Inputs

Solve for

Power (

1-β

)

Sample Size (

n

)

Type I error rate (

α

)

no event-event (

p₀₁

)

event-no event (

p₁₀

)

X-Axis Variable

Min

Max

Calculate Sample Size Needed to Compare Paired Proportions: McNemar's Z-test, 1-Sided

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.

\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,

n_{ij}

represents the number of pairs having

i

successes in Group 'A' and

j

successes in Group 'B'. The corresponding proportions are denoted

p_{ij}

, with table

\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:

H_0

: Both groups have the same success probability

H_1

: The success probability is greater for one group than the other Mathematically, this can be represented as

H_0: p_{10} = p_{01}

H_1: p_{10} < p_{01}

H_0: p_{10} = p_{01}

H_1: p_{10} > p_{01}

In the formulas below, we use the notation that

p_{disc} = p_{10} + p_{01}

and

p_{diff} = p_{10} - p_{01}

Formulas

This calculator uses the following formulas to compute sample size and power, respectively:

n = \left( \frac{z_{1-\alpha} \sqrt{p_{disc}} + z_{1-\beta} \sqrt{p_{disc} - p_{diff}^2}}{p_{diff}} \right)^2

1-\beta = \Phi \left( \frac{|p_{diff}| \sqrt{n} - z_{1-\alpha} \sqrt{p_{disc}}}{\sqrt{p_{disc} - p_{diff}^2}} \right)

where:

n

is sample size

\Phi

is the standard Normal distribution function

\Phi^{-1}

is the standard Normal quantile function

\alpha

is Type I error

\beta

is Type II error, meaning

1-\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)*sqrt(pdisc)+qnorm(1-beta)*sqrt(pdisc-pdiff^2))/pdiff)^2)
8ceiling(n) # 23
9x=(abs(pdiff)*sqrt(n)-qnorm(1-alpha)*sqrt(pdisc))/sqrt(pdisc-pdiff^2);
10(Power = pnorm(x))

References

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