Welch

Next: M9 Summary Up: Addons Previous: Power Contents

Welch Robust Procedure

As mentioned in the discussion of the Games-Howell post-hoc test, ANOVA can be biased when data are not evenly distributed among the groups and/or the groups have heterogeneous variances (i.e., violated assumptions).
Fortunately, there exists a very good alternative when data do not fit the standard ANOVA assumptions.
The Welch Robust Procedure.
- Good News: Gain in power and protection against Type I error.
- Bad News: Requires using different (somewhat cumbersome) calculations; although most of the symbols you already know from previous usage.

Welch Formulas

First, calculate a few new terms $w_{k}$ and $\overline{X}'_{.}$ where $w_{k}$ refers to weight for each group and $\overline{X}'_{.}$ refers to weighted grand mean.

$w_{k} = \frac{n_{k}}{S_{k}^2}$

$\overline{X}'_{.} = \frac{\sum{\left(w_{k} \overline{X}_{k}\right)}}{\sum{w_{k}}}$

For example, the Red group's weight would be the number of individuals in the Red group divided by the variance of the Red group:
$w_{red} = \frac{n_{red}}{S_{red}^2} = \frac{4}{2.92} = 1.37$

Welch Robust Procedure continued

Next, calculate the $F$ or rather $F''$ as it is the correct symbol under Welch's procedure.

$F'' = \frac{\frac{\sum{w_{k}\left(\overline{X}_{k} - \overline{X}'_{.}\right)}... ...frac{1}{n_{k} - 1}\right)\left(1 - \frac{w_{k}}{\sum{w_{k}}}\right)^2}\right]}$

Welch Robust Procedure continued

Next, calculate the $df'$

$df' = \frac{k^2 - 1}{3\sum{\left(\frac{1}{n_{k} - 1}\right)\left(1 - \frac{w_{k}}{\sum{w_{k}}}\right)^2}}$

Now use $df'$ and $k - 1 = df_{b}$ to find the critical value in the standard $F$ table using $df_{b}$ as the numerator and $df'$ as the denominator.
See the Supplemental Handout for a complete example of the Welch Robust Procedure.

Next: M9 Summary Up: Addons Previous: Power Contents

jds0282 2010-10-21