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I am trying to understand the independent t-test formula. It is to test if A and B have the same mean, assuming A and B have the same variance. The t is calculated by the following formula:

$A = A_1, A_2, A_3, ..., A_{n_A}$

$B = B_1, B_2, B_3, ..., B_{n_B}$

$S = \frac{(n_A-1)s_A^2+(n_B-1)s_B^2}{n_A+n_B-2}$

$t = \frac{m_A - m_B}{\sqrt{\frac{S}{n_A}+\frac{S}{n_B}}}$

Is it possible to use the unbiased variance of A + B instead of the $S$?

$A + B = A_1, A_2, A_3, ..., A_{n_A}, B_1, B_2, B_3, ..., B_{n_B}$

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    $\begingroup$ Reconsider your premises: That $s^{2}$ in your post is an unbiased estimate of the variance, for two groups that might have different means. $\endgroup$
    – Glen_b
    Oct 3 at 23:27
  • $\begingroup$ Thank you for pointing it out. What should I use? Tried using $S$ for now $\endgroup$
    – Jihyun
    Oct 4 at 0:54
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    $\begingroup$ I don't understand your question. What's the problem with using $s^2$? $\endgroup$
    – Glen_b
    Oct 4 at 1:04

1 Answer 1

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It's a bad idea. If the means in the two groups are different, the variance computed from the two groups combined will include the between-group variation and so will be larger than the true within-group variance $s^2$.

For example: two groups of size 30, where A has mean 0 and B has mean 5

> a<-rnorm(30)
> b<-rnorm(30,mean=5)
> var(a)
[1] 0.5855081
> var(b)
[1] 0.858661
> var(c(a,b))
[1] 7.901566

Under the null hypothesis you would still get an unbiased estimator of $s^2$, but when the null is false you will overestimate $s^2$ and so lose power compared to the standard $t$-test

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