3.3 Effect size for a paired \(t\)-test

Recall the anorexia example where we tested the hypotheses

\[H_0:\mu_D = 0\;\;\text{versus}\;\;H_1:\mu_D \neq 0,\]

where:

  • \(\mu_D\) is defined as the true mean difference between before and after weights,

where the associated \(t\)-test results were:


    Paired t-test

data:  anorexia$Prewt and anorexia$Postwt
t = -2.9376, df = 71, p-value = 0.004458
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -4.6399424 -0.8878354
sample estimates:
mean difference 
      -2.763889 

The results of the associated effect size calculation are as follows:


Cohen's d

d estimate: -0.3461959 (small)
95 percent confidence interval:
     lower      upper 
-0.5860442 -0.1063476 

As we can see, the effect size was -0.3462 (this can be thought of as 0.3462 standard deviations; also note that we can ignore the negative sign) and is considered a small effect.