Solved – Should I use Welch’s (1947) approximate degrees of freedom or Satterthwaite’s (1946)

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_yright)^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_yright)^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.

Contents

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.