) 1021--1057], multilinear forms [Ann. Statist. 29 (2001) 328--371], $\bar{\chi}^2$ [Statist. Probab. Lett 32 (1997) 367--376, Ann. Statist. 25 (1997) 2368--2387] and $\chi^2$ scale space [Adv. in Appl. Probab. 33 (2001) 773--793]. The trick involves approaching the problem from the point of view of Roy's union-intersection principle. The results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma.