  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.
