Suppose I have two independent samples from $N(\mu_m,\sigma^2)$ and $N(\mu_n,\sigma^2)$ (common variance), of sizes $m$ and $n$, and I want to test $H_0:\mu_m=\mu_n$.

Instead of the pooled two-sample $t$-test, I want to ask about a statistic built from a different intuition. The pooled variance $s^2_{m+n}$ feels "polluted" to me: it blends both groups' scatter with the gap between their means. So instead, let me anchor group $m$'s deviations to the global mean.

Let $\mu_{m+n}=\dfrac{m\bar x_m + n\bar x_n}{m+n}$ be the grand mean of all $m+n$ points, and define

$$ s'^2_m := \frac{1}{m-1}\sum_{i=1}^{m}\bigl(x_i-\mu_{m+n}\bigr)^2, \qquad s^2_m := \frac{1}{m-1}\sum_{i=1}^{m}\bigl(x_i-\bar x_m\bigr)^2 . $$

My proposed statistic is the ratio

$$ R := \frac{s'^2_m}{s^2_m}. $$

The idea: under $H_0$, $\bar x_m \approx \mu_{m+n}$, so $s'^2_m\approx s^2_m$ and $R\approx 1$; under the alternative, $\mu_{m+n}$ sits between the two group means, $\bar x_m$ is pulled away from it, and $R$ grows. Large $R$ rejects.

My questions:

  1. Is $R$ a valid test statistic, and what is its exact null distribution? Using the identity $\sum_i (x_i-\mu_{m+n})^2 = \sum_i(x_i-\bar x_m)^2 + m(\bar x_m-\mu_{m+n})^2$, I get $$ R = 1 + \frac{m\,(\bar x_m-\mu_{m+n})^2}{\sum_i (x_i-\bar x_m)^2}. $$ Are the numerator and denominator independent here (so that $R$ has a tractable distribution), and if so what is it?

  2. This statistic is deliberately asymmetric — it uses group $m$'s scatter as the noise yardstick and only $m$'s deviation as signal; group $n$ enters only through $\mu_{m+n}$. Is this asymmetry a defect, or can it be justified? (e.g. weighting by sample size / "credibility". m >> 0 means I have a greater "credibility", and R scales up with $m$) How does it compare to the symmetric pooled $t$/$F$ test in terms of power and noncentrality?

  3. More generally, is there a clean way to see what I lose by using $R$ instead of the pooled statistic, given that the pooled variance also mixes between- and within-group variation?

I'd appreciate either a derivation of the null distribution of $R$ or a reference where this kind of "deviation-from-grand-mean over within-group" ratio is analyzed.