Abstract
Our data are random fields of multivariate normal 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 non-zero 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 & Worsley (1999a) for Hotelling's T^2, but also random fields of Roy's maximum root, maximum canonical correlations (Cao & Worsley, 1999b), \bar\chi^2 (Lin & Lindsay, 1997; Takemura & Kuriki, 1997), multilinear forms (Kuriki & Takemura, 2001), and \chi^2 scale-space (Worsley, 2001). 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 non-missile trauma, and to detecting functional magnetic resonance imaging (fMRI) activation allowing for unknown latency of the hemodynamic response.