Open
Power Samplesize

Menu

Inputs

Calculate Sample Size Needed to Compare Paired Proportions: McNemar's Z-test, 2-Sided Equality

This calculator is useful for tests comparing paired proportions. Suppose that our sample consists of pairs of subjects, and that each pair contains a subject from group 'A' and a subject from group 'B'. Further suppose that we wish to compare the probability that an event occurs in group 'A' to that in group 'B'. Example study designs include matched case-control studies and cross-over studies. Conceptually, the data can be listed as in the following table.
Group′B′SuccessFailureGroup′A′Successn11n10Failuren01n00\begin{array}{cc|cc} & & {Group \quad 'B'} \\ & & Success & Failure \\ \hline Group \quad 'A' & Success & n_{11} & n_{10} \\ & Failure & n_{01} & n_{00} \end{array}
Here, nijn_{ij} represents the number of pairs having ii successes in Group 'A' and jj successes in Group 'B'. The corresponding proportions are denoted pijp_{ij}, with table
Group′B′SuccessFailureGroup′A′Successp11p10Failurep01p00\begin{array}{cc|cc} & & {Group \quad 'B'} \\ & & Success & Failure \\ \hline Group \quad 'A' & Success & p_{11} & p_{10} \\ & Failure & p_{01} & p_{00} \end{array}
Interest is in comparing the following hypotheses:
H0:Both groups have the same success probabilityH_0: \text{Both groups have the same success probability}
H1:The success probability is not equal between the GroupsH_1: \text{The success probability is not equal between the Groups}
Mathematically, this can be represented as
H0:p10=p01H_0:p_{10}=p_{01}
H1:p10≠p01H_1:p_{10}\neq p_{01}
In the formulas below, we use the notation that
pdisc=p10+p01p_{disc}=p_{10}+p_{01}
and
pdiff=p10−p01p_{diff}=p_{10}-p_{01}

Formulas

This calculator uses the following formulas to compute sample size and power, respectively:
n=(z1−α/2pdisc+z1−βpdisc−pdiff2pdiff)2n=\left(\frac{z_{1-\alpha/2}\sqrt{p_{disc}}+z_{1-\beta}\sqrt{p_{disc}-p_{diff}^2}}{p_{diff}}\right)^2
1−β=Φ(pdiffn−z1−α/2pdiscpdisc−pdiff2)1-\beta=\Phi\left(\frac{p_{diff}\sqrt{n}-z_{1-\alpha/2}\sqrt{p_{disc}}}{\sqrt{p_{disc}-p_{diff}^2}}\right)
where nn is sample size Φ\Phi is the standard Normal distribution function Φ−1\Phi^{-1} is the standard Normal quantile function α\alpha is Type I error β\beta is Type II error, meaning 1−β1-\beta is power

R Code

1p01=0.45
2p10=0.05
3alpha=0.05
4beta=0.20
5pdisc=p10+p01
6pdiff=p10-p01
7(n=((qnorm(1-alpha/2)*sqrt(pdisc)+qnorm(1-beta)*sqrt(pdisc-pdiff^2))/pdiff)^2)
8ceiling(n) # 23
9x1=( pdiff*sqrt(n)-qnorm(1-alpha/2)*sqrt(pdisc))/sqrt(pdisc-pdiff^2);
10x2=(-pdiff*sqrt(n)-qnorm(1-alpha/2)*sqrt(pdisc))/sqrt(pdisc-pdiff^2);
11(Power = pnorm(x1)+pnorm(x2))

References

Connor R. J. 1987. Sample size for testing differences in proportions for the paired-sample design. Biometrics 43(1):207-211. page 209.