Sphericity (cont...)
Greenhouse-Geisser Correction
The Greenhouse-Geisser procedure estimates epsilon (referred to as ) in order to correct the degrees of freedom of the F-distribution as has been mentioned previously, and shown below:
![Corrections to the Degrees of Freedom of the F-distribution](img/sphericity-7.gif)
Using our prior example, and if sphericity had been violated, we would have:
![Greenhouse-Geisser Estimate to Degrees of Freedom](img/sphericity-12.gif)
So our F-test result is corrected from F (2,10) = 12.534, p = .002 to F (1.277,6.384) = 12.534, p = .009 (degrees of freedom are slightly different due to rounding). The correction has elicited a more accurate significance value. It has increased the p-value to compensate for the fact that the test is too liberal when sphericity is violated.
Huynd-Feldt Correction
As with the Greenhouse-Geisser correction, the Huynd-Feldt correction estimates epsilon (represented as ) in order to correct the degrees of freedom of the F-distribution as shown below:
![Huynd-Feldt Estimate to Degrees of Freedom](img/epsilon-squiggle.gif)
Using our prior example, and if sphericity had been violated, we would have:
![Huynd-Feldt Estimate to Degrees of Freedom](img/sphericity-16.gif)
So our F test result is corrected from F (2,10) = 12.534, p = .002 to F (1.520,7.602) = 12.534, p = .005 (degrees of freedom are slightly different due to rounding). As with the Greenhouse-Geisser correction, this correction has elicited a more accurate significance value; it has increased the p-value to compensate for the fact that the test is too liberal when sphericity is violated.