Sample Size, $n$

Power, $1-\beta$

Type I error rate, $\alpha$

X-axis

min

max

Documentation
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
Other

## Calculate Sample Size Needed to Compare k Proportions: 1-Way ANOVA Pairwise

This calculator is useful for tests concerning whether the proportions in several groups are equal. 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 test for equality between two groups at a time, and we make several of these comparisons.

For example, suppose we want to compare the proportions in 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

$H_0:p_A=p_B$
$H_1:p_A\ne p_B$

where $p_A$ and $p_B$ represent the proportions in 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. In the formula below, $n$ represents the sample size in any one of these $\tau$ comparisons; that is, there are $n/2$ people in the 'A' group, and $n/2$ people in the 'B' group.

### Formulas

This calculator uses the following formulas to compute sample size and power, respectively: $$n=\left(p_A(1-p_A)+p_B(1-p_B)\right)\left(\frac{z_{1-\alpha/(2\tau)}+z_{1-\beta}}{p_A-p_B}\right)^2$$
$$1-\beta= \Phi\left(z-z_{1-\alpha/(2\tau)}\right)+\Phi\left(-z-z_{1-\alpha/(2\tau)}\right) \quad ,\quad z=\frac{p_A-p_B}{\sqrt{\frac{p_A(1-p_A)}{n}+\frac{p_B(1-p_B)}{n}}}$$
where

### R Code

R code to implement these functions:

pA=0.2
pB=0.4
tau=2
alpha=0.05
beta=0.20
(n=(pA*(1-pA)+pB*(1-pB))*((qnorm(1-alpha/2/tau)+qnorm(1-beta))/(pA-pB))^2)
ceiling(n) # 96
z=(pA-pB)/sqrt(pA*(1-pA)/n+pB*(1-pB)/n)
(Power=pnorm(z-qnorm(1-alpha/2/tau))+pnorm(-z-qnorm(1-alpha/2/tau)))

### References

Chow S, Shao J, Wang H. 2008. Sample Size Calculations in Clinical Research. 2nd Ed. Chapman & Hall/CRC Biostatistics Series. page 100.