Paired Binary Data
Design and notation
In a matched-pairs design both treatments are applied to the same unit, as in a cross-over trial, a comparison of two diagnostic procedures on the same patients, or a trial that treats one eye of each patient with each treatment. A pair has one of four outcomes. Let \(p_{10}\) be the probability that only treatment 1 responds and \(p_{01}\) the probability that only treatment 2 responds. The difference of the response probabilities of the two treatments is \(\delta = p_{10} - p_{01}\), and the probability of a discordant pair is \(\psi = p_{10} + p_{01}\). Among \(N\) pairs, let \(a\) and \(b\) be the numbers of the two kinds of discordant pairs. All tests of the package depend on the data only through \((a, b)\).
The Paired* functions follow the Binary*
functions for two independent groups: PairedRR(),
PairedPower() and PairedSampleSize() for the
fixed design, and PairedPowerBSSR(),
PairedBSSR(), PairedTypeIErrorBSSR() and
PairedAlphaAdjBSSR() for blinded sample size
re-estimation.
Tests
Six tests are available. 'McNemar' refers \(Z = (a - b) / \sqrt{a + b}\) to the
standard normal distribution, and 'McNemar-cc' refers \((|a - b| - 1) / \sqrt{a + b}\) to it, with
the continuity correction of formula (2) of Fagerland et al. (2013), and
\((a - b - 1) / \sqrt{a + b}\) for the
alternative 'greater' and \((b -
a - 1) / \sqrt{a + b}\) for 'less'.
'McNemar-exact' refers \(a\) to the binomial distribution with \(a + b\) trials and probability 1/2, and
'McNemar-midP' is its mid-p version (Fagerland et al.,
2013). 'McNemar-uncond' orders the outcomes by \(Z\) and maximizes the null tail probability
over \(\psi\) (Suissa and Shuster,
1991; Berger and Sidik, 2003), optionally with the Berger-Boos
procedure. 'Paired-Score' is the score test of Nam (1997)
and Tango (1998) for the null hypothesis \(\delta = -\text{margin}\), which reduces to
'McNemar' with a margin of 0.
The example of Berger and Sidik (2003, Section 4) has \(n = 30\) pairs with \(x_{10} = 11\) and \(x_{01} = 4\) and tests \(p_{10} \le p_{01}\) against \(p_{10} > p_{01}\).
pv <- function(...) attr(PairedRR(N = 30, alpha = 0.05, ...), 'p.value')[12, 5]
p.bs <- c(exact = pv(Test = 'McNemar-exact'),
asymptotic = pv(Test = 'McNemar'),
unconditional = pv(Test = 'McNemar-uncond', ref.pvalue = TRUE),
berger.boos = pv(Test = 'McNemar-uncond', ref.pvalue = TRUE, bb.gamma = 0.0005))
round(p.bs, 4)
#> exact asymptotic unconditional berger.boos
#> 0.0592 0.0354 0.0494 0.0382The p-values 0.0592, 0.0354, 0.0494, 0.0382 are those of the tests CB, M, Z and \(Z_C\) in Table 3 of Berger and Sidik (2003). The rejection region of a test is a triangle of possible outcomes.
Power and number of pairs
The power is the probability of the rejection region under the multinomial distribution of \((a, b, N - a - b)\), computed by enumeration. The normal approximation to the number of pairs is formula (9) of Nam (1997), which is formula (3) of Connor (1987) when the margin is 0. In the example of Connor (1987, p. 210) the response probabilities are 0.8 and 0.7 and \(\psi = 0.3\), so \(p_{10} = 0.2\) and \(p_{01} = 0.1\).
n.std <- PairedSampleSize(p10 = 0.2, p01 = 0.1, alpha = 0.05, tar.power = 0.9,
Test = 'McNemar', method = 'standard')
n.exact <- PairedSampleSize(p10 = 0.2, p01 = 0.1, alpha = 0.05, tar.power = 0.9,
Test = 'McNemar-exact')
n.midp <- PairedSampleSize(p10 = 0.2, p01 = 0.1, alpha = 0.05, tar.power = 0.9,
Test = 'McNemar-midP')
c(standard = n.std$N, exact = n.exact$N, midp = n.midp$N)
#> standard exact midp
#> 254 269 252The normal approximation gives 254 pairs, as in Connor (1987). The exact search needs 269 pairs for the exact conditional test and 252 for its mid-p version.
Blinded sample size re-estimation
The number of discordant pairs can be counted without unblinding: a
pair is discordant whatever treatment responded. At an interim analysis
after \(n_{1}\) pairs, the blinded
estimate of \(\psi\) is \(D_{1} / n_{1}\), where \(D_{1}\) is the number of discordant pairs.
The assumed difference Delta.A is kept, and the estimate is
raised to \(|\Delta_{A}|\) when it is
smaller. The number of pairs is then re-estimated by the normal
approximation (ss.method = 'standard'), by the exact search
('exact'), or by the smallest number at which the
conditional power given \(D_{1}\)
reaches the target ('conditional.power'), the rule of
Schepers et al. (2026). The re-estimation depends on the data only
through \(D_{1}\), and given the
numbers of discordant pairs the direction of each discordant pair is
independent of them, so the power is computed exactly.
Schepers et al. (2026) plan the exact conditional test at the two-sided level 0.05 with \(\Delta_{A} = 0.15\) and \(\psi = 0.2\).
plan <- PairedSampleSize(p10 = 0.175, p01 = 0.025, alpha = 0.05, tar.power = 0.8,
Test = 'McNemar-exact', alternative = 'two.sided',
search = 'smallest')
plan$N
#> [1] 72res <- PairedPowerBSSR(psi = c(0.15, 0.2, 0.3), Delta.T = 0.15, Delta.A = 0.15,
N = plan$N, n.interim = 30, alpha = 0.05, tar.power = 0.8,
Test = 'McNemar-exact', alternative = 'two.sided',
ss.method = 'conditional.power', N.max = 200)
res
#> Blinded sample size re-estimation for paired binary data
#>
#> Test : McNemar-exact
#> Alternative : two.sided
#> Re-estimation : conditional power (unrestricted)
#> Pairs : N = 72 planned, 30 at the interim analysis
#> Effect : assumed 0.15, true 0.15
#> Alpha : 0.05, target power 0.8
#>
#> psi p10 p01 power.BSSR power.TRAD E.N
#> 0.15 0.150 0.000 0.8946 0.9684 63.98
#> 0.20 0.175 0.025 0.7415 0.8056 74.97
#> 0.30 0.225 0.075 0.7443 0.5824 109.72With \(\psi = 0.3\) the fixed design of 72 pairs has power 0.582, and the re-estimation design 0.744 with 109.7 pairs on average. With \(\psi = 0.15\) the fixed design has power 0.968 and the re-estimation design 0.895.
PairedBSSR() performs the re-estimation of a single
trial. With 5 discordant pairs among 30 interim pairs:
PairedBSSR(n1 = 30, D1 = 5, Delta.A = 0.15, alpha = 0.05, tar.power = 0.8,
Test = 'McNemar-exact', alternative = 'two.sided',
ss.method = 'conditional.power', N.max = 200)
#> Blinded sample size re-estimation from paired interim data
#>
#> Test : McNemar-exact
#> Alternative : two.sided
#> Re-estimation : conditional power (unrestricted)
#> Assumed effect : 0.15
#> Alpha : 0.05, target power 0.8
#>
#> Interim data
#> Pairs : n1 = 30, discordant D1 = 5
#> Discordance : estimate 0.1667, used 0.1667
#> Planning values : p10 = 0.1583, p01 = 0.0083
#>
#> Re-estimation
#> Re-estimated N : 53
#> Still to enrol : 23 pairs
#> Final N : 53
#> Power at final N : 0.7499
#> Conditional power: 0.8077Type I error rate
The exact conditional test keeps its level after any blinded
re-estimation, since the number of pairs in which only treatment 1
responds stays binomial given the total number of discordant pairs
(Schepers et al., 2026). The other tests have a type I error rate that
depends on \(\psi\), and the
re-estimation can change it. PairedTypeIErrorBSSR()
computes the rate on the boundary \(p_{10} =
p_{01}\) and certifies its largest value. In the example the
final number of pairs is at most 150.
tie <- PairedTypeIErrorBSSR(Delta.A = 0.15, N = plan$N, n.interim = 30, alpha = 0.025,
tar.power = 0.8, Test = 'McNemar', ss.method = 'standard',
N.max = 150)
attr(tie, 'max')
#> Design psi TIE bound
#> 1 BSSR 0.9790955 0.02576064 0.02576064
#> 2 TRAD 0.3351974 0.02673970 0.02673970
plot(tie)
The largest type I error rate of the asymptotic McNemar test at the
one-sided level 0.025 is 0.0258 for the re-estimation design and 0.0267
for the fixed design. PairedAlphaAdjBSSR() lowers the
nominal level until the rate is controlled.
adj <- PairedAlphaAdjBSSR(Delta.A = 0.15, N = plan$N, n.interim = 30, alpha = 0.025,
tar.power = 0.8, Test = 'McNemar', ss.method = 'standard',
N.max = 150)
adj
#> Adjusted significance level for blinded sample size re-estimation
#>
#> Test : McNemar
#> Alternative : greater
#> Adjusted part : final analysis only
#> Maximum : certified over psi in [0, 1]
#> Target level : 0.025
#>
#> Design max.TIE alpha.adj max.TIE.adj
#> BSSR 0.0257606 0.0242598 0.0246174
#> Fixed sample 0.0267397 0.0244499 0.0247896Non-inferiority and equivalence
The score test takes a margin. With
alternative = 'greater' and margin = 0.05 it
tests \(\delta \le -0.05\) against
\(\delta > -0.05\). Nam (1997,
Section 5) plans a trial with \(p_{01} =
0.05\), \(\delta_{0} = -0.05\)
and \(\delta_{1} = 0\) at the one-sided
level 0.05 with power 0.8.
ni <- PairedSampleSize(p10 = 0.05, p01 = 0.05, alpha = 0.05, tar.power = 0.8,
Test = 'Paired-Score', margin = 0.05)
c(normal = attr(ni, 'n.raw'), exact = ni$N)
#> normal exact
#> 273.8196 269.0000Formula (9) of Nam (1997) gives 273.8 pairs, and the exact power of
the score test reaches 0.8 with 269 pairs. With
alternative = 'equivalence' and
margin = c(lower, upper) the null hypothesis \(\delta \le -\text{lower}\) or \(\delta \ge \text{upper}\) is tested by two
one-sided score tests. In a re-estimation design the type I error rate
is evaluated on both boundaries.
References
Berger, R. L. and Sidik, K. (2003). Exact unconditional tests for a 2 × 2 matched-pairs design. Statistical Methods in Medical Research, 12, 91-108.
Connor, R. J. (1987). Sample size for testing differences in proportions for the paired-sample design. Biometrics, 43, 207-211.
Fagerland, M. W., Lydersen, S. and Laake, P. (2013). The McNemar test for binary matched-pairs data: mid-p and asymptotic are better than exact conditional. BMC Medical Research Methodology, 13, 91.
Nam, J. (1997). Establishing equivalence of two treatments and sample size requirements in matched-pairs design. Biometrics, 53, 1422-1430.
Schepers, M., Brannath, W., Hoffmann, E., Stingl, J. and Schmidtmann, I. (2026). Blinded sample size review for McNemar’s test based on primary and surrogate endpoints. arXiv:2608.17784.
Suissa, S. and Shuster, J. J. (1991). The 2 x 2 matched-pairs trial: exact unconditional design and analysis. Biometrics, 47, 361-372.
Tango, T. (1998). Equivalence test and confidence interval for the difference in proportions for the paired-sample design. Statistics in Medicine, 17, 891-908.