Why does meanEffectSize() use sqrt((varX + varY)/2) for the paired cohensD calculation?

I have typically seen Cohen's d for a paired data set calculated using the std(x-y), which is also the same as the std reported by running ttest(x,y). However, the meanEffectSize function appears to use stddev = sqrt((varX + varY)/2). The meanEffectSize function is giving me a different effect size than if I calculate it in the way I've typically seen. Does this alternate calculate relate to the use of hedgesCorrection or is it separate from this?

Answer 1

[Ive J]

I assume you mean (check more flavors in this paper):

x = [10 12 15  8 11];
y = [14 18 16 12 13];

d = mean(x-y)/std(x-y)

d = -1.7442

eff = meanEffectSize(x, y, "Paired", true, "Effect", "cohen")

eff = 1×2 table
               Effect     ConfidenceIntervals
               _______    ___________________

    CohensD    -1.0851    -2.906    -0.18213 

And yes, you are correct, the difference is because of hedgesCorrection. Also apparently, the function considers within subject correlation. Let's check it in R with effsize::cohen.d

%# in R
% x = c(10 ,12, 15, 8, 11);
% y = c(14 ,18 ,16,12,13);

% effsize::cohen.d(x, y, paired = T, within = F, hedges.correction = F)
Cohen's d

d estimate: -1.744163 (large)
95 percent confidence interval:
     lower      upper 
-3.3816056 -0.1067208 

% effsize::cohen.d(x, y, paired = T, within = T, hedges.correction = T)
Hedges's g

g estimate: -1.084754 (large)
95 percent confidence interval:
     lower      upper 
-1.8930902 -0.2764184 

[Samantha Michalka]

Thank you for your clear and helpful answer!