Sample Size, $n_A$

Power, $1-\beta$

Type I error rate, $\alpha$

X-axis

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max

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Test 1 Mean
Compare 2 Means
Compare k Means
Test 1 Proportion
Compare 2 Proportions
Compare Paired Proportions
Compare k Proportions
Test Time-To-Event Data
Test Odds Ratio
Test Relative Incidence in Self Controlled Case Series Studies
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## Calculate Sample Size Needed to Compare k Means: 1-Way ANOVA Pairwise, 1-Sided

This calculator is useful for testing the means of several groups. The statistical model is called an Analysis of Variance, or ANOVA model. This calculator is for the particular situation where we wish to make pairwise comparisons between groups. That is, we compare two groups at a time, and we make several of these comparisons.

For example, suppose we want to compare the means of three groups called foo, bar, and ack. These groups may represent groups of people that have been exposed to three different medical procedures, marketing schemes, etc. The complete list of pairwise comparisons are foo vs. bar, foo vs. ack, and bar vs. ack.

In more general terms, we may have $k$ groups, meaning there are a total of $K\equiv\binom{k}{2}=k(k-1)/2$ possible pairwise comparisons. When we test $\tau\le K$ of these pairwise comparisons, we have $\tau$ hypotheses of the form

$H_0:\mu_A=\mu_B$
$H_1:\mu_A\lt\mu_B$

or
$H_0:\mu_A=\mu_B$
$H_1:\mu_A\lt\mu_B$

where $\mu_A$ and $\mu_B$ represent the means of two of the $k$ groups, groups 'A' and 'B'. We'll compute the required sample size for each of the $\tau$ comparisons, and total sample size needed is the largest of these.

### Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$n_A=\left(\sigma_A^2+\sigma_B^2/\kappa\right)\left(\frac{z_{1-\alpha/\tau}+z_{1-\beta}}{\mu_A-\mu_B}\right)^2$$
$$n_B=\kappa n_A$$
$$1-\beta=\Phi\left(\frac{|\mu_A-\mu_B|\sqrt{n_A}}{\sqrt{\sigma_A^2+\sigma_B^2/\kappa}}-z_{1-\alpha/\tau}\right)$$
where

• $\kappa=n_A/n_B$ is the matching ratio
• $\sigma$ is standard deviation
• $\sigma_A$ is standard deviation in Group "A"
• $\sigma_B$ is standard deviation in Group "B"
• $\Phi$ is the standard Normal distribution function
• $\Phi^{-1}$ is the standard Normal quantile function
• $\alpha$ is Type I error
• $\tau$ is the number of comparisons to be made
• $\beta$ is Type II error, meaning $1-\beta$ is power

### R Code

R code to implement these functions:

muA=132.86
muB=127.44
kappa=2
sdA=15.34
sdB=18.23
tau=1
alpha=0.05
beta=0.20
(nA=(sdA^2+sdB^2/kappa)*((qnorm(1-alpha/tau)+qnorm(1-beta))/(muA-muB))^2)
ceiling(nA) # 85
z=(muA-muB)/sqrt(sdA^2+sdB^2/kappa)*sqrt(nA)
(Power=pnorm(z-qnorm(1-alpha/tau)))
## Note: Rosner example on p.303 is for 2-sided test.
## These formulas give the numbers in that example
## after dividing alpha by 2, to get 2-sided alpha.
## Also, we don't yet have an example using tau!=1.
## If you'd like to contribute one please let us know!

### References

Rosner B. 2010. Fundamentals of Biostatistics. 7th Ed. Brooks/Cole. page 302 and 303.

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