Exact F

Rao's R statistic reduces to an exact F-variate when p = 1 or 2, or when k = 2 or 3.

General Formulas

and m = N - 1 -(p + k)/2

The Case When k = 2

If k = 2 then n_h = k - 1 = 1, and

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m = N - 1 -(p + k)/2
m = N - 1 -(p + 2)/2
m = N + (-p - 2 - 2)/2
m = N + (-p - 4)/2
m = (2N - p - 4)/2

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R is distributed as F with p and N-p-1 degrees of freedom.

The Case When k = 3

If k = 3 then n_h = k - 1 = 2, and

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m = N - 1 -(p + k)/2
m = N - 1 -(p + 3)/2
m = N + (-p - k -2)/2
m = N + (-p - 5)/2
m = (2N - p - 5)/2

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R is distributed as F with 2p and 2(N-p-2) degrees of freedom.

The Case When p = 1

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m = N - 1 -(p + k)/2
m = N - 1 -(1 + k)/2
m = N +(-1 - k - 2)/2
m = N +(-k - 3)/2
m = (2N - k - 3)/2

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R is distributed as F with k-1 and N-k degrees of freedom.

The Case When p = 2

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m = N - 1 -(p + k)/2
m = N - 1 -(2 + k)/2
m = N +(-2 - k - 2)/2
m = N +(-k - 4)/2
m = (2N - k - 4)/2

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R is distributed as F with 2(k-1) and 2(N-k-1) degrees of freedom.

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