Re: help: paired t-test for 2-d variables

From: Ray Koopman (koopman_at_sfu.ca)
Date: 06/26/04 

Date: 26 Jun 2004 11:59:30 -0700

Gotta watch those late-night posts. At 2:59:58am, I wrote in message
news:<242be6f5.0406260159.4686ca8e@posting.google.com>...
> [...]
> Finally, get T_0^2 = m'.(S^-1).m * N(N-2)/(2(N-1)),
> where S^-1 is the inverse of S,
> the dot denotes matrix multiplication,
> and the prime denotes matrix transposition.
> ("T_0^2" is a T with a subscript 0 and superscript 2;
> in words, it's "T-zero-square".)
>
> If the null is true then T_0^2 should be distributed as F(2,N-2);
> you reject the null if the observed T_0^2 is too big, just as in anova.

That will get you the right answer, but the nomenclature is wrong.

T^2 = m'.(S^-1).m * N.

The critical value of T^2 for testing the null is called T_0^2.

T_0^2 = (2(N-1)/(N-2)) * F(2,N-2,alpha),

where F(2,N-2,alpha) is the upper-alpha critical value
in the F(2,N-2) distribution; i.e., the value that has
100*alpha percent of the distribution above it.

You reject the null if T^2 > T_0^2,
which is equivalent to the procedure that I gave in my previous post,
but the quantity that I called T_0^2 I should have called F,
because it's compared directly to the critical F.