t-test & Welch-test [Power / Sample Size]

posted by Helmut Homepage – Vienna, Austria, 2019-09-14 00:16  – Posting: # 20596
Views: 727

Hi Rocco,

» Where does the formula on slide 10.83 in bebac.at/lectures/Leuven2013WS2.pdf for CI for parallel design come from? I cannot seem to find reference anywhere.

Honestly, I don’t remember why I simplified the commonly used formula.
Algebra:$$s\sqrt{\tfrac{n_1+n_2}{n_1n_2}}=\sqrt{s^2(1/n_1+1/n_2)}\;\tiny{\square}$$ Comparison with the data of the example.The formula for Satterthwaite’s approximation of the degrees of freedom given in slide 11 contained typos (corrected in the meantime). Of course,$$\nu\approx\frac{\left(\frac{{s_{1}}^{2}}{n_1}+\frac{{s_{2}}^{2}}{n_2}\right)^2}{\frac{{s_{1}}^{4}}{n{_{1}}^{2}(n_1-1)}+\frac{{s_{2}}^{4}}{n{_{2}}^{2}(n_2-1)}}$$Satterthwaite’s approximation adjusts both for unequal variances and group sizes. The conventional t-test is fairly robust against the former but less so for the latter.
In the R function t.test() var.equal = FALSE is the default because:

Cheers,
Helmut Schütz
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