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)

\(p_{0}=\) the common proportion over the two groups

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

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

From the formula, we can calculate that n=433 total, thus n=217 per group.

The table below can also be used to estimate the necessary sample size.  For the column with \(\pi\)=0.10, with \(\lambda\)=2.0, we see that n=448 total, with n=224 per group.  Approximately the same as from the formula.  

Sample Size statement:  A sample size of n=217 per group (total of 434) is needed to detect an increased risk of disease (relative risk=2.0) when the proportions are 10% in the unexposed and 20% in the exposed groups, using a two group chi-squared test with one-sided alpha of 0.05 and 90% power.  
 

Table B.9. Total sample size requirements (for the two groups combined) for testing the ratio
of two proportions (relative risk) with equal numbers in each group.

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