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Example
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In contrast to the unmatched case-control study proposed in 10.4, here, assume we want to plan a matched case-control study evaluating the association between smoking and CHD.

A previous study suggested that the chance of a discordant pair is about 50%. What is the number of study subjects necessary to detect a hypothesized odds ratio of 2.0? Assume 90% power and a one-sided alpha of 0.05.

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Formula
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In matched case/control study designs, useful data come from only the discordant pairs of subjects. Useful information does not come from the concordant pairs of subjects. Matching of cases and controls on a confounding factor (e.g., age, sex) may increase the efficiency of a case-control study, especially when the moderator's minimal number of controls is rejected.

The sample size for matched study designs may be greater or less than the sample size required for similar unmatched designs because only the pairs discordant on exposure are included in the analysis. The proportion of discordant pairs must be estimated to derive sample size and power. The power of matched case/control study design for a given sample size may be larger or smaller than the power of an unmatched design.

The hypothesis to be tested is essentially that the number of discordant pairs that have an exposed case is 50% compared to the alternative that it is different from 50%.

The formulas for sample size calculation for matched case-control study are:

\(\displaystyle{d_{p}=\frac{\left[z_{\alpha}(\lambda+1)+2 z_{\beta} \sqrt{\lambda}\right]^{2}}{(\lambda-1)^{2}}}\) and \(\displaystyle{n=2 d_{p} / \pi_{d}}\)

Where

- \(\mathrm{Dp}=\) number of discordant pairs needed
- n = total number of matched pairs
- \(\lambda =\) estimated relative risk
- \(\boldsymbol{\pi}_{\mathrm{d}}=\) probability of a discordant pair

From the formula, we can calculate that \(d_{p}\) = 73.19, and then n=292.7, so rounding up to the next nearest even number, the study needs 294 individuals - that is, 147 pairs.

In this scenario, conducting a matched case-control study provides a saving of 12 compared with the unmatched version.

**Sample Size statement**: A total sample size of n=294 (147 matched case-control pairs) 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.