# Statistics: Multivariate tests

### From MathWiki

Multivariate tests can be expressed as functions of eigenvalues of a hypothesis matrix, 'H', relative to an error matrix, 'E'. Letting *T* = *H* + *E* we can express tests as follows:

## Wilks' test

Wilks' test rejects for small values of , equivalently for large values of .

Let Λ be the diagonal matrix of eigenvalues of *H* relative to *E*. There is a matrix *A* such that *E* = *A**A*', *H* = *A*Λ*A*' and *T* = *A*(Λ + *I*)*A*'. We can order the eigenvalues: . The rank of *H* is equal to the number of non-zero λ's.

Wilk's test rejects for large values of .

If we consider drawing the ellipse in a metric that makes a unit sphere, the radii of principal axes of are given by with .

Thus Wilk's test rejects for large values of the volume of or, equivalently, for large values of the volume of relative to the volume of in the original metric.

## Links

- SAS documentation on multivariate tests (
*http://post.queensu.ca:8080/SASDoc/getDoc/en/statug.hlp/introreg_sect21.htm*)