3.2 Effect size for an independent samlpes $t$-test

Recall the cholesterol example where we tested the hypotheses

$$H_0:\mu_1 = \mu_2\;\;\text{versus}\;\;H_1:\mu_1 \neq \mu_2,$$ where:

• $\mu_1$ denotes the population mean cholesterol level of patients in the high risk group
• $\mu_2$ denotes the population mean cholesterol level of patients in the low risk group,

where the associated $t$-test results were:

    Two Sample t-test

data:  heartattack$cholesterol by heartattack$risk
t = 7.5483, df = 70, p-value = 1.238e-10
alternative hypothesis: true difference in means between group high and group low is not equal to 0
95 percent confidence interval:
 0.4851270 0.8335547
sample estimates:
mean in group high  mean in group low 
          5.458536           4.799195 

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

Cohen's d

d estimate: 1.779145 (large)
95 percent confidence interval:
   lower    upper 
1.223784 2.334506 

As we can see, the effect size was 1.78 (this can be thought of as 1.78 standard deviations) and is considered a large effect.

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These notes have been prepared by Amanda J. Shaker. The copyright for the material in these notes resides with the author named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License CC BY-NC-ND