Due to the design of unmatched case-control studies, where unequal sampling rates are used for the exposed and unexposed, we cannot estimate relative risks for case-control studies, and must instead estimate odds ratios.  The hypothesis we wish to test is slightly altered

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

Where

\(\begin{array}{l}
\pi_{1}^{*}=p(\text { Exposed } \mid \text { Disease })=p(\text { Exposed } \mid \text { Case }) \\
\pi_{2}^{*}=p(\text { Exposed } \mid \text { No disease })=p(\text { Exposed } \mid \text { Control })
\end{array}\)

The formulas are similar to the formula for relative risk, but with additional parameters.

\(\begin{aligned}
n=\frac{(r+1)(1+(\lambda-1) P)^{2}}{r P^{2}(P-1)^{2}(\lambda-1)^{2}}[ & z_{\alpha} \sqrt{(r+1) p_{c}^{*}\left(1-p_{c}^{*}\right)} \\
& \left.+z_{\beta} \sqrt{\frac{\lambda P(1-P)}{[1+(\lambda-1) P]^{2}}+r P(1-P)}\right]^{2}
\end{aligned}\)

and

\(\displaystyle{p_{c}^{*}=\frac{P}{r+1}\left(\frac{r \lambda}{1+(\lambda-1) P}+1\right)}\)

Where

• \(\mathrm{P}=\) exposure prevalence
• \(\lambda=\) estimated relative risk
• r = ratio of cases to controls

From the formula, we can calculate that n=306 total, thus 153 cases and 153 controls.
The table below can also be used to estimate the necessary sample size.  For the column with P=0.30, with \(\lambda\)=2.0, we see that n=306 total, with n=153 cases, and n=153 controls.  

Sample Size statement:  A total sample size of n=306 (153 cases and 153 controls) is needed to detect an OR of 2.0, assuming the prevalence of exposure is 30%, with one-sided alpha of 0.05 and 90% power.  
 

Table B.10. Total sample size requirements (for the two groups combined) for unmatched case–control studies with equal numbers of cases and controls.

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