Statistics: Multivariate tests

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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.

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