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

For our example, \(p_{c}{ }^{*}=0.3808\) and n= 375.6 total patients - that is 188 cases and 188 controls.

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