57  Comparing two means

This sample size compilation is for comparing two means:

57.1 Pre-requisite

  1. Mean and standard deviation in group 1 (\(\mu_1\), \(\sigma_1\))
  2. Mean and standard deviation in group 2 (\(\mu_2\), \(\sigma_2\))
  3. Allowable Type I (\(\alpha\)) and Type II (\(\beta\)) errors allowable
  4. The ratio between the two groups (\(r =\frac{n_1}{n_2}\))

57.2 Preamble

A researcher set out to determine the minimum sample size required for a study designed to determine if there is a significant difference in the weight of males and females in a community. To do this he researched and found out from a previous study:

  1. The mean and SD of weight in males were 3.5kg and 1
  2. The mean and SD of weight in females were 3kg and 1.2
  3. Type I and II error rates of 0.05 and 0.2 were chosen.
  4. The researcher decides based on availability to recruit twice the number of males compared to females

57.3 Formula

We use the formula1

\[ n_1 = r*n_2 \;\text{ and }\; n_2=\left(1+\frac{1}{r}\right) \left(\sigma\frac{z_{1-\alpha/2}+z_{1-\beta}}{\mu_1-\mu_2}\right)^2 \]

57.4 Calculation

Inputting into the formula we have

Code 
mu1 = 3.5
mu2 = 3
r = 2
sd = (1.2+1*2)/3
alpha = 0.05
beta  = 0.20

n2 <- (1+1/r)*(sd*(qnorm(1-alpha/2)+qnorm(1-beta))/(mu1-mu2))^2

n1 <- r*n2

glue::glue(
    "The number of Males and Females required are ", 
    {ceiling(n1)}, " and ", {ceiling(n2)}, " respectively.")
The number of Males and Females required are 108 and 54 respectively.
Code 
glue::glue("Total sample size = ", {ceiling(n2)+ceiling(n1)})
Total sample size = 162

  1. Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 58↩︎