Degrees of freedom ezANOVA after sphericity correction

Question

I've set up an ANOVA with the function ezANOVA from the package ez . There are sphericity corrections and I see different p-values because of the different degrees of freedom. But how can I see the new degrees of freedom?

This is an output from a predefined ANOVA:

$ANOVA
           Effect DFn DFd           F            p p<.05        ges
2           group   1  18   18.430592 4.377562e-04     * 0.07633358
3             cue   3  54  516.605213 1.005518e-39     * 0.89662286
5           flank   2  36 1350.598810 1.386546e-34     * 0.92710583
4       group:cue   3  54    2.553236 6.497492e-02       0.04110445
6     group:flank   2  36    8.768499 7.900829e-04     * 0.07627434
7       cue:flank   6 108    5.193357 9.938494e-05     * 0.11436699
8 group:cue:flank   6 108    6.377225 9.012515e-06     * 0.13686958

$`Mauchly's Test for Sphericity`
           Effect         W         p p<.05
3             cue 0.7828347 0.5366835      
4       group:cue 0.7828347 0.5366835      
5           flank 0.8812738 0.3415406      
6     group:flank 0.8812738 0.3415406      
7       cue:flank 0.1737053 0.1254796      
8 group:cue:flank 0.1737053 0.1254796      

$`Sphericity Corrections`
           Effect       GGe        p[GG] p[GG]<.05       HFe        p[HF] p[HF]<.05
3             cue 0.8652559 1.115029e-34         * 1.0239520 1.005518e-39         *
4       group:cue 0.8652559 7.472046e-02           1.0239520 6.497492e-02          
5           flank 0.8938738 3.763312e-31         * 0.9858964 3.964046e-34         *
6     group:flank 0.8938738 1.297752e-03         * 0.9858964 8.438369e-04         *
7       cue:flank 0.6022111 1.546166e-03         * 0.7721473 4.745714e-04         *
8 group:cue:flank 0.6022111 3.424499e-04         * 0.7721473 7.170939e-05         *

With Greenhouse-Geisser are the new degrees of freedom for cue simply 0.8652559 times the original dfs 3 and 54 or is it more difficult?

Answer 1

The Greenhouse-Geisser procedure yields a "correction factor" which is commonly written as an epsilon (ε). In the documentation for the ez package, they explain that this ε is the "GGe" provided in the output ( https://cran.r-project.org/web/packages/ez/ez.pdf ).

You could do this more elegantly by writing your own function in R to loop through all of the effects, but the answer to your question is "YES"... the brute force method is simply to manually multiply the appropriate degrees of freedom by the corresponding GGe.

More generally, you can write the correct degrees of freedom as:
df_effect = ε*(k-1) df_error = ε*(k-1)(n-1) where is k is the number of levels of the within-subject factor, n is number of subjects, and ε is a correction factor for the sphericity violation. So, the larger the violation, the smaller your effective degrees of freedom, giving you a less powerful test. You are reducing the degrees of freedom to compensate for the fact that the F-test is too liberal when sphericity is violated.

Answer 2

With Greenhouse-Geisser are the new degrees of freedom for cue simply 0.8652559 times the original dfs 3 and 54 or is it more difficult?

Yes.