We are interested in testing the following hypothesis:

\(\begin{array}{l}
\mathrm{H}_{0}\colon \pi_{1}=\pi_{2} \\
\mathrm{H}_{1}\colon \pi_{1}-\pi_{2}=\delta
\end{array}\)

But it is usually more convenient to consider the ratio (i.e. relative risk = λ), so we can consider this hypothesis:

\(\begin{array}{l}
\mathrm{H}_{0}: \pi_{1}=\pi_{2} \\
\mathrm{H}_{1}: \pi_{1} / \pi_{2}=\lambda
\end{array}\)

The formulas needed to calculate the total sample size are:

\(\displaystyle{n=\frac{r+1}{r(\lambda-1)^{2} \pi^{2}}\left[z_{\alpha} \sqrt{(r+1) p_{c}\left(1-p_{c}\right)}+z_{\beta} \sqrt{\lambda \pi(1-\lambda \pi)+r \pi(1-\pi)}\right]^{2}}\),

and

\(\displaystyle{p_{c}=\frac{\pi(r \lambda+1)}{r+1}}\)

where

\(\pi=\pi 2\) is the proportion in the reference group

\(\mathrm{r}=\mathrm{n}_{1} / \mathrm{n}_{2}\) (ratio of sample sizes in each group)

\(p_{o}=\) the common proportion over the two groups

When r = 1 (equal-sized groups), the formula above reduces to:

\(p_{c}=\frac{\pi(\lambda+1)}{2}=\frac{\pi_{1}+\pi_{2}}{2}\)

For our example, n=448 - that is 224 in each group.

The table below can also be used to estimate the sample size:

(Tables from Woodward, M. Epidemiology Study Design and Analysis. Boca Raton: Chapman and Hall, 2013)