# Statistics: Multivariate tests

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 $\frac{|E|}{|E+H|}$, equivalently for large values of $\frac{|T|}{|E|}$.

Let Λ be the diagonal matrix of eigenvalues of H relative to E. There is a matrix A such that E = AA', H = AΛA' and T = A(Λ + I)A'. We can order the eigenvalues: $\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_p \ge 0$. The rank of H is equal to the number of non-zero λ's.

Wilk's test rejects for large values of $\prod (1+\lambda_i)$.

If we consider drawing the ellipse $\mathcal{E}_T = 0 \oplus \sqrt{T}$ in a metric that makes $\mathcal{E}_E = 0 \oplus \sqrt{E}$ a unit sphere, the radii of principal axes of $\mathcal{E}_T$ are given by $r_1, r_2 , \ldots , r_p$ with $r_i = \sqrt{1+\lambda_i}$.

Thus Wilk's test rejects for large values of the volume of $\mathcal{E}_T$ or, equivalently, for large values of the volume of $\mathcal{E}_T$ relative to the volume of $\mathcal{E}_E$ in the original metric.