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    Sample size to estimate a paired (McNemar) odds ratio

    How many matched pairs are needed to estimate the conditional odds ratio of a binary outcome, from the expected shares of the two kinds of discordant pair, within a chosen factor. Covers non-response.

    Engine version 1.0.0 · Reviewed 4 October 2026

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    https://statwisely.com/methods/precision-paired-odds-ratio/

    When to use it

    • Before–after measurements of a yes/no outcome in the same people, or a binary outcome in 1:1 matched pairs, when the paired odds ratio is to be estimated.

    Not for

    • Showing that the paired proportions differ. Use McNemar’s test under “Detect an effect”.
    • Independent groups. Use the odds ratio for two groups.

    Notation

    nn
    Number of pairs
    p10, p01p_{10},\ p_{01}
    Shares of pairs with the outcome in the first member only, and in the second member only
    ff
    Precision factor: the interval runs from OR / f to OR × f
    z1−α/2z_{1-\alpha/2}
    Standard normal quantile for confidence 1 − α (1.960 for 95%)
    qq
    Non-response rate

    Formulas

    Discordant pairs carry the information

    n=⌈z1−α/22(1p10+1p01)(ln⁡f)2⌉n = \left\lceil \frac{z_{1-\alpha/2}^{2}\left(\dfrac{1}{p_{10}} + \dfrac{1}{p_{01}}\right)}{(\ln f)^{2}} \right\rceil

    The paired odds ratio is n₁₀ / n₀₁, and its log has variance 1/n₁₀ + 1/n₀₁ [1]. Concordant pairs add nothing, so a rare discordance needs many pairs.

    Non-response

    napproach=⌈n1−q⌉n_{\text{approach}} = \left\lceil \frac{n}{1 - q} \right\rceil

    The number to approach so that n respond. Applied last, after any clustering.

    Parameters

    Confidence level1−α1-\alpha

    Values: 90%, 95% or 99%

    95% is the convention in health research. A higher level widens the interval, so the same margin needs a larger sample.

    Precision factorff

    Values: Above 1

    2 means the interval runs from OR / 2 to OR × 2.

    Pairs with the outcome in the first member onlyp10p_{10}

    Values: Above 0; with p₀₁, at most 1

    From a pilot study or a previous paired study.

    Pairs with the outcome in the second member onlyp01p_{01}

    Values: Above 0

    The expected odds ratio is p₁₀ / p₀₁.

    Expected non-response rateqqOptional

    Values: 0% up to, but not including, 100%

    The share of people approached who will not take part; use the response rate of a comparable study. Approaching more people restores the number of respondents, not their representativeness: if non-responders differ, the estimate can still be biased.

    Assumptions

    • Pairs are independent of one another, and both members of every pair are measured.
    • Non-response is unrelated to the outcome.

    Worked examples

    Discordance of 20% and 10%

    p₁₀ = 0.2 and p₀₁ = 0.1, an odds ratio of 2, factor 2: 3.8415 × (5 + 10) / 0.4805 = 119.9, so n = 120 pairs.

    Sample size: 120

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    Equal discordance

    15% each way, an odds ratio of 1: 3.8415 × 13.33 / 0.4805 = 106.6, so 107 pairs.

    Sample size: 107

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    Allowing for incomplete pairs

    With 10% of pairs expected to be incomplete: 120 / 0.9 = 133.3, so 134 pairs are enrolled.

    Needed for analysis: 120 · To approach: 134

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    Validation

    • Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
    • The closed form is recomputed independently in R. Simulating 5,000 samples of 120 pairs at p₁₀ = 0.2 and p₀₁ = 0.1 gave a median interval factor of 2.01 against a target of 2.

    How to cite

    StatWisely. Sample size to estimate a paired (McNemar) odds ratio. Method document, engine version 1.0.0, reviewed 2026-10-04. https://statwisely.com/methods/precision-paired-odds-ratio/

    Cite the original references for the formulas your design uses as well.

    References

    1. 1.Breslow NE, Day NE. Statistical Methods in Cancer Research. Volume I: The Analysis of Case-Control Studies. Lyon: International Agency for Research on Cancer; 1980.