# Is $T^2$ equivalent to 1 Way MANOVA when we have just two populations?

When we're in the presence of two populations, for the same assumptions (random samples, multivariate distribution, same covariance matrix), and we want to test the mean vectors, we have two options:

• $T^2$ statistic (a Mahalanobis distance function)
• A statistic,function of the Wilk's lambda (MANOVA)

Both statistics have the same distribution $F_{p,n-p-1}$.

I was thinking that both should be equivalent. However, I don't see how the MANOVA statistic which involves a fraction of determinants can be made to be equal to the $T^2$ statistic.

Any help would be appreciated.

• The answer to the title question is: Yes. – amoeba says Reinstate Monica Apr 25 '17 at 15:03
• Regarding the math, MANOVA looks at $\Sigma_W^{-1}\Sigma_B$, but for two classes $\Sigma_B$ is rank-one, so this whole expression reduces to T2 which is $(\mu_1-\mu_2)^\top\Sigma_W^{-1}(\mu_1-\mu_2)$. Or something along these lines. – amoeba says Reinstate Monica Apr 25 '17 at 15:06