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Power of Blinded Sample Size Re-Estimation for Paired Binary Data

Source: R/PairedPowerBSSR.R
PairedPowerBSSR.Rd

Calculates the exact power and the distribution of the final number of pairs of a matched-pairs design with blinded sample size re-estimation, together with the power of the fixed design with the planned number of pairs. The number of pairs is re-estimated at an interim analysis from the number of discordant pairs, without the direction in which they are discordant, and the final analysis uses all pairs.

Usage

PairedPowerBSSR(
  psi,
  Delta.T,
  Delta.A,
  N,
  omega = NULL,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  alternative = c("greater", "less", "two.sided", "equivalence"),
  n.grid = 100,
  bb.gamma = 0,
  ss.method = c("exact", "standard", "conditional.power"),
  ss.Test = Test,
  ss.alpha = alpha,
  search = c("crossing", "smallest", "stable"),
  search.limit = c(2, 50),
  N.min = NULL,
  N.max = NULL,
  n.interim = NULL,
  margin = 0,
  ref.pvalue = FALSE
)

Arguments

psi

True probability p10 + p01 of a discordant pair. A vector is allowed

Delta.T

True difference p10 - p01, a single value

Delta.A

Assumed difference p10 - p01 used in the re-estimation

N

Planned number of pairs, which is the number of pairs of the fixed design

omega

Fraction of N at the interim analysis, rounded up. Supply either omega or n.interim

alpha

Level of significance for the alternative specified by alternative

tar.power

Target power of the re-estimation

Test

Type of statistical test. Options: 'McNemar', 'McNemar-cc', 'McNemar-exact', 'McNemar-midP', 'McNemar-uncond' or 'Paired-Score'

restricted

Logical. If TRUE, the final number of pairs is at least N. Default is FALSE

alternative

Direction of the alternative hypothesis. Options: 'greater' (default) for p10 - p01 > -margin, 'less' for p10 - p01 < margin, 'two.sided' for p10 != p01, or 'equivalence' for -lower < p10 - p01 < upper

n.grid

Number of grid points used to search over the probability of a discordant pair in the test 'McNemar-uncond'. Default is 100

bb.gamma

Confidence level parameter of the Berger-Boos procedure for the test 'McNemar-uncond'. The default of 0 disables the procedure

ss.method

Re-estimation rule. 'exact' (default) and 'standard' re-estimate the number of pairs of PairedSampleSize with the method of that name. 'conditional.power' takes the smallest number of pairs at which the conditional power given the interim number of discordant pairs attains tar.power, the rule of Schepers et al. (2026); it requires N.max. See Details

ss.Test

Test used in the re-estimation. Default is Test

ss.alpha

Level used in the re-estimation. Default is alpha

Search of the exact re-estimation, see PairedSampleSize; it does not go beyond N.max

search.limit

Limit of the search 'stable', see PairedSampleSize

N.min

Smallest final number of pairs, or NULL

N.max

Largest final number of pairs, or NULL

n.interim

Number of pairs at the interim analysis. Supply either omega or n.interim

margin

Margin on the scale of the difference p10 - p01. The default of 0 gives a test of superiority, and a value other than 0 tests the null hypothesis p10 - p01 <= -margin against p10 - p01 > -margin when alternative is 'greater', and p10 - p01 >= margin against p10 - p01 < margin when it is 'less'. A negative value tests for superiority by more than its absolute value. A margin other than 0 requires Test = 'Paired-Score' and a one-sided alternative. For alternative = 'equivalence' the margin is the pair c(lower, upper), both in (0, 1), or a single value used for both, and the null hypothesis p10 - p01 <= -lower or p10 - p01 >= upper is tested

ref.pvalue

Logical. If TRUE, the maximization over the probability of a discordant pair in the test 'McNemar-uncond' is refined between the grid points, see Details. Default is FALSE

Value

An object of class bbssr_ppowerbssr, a data frame with one row for each element of psi that gives probabilities p10 and p01 in [0, 1], containing:

psi, p10, p01

True probabilities of a discordant pair and of its two kinds

power.BSSR

Power of the design with re-estimation

power.TRAD

Power of the fixed design with N pairs

E.N

Expected final number of pairs

The attribute reestimation holds the final number of pairs for every interim number of discordant pairs, N.dist the distribution of the final number of pairs in each scenario, and the other attributes the design settings.

Details

Among the n1 pairs of the interim analysis, the number D1 of discordant pairs is binomial with n1 trials and probability psi, and the blinded estimate of psi is D1 / n1. The assumed difference Delta.A is kept, and the estimate is raised to |Delta.A| when it is smaller, since psi >= |p10 - p01|. The planning probabilities are then p10 = (psi.plan + Delta.A) / 2 and p01 = (psi.plan - Delta.A) / 2, as in Xu and Yu (2013) and Schepers et al. (2026).

The final number of pairs depends on the interim data only through D1. Given the numbers of discordant pairs of the two stages, the number of pairs in which only treatment 1 responds is binomial with the total number of discordant pairs and probability p10 / psi, whatever the re-estimation did. The power is therefore sum over D1 and D2 of b(D1; n1, psi) b(D2; N(D1) - n1, psi) g(D1 + D2), where g(D) is the probability that the final analysis with N(D1) pairs rejects given D discordant pairs. It is computed exactly by enumeration.

With ss.method = 'conditional.power' the final number of pairs is the smallest number N between the bounds at which sum over D2 of b(D2; N - n1, psi.plan) g_N(D1 + D2), computed with the planning probabilities and the test ss.Test at the level ss.alpha, attains tar.power. With the exact conditional McNemar test this is the conditional power (4) of Schepers et al. (2026), which they evaluate with p* = 0.5 + Delta / (2 psi) at the blinded estimate of psi. When no number up to N.max attains the target, N.max is used, a choice the article leaves open.

The exact conditional McNemar test keeps its level whatever blinded rule is used, 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 can exceed their nominal level after re-estimation, see PairedTypeIErrorBSSR.

References

Schepers M, Brannath W, Hoffmann E, Stingl J, Schmidtmann I (2026). Blinded sample size review for McNemar's test based on primary and surrogate endpoints. arXiv:2608.17784.

Xu J, Yu M (2013). Sample size determination and re-estimation for matched pair designs with multiple binary endpoints. Biometrical Journal, 55(3), 430-443.

Examples

res <- PairedPowerBSSR(psi = c(0.15, 0.2, 0.3), Delta.T = 0.15, Delta.A = 0.15, N = 72,
                       n.interim = 30, alpha = 0.05, tar.power = 0.8,
                       Test = 'McNemar-exact', alternative = 'two.sided',
                       ss.method = 'standard')
print(res)
#> Blinded sample size re-estimation for paired binary data
#> 
#>   Test             : McNemar-exact
#>   Alternative      : two.sided
#>   Re-estimation    : normal approximation (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     0.7851     0.9684  59.33
#>  0.20 0.175 0.025     0.7174     0.8056  71.22
#>  0.30 0.225 0.075     0.7190     0.5824 102.98

# \donttest{
# Conditional power rule of Schepers et al. (2026) with at most 150 pairs
res <- PairedPowerBSSR(psi = c(0.15, 0.2, 0.3), Delta.T = 0.15, Delta.A = 0.15, N = 72,
                       n.interim = 30, alpha = 0.05, tar.power = 0.8,
                       Test = 'McNemar-exact', alternative = 'two.sided',
                       ss.method = 'conditional.power', N.max = 150)
print(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  74.92
#>  0.30 0.225 0.075     0.7428     0.5824 108.17
# }