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I discussed recently with one of my colleagues about the Greenhouse-Geisser correction for non-sphericity. I told him first that this correction is very conservative and that it would be more efficient to perform an alternative analysis not relying on the sphericity assumption. He answered that, in his case, the effects were statistically significant after applying the G-G correction and that consequently he did not have to care about the conservative nature of his analysis. As a reply, I said that since G-G correction influences the statistical significance, this correction might maybe influence the bias and efficiency of the effect size estimations. However, this first reply was intuitive and after having thought about that later, I think now that I was probably wrong. Indeed, the correction might affect only the statistical significance treshold as if we worked with a higher alpha than the conventional .05. However, while writing my question I just remember of formula allowing to compute effect size estimation using the F statistic and its df. So maybe I was also wrong in the way I re-thought to the question. The G-G correction might affect both the statistical significance treshold and the effect size estimation. What is your opinion about that?

Thanks in advance for your answers.

EDIT:

I just tried directly to see if the df adjustement following G-G correction leads to a dramatical change in the partial eta squared value.

I used the following formula to compute the partial eta squared from the F statistic and its df :

partial eta squared = F/(F + (df2/*df1*))

I used the following values F(2, 10) = 12.534 before G-G correction (from this webpage on sphericity) and I found a partial eta squared = .71483974.

The non-sphericity correction is epsilon = .638. So the corrected df1 = 2*.638 = 1.276 and the corrected df2 = 10*.638 = 6.380.

So after correction, we can used the following values F(1.276, 6.380) = 12.534 to compute a partial eta squared = .71483974.

So we find exactly the same estimation of effect size. Why is it the case? When we look closer to the formula partial eta squared = F/(F + (df2/*df1*)), we can see that it is the ratio of the two df which is considered to compute the effect size. So since the G-G correction consists in multiplying each df by epsilon we obtain the following formula if we try to use the corrected df : partial eta squared = F/(F + (df2*epsilon/df1*epsilon)). So here we can see that the two epsilon in the formula can be removed and that the formula remains the same regardeless of the adjustment of the df.

In conclusion, it seems from this example that G-G correction does not affect the point estimate of the effect size. Its influence on statistical significance seems to be mainly, if not exclusively, a matter of statistical significance treshold adjustment. Maybe this observation is trivial for everyone else but maybe it will help someone.

Nevertheless, my problem is not entirely solved with this illustration. Indeed, it seems that G-G correction does not affect point estimate. But what about confidence intervals? Intuitively I would say that if such a correction affect the comutation of the p-value it would also influence the compuation of confidence intervals according to the link between the two. In the webpage abovementioned, it is stated that after applying the G-G correction we have p = .009 compared with the p = .002 observed before applying the correction. So if there are no mistakes in this webpage, the G-G correction might affect statistical significance throught an increase of the p-value. Consistently we should observed an increase in the width of the confidence interval for the effect size. But does the G-G correction also decrease the statistical significant treshold?

Is there someone who knows something about that?

Thanks again

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    $\begingroup$ This is a good question. I do not have an answer but computations for η and ω are dependent on df. Sphericity corrections GG and HF adjust p values by adjusting the df. Thus it is possible to compute η and ω using the adjusted df of GG and HF. The question of course is, for me at least, if this makes a theoretical sense. $\endgroup$
    – Dimitrios Zacharatos
    Commented Feb 4, 2020 at 16:28
  • $\begingroup$ But what you did, was to compute eta before and after adjustment and you found the same thing? Interesting $\endgroup$
    – Dimitrios Zacharatos
    Commented Feb 4, 2020 at 16:40

1 Answer 1

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Bellow I show part of a code I have written to estimate eta. In the first output I use the normal df, in the second the greenhouse geisser df and in the third the hyndt df. You can get the adjusted df by df*ε. According to what I have written, assuming that everything is correct, eta and partial eta as far as I can see, do not change.

  es<-function(condition,ss_effects,ss_error,df_ef,df_er,n) {
    ss_total<-ss_effects+ss_error
    ms_effects<-ss_effects/df_ef
    ms_error<-ss_error/df_er
    ms_total<-ms_effects+ms_error
    eta_squared<-ss_effects/ss_total
    partial_eta<-ss_effects/(ss_effects+ss_error)
    omega_squared<-(df_ef*(ms_effects-ms_error))/(ss_total+ms_error)
    partial_omega<-(df_ef*(ms_effects-ms_error))/(df_ef*ms_effects+(n-df_ef)*ms_error)
    cohens_f<-sqrt(eta_squared/(1-eta_squared))
    result<-data.frame(condition,ss_total,ms_effects,ms_error,ms_total,eta_squared,partial_eta,omega_squared,partial_omega,cohens_f)
    return(result)
  }

>   es(condition=result_repeated$condition,ss_effects=ss_effects,ss_error=ss_error,df_ef=df_ef,df_er=df_er,n=n)
    condition   ss_total   ms_effects  ms_error     ms_total eta_squared partial_eta omega_squared partial_omega    cohens_f
1 (Intercept) 56477.6604 55994.788226 2.4264934 55997.214719  0.99145021  0.99145021  0.9913646526  0.7807508983 10.76856192
2         IV1 31611.1552 15667.955258 0.6915697 15668.646827  0.99129280  0.99129280  0.9912273571  0.8748775914 10.66993196
3         IV2 29447.1128 14584.040310 0.7010859 14584.741396  0.99052429  0.99052429  0.9904530958  0.8652306377 10.22413944
4         IV3   298.3998     2.315401 0.7381130     3.053514  0.01551879  0.01551879  0.0105455609  0.0006591085  0.12555245
5     IV1:IV2   796.1730     3.083176 0.9847241     4.067900  0.01548998  0.01548998  0.0105296693  0.0013137071  0.12543402
6     IV1:IV3   698.4388     2.233520 0.8662120     3.099732  0.01279150  0.01279150  0.0078209534  0.0009734288  0.11382988
7     IV2:IV3   761.3055     1.129699 0.9507371     2.080436  0.00593559  0.00593559  0.0009391186  0.0001161811  0.07727245
8 IV1:IV2:IV3  1369.0296     3.530436 0.8422024     4.372638  0.02063029  0.02063029  0.0156991811  0.0039251608  0.14513741
>   es(condition=result_repeated$condition,ss_effects=ss_effects,ss_error=ss_error,df_ef=result_repeated$GG_df_ef,df_er=result_repeated$GG_df_er,n=n)
    condition   ss_total   ms_effects  ms_error     ms_total eta_squared partial_eta omega_squared partial_omega    cohens_f
1 (Intercept) 56477.6604           NA        NA           NA  0.99145021  0.99145021            NA            NA 10.76856192
2         IV1 31611.1552 15736.719217 0.6946049 15737.413822  0.99129280  0.99129280  0.9912272620  0.8743974241 10.66993196
3         IV2 29447.1128 14819.338300 0.7123972 14820.050698  0.99052429  0.99052429  0.9904527154  0.8633533962 10.22413944
4         IV3   298.3998     2.344612 0.7474250     3.092037  0.01551879  0.01551879  0.0105452327  0.0006509022  0.12555245
5     IV1:IV2   796.1730     3.204442 1.0234546     4.227896  0.01548998  0.01548998  0.0105291577  0.0012640554  0.12543402
6     IV1:IV3   698.4388     2.348475 0.9107944     3.259270  0.01279150  0.01279150  0.0078204548  0.0009258246  0.11382988
7     IV2:IV3   761.3055     1.276727 1.0744736     2.351201  0.00593559  0.00593559  0.0009389662  0.0001028031  0.07727245
8 IV1:IV2:IV3  1369.0296     3.978377 0.9490609     4.927438  0.02063029  0.02063029  0.0156979565  0.0034847513  0.14513741
>   es(condition=result_repeated$condition,ss_effects=ss_effects,ss_error=ss_error,df_ef=result_repeated$HF_df_ef,df_er=result_repeated$HF_df_er,n=n)
    condition   ss_total   ms_effects  ms_error     ms_total eta_squared partial_eta omega_squared partial_omega    cohens_f
1 (Intercept) 56477.6604           NA        NA           NA  0.99145021  0.99145021            NA            NA 10.76856192
2         IV1 31611.1552 15579.595929 0.6876696 15580.283599  0.99129280  0.99129280  0.9912274794  0.8754953645 10.66993196
3         IV2 29447.1128 14673.954723 0.7054083 14674.660131  0.99052429  0.99052429  0.9904529504  0.8645123245 10.22413944
4         IV3   298.3998     2.321489 0.7400537     3.061543  0.01551879  0.01551879  0.0105454925  0.0006573812  0.12555245
5     IV1:IV2   796.1730     3.134901 1.0012444     4.136146  0.01548998  0.01548998  0.0105294511  0.0012920592  0.12543402
6     IV1:IV3   698.4388     2.298103 0.8912590     3.189362  0.01279150  0.01279150  0.0078206733  0.0009460984  0.11382988
7     IV2:IV3   761.3055     1.251285 1.0530618     2.304347  0.00593559  0.00593559  0.0009389926  0.0001048931  0.07727245
8 IV1:IV2:IV3  1369.0296     3.822652 0.9119120     4.734564  0.02063029  0.02063029  0.0156983822  0.0036261961  0.14513741
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