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I'm confused as to the correct formula for approximate degrees of freedom to use for Welch's t-test. Satterthwaite's (1946) formula is the most commonly cited formula, but Welch gave an alternative in 1947. I'm not sure which is preferable (or used by most statistical software).

Satterthwaite's formula: $$\frac{\left(s_x^2/n_x +s_y^2/n_y\right)^2}{(s_x^2/n_x )^2/(n_x-1)+(s_y^2/n_y )^2/(n_y-1)}$$

Welch's formula: $$-2+ \frac{\left(s_x^2/n_x +s_y^2/n_y\right)^2}{(s_x^2/n_x )^2/(n_x+1)+(s_y^2/n_y )^2/(n_y+1)}$$

References:

  • Satterthwaite, F.E. (1946). "An Approximate Distribution of Estimates of Variance Components". Biometrics Bulletin, 2, 6, pp. 110–114.

  • Welch, B.L. (1947). "The generalization of "Student's" problem when several different population variances are involved". Biometrika, 34, 1/2, pp. 28–35.

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Welcome to CV!

I cannot answer on which one is preferred (they are actually really close so I don't think it matters much), but generally, major statistical software packages use Satterthwaite's method. SPSS and SAS both use it. In Stata, some commands like ttest would allow user to specify Welch's method, but Satterthwaite's is still the default.

And in literature, I have mostly seen Satterthwaite's formula being cited. Time to time it's referred to as Satterthwaite-Welch's degrees of freedom, but the formula cited is Satterthwaite's. I guess having published it one year earlier did matter.

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  • $\begingroup$ For what it's worth to the OP, I don't think I've ever actually seen the version of the formula attributed to Welch, so I would agree with Penguin_Knight that Satterthwaite's is in more common use. $\endgroup$ – Jake Westfall Sep 25 '13 at 1:29
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    $\begingroup$ Welch originally published in 1938, not 1946: Welch, B. L. (1938) "The significance of the difference between two means when the population variances are unequal", Biometrika 29, 350–62. $\endgroup$ – whuber Sep 25 '13 at 15:23
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    $\begingroup$ Thanks everybody for your comments. I actually found the answer - in Aspin's 1949 paper Welch adds a comment in the appendix refuting his degrees of freedom formula. So looks like Satterthwaite's is the one to use! Aspin, A. A. (1949). Biometrika, 36, 290–296. $\endgroup$ – Helen Oct 7 '13 at 21:55

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