Suppose we have $z = \frac{u/p}{v/q} \sim F(p, q, \lambda)$ whereby u is distributed as a non-central chi-square with parameters p and $\lambda$. The power of an F-Test is the probability of rejecting $H_0$ for a given value of $\lambda$. If $F_\alpha$ is the upper $\alpha$ percentage point of the central $F$ distribution, then the power, $P(p,q,\alpha,\lambda)$ can be defined as $P(p,q,\alpha, \lambda) = Prob(z\geq F_\alpha)$. According to my textbook, if p increases, $P(p,q,\alpha, \lambda)$ decrease, and if $q, \alpha \text{ or } \lambda$ increases, $P(p,q, \alpha, \lambda)$ increases. How to understand the effects of the change in parameters value on the power of F-test? What is the proof for it?
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To clarify, I'm assuming it should be $v/q$ in the denominator of $z$? And $F_\alpha$ from the central F has degrees of freedom $p$ and $q$? – dmsul Sep 21 '16 at 13:30
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@dmsul yes sorry for typo – user10024395 Sep 21 '16 at 14:23