Type I Error Rate of Blinded Sample Size Re-Estimation for Paired Binary Data

Calculates the exact type I error rate of a matched-pairs design with blinded sample size re-estimation, and of the fixed design with the planned number of pairs, over the probability of a discordant pair on the boundary of the null hypothesis, together with the largest rate.

Usage

PairedTypeIErrorBSSR(
  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,
  psi = seq(0, 1, by = 0.01),
  maximize = c("certified", "refined", "grid"),
  margin = 0,
  ref.pvalue = FALSE
)

Arguments

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: 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
psi: Values of the probability p10 + p01 of a discordant pair at which the rate is evaluated. Values that do not lie on the boundary of the null hypothesis, below |margin|, are dropped. Default is seq(0, 1, by = 0.01)
maximize: How the largest rate is found. 'certified' (default) takes the largest value on the grid and certifies the maximum over the interval spanned by the grid with an upper bound, 'refined' refines the largest values on the grid between neighbouring grid points, and 'grid' takes the largest value on the grid. See Details
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_ptie, a data frame with one row for each value of psi on the boundary, containing psi, the probabilities p10 and p01 on the boundary, and the rates TIE.BSSR of the design with re-estimation and TIE.TRAD of the fixed design. For a test of equivalence the rates on the two boundaries are stacked, with the column boundary. The attribute max holds the largest rates TIE of both designs, the values psi at which they occur, the upper bounds bound of the certified maximization (otherwise NA) and, for a test of equivalence, the boundary. The attribute reestimation holds the final number of pairs for every interim number of discordant pairs, and the other attributes the design settings.

Details

The type I error rate is evaluated on the boundary p10 - p01 = d0 of the null hypothesis, with d0 = 0 for a test of superiority, d0 = -margin for alternative = 'greater' and d0 = margin for 'less', as for two independent groups. For a fixed design, Berger and Sidik (2003, Theorem 1) show that the largest rate over the null hypothesis of superiority lies on this boundary for rejection regions that are Barnard convex. A test of equivalence has two boundaries, -lower and upper, and the largest rate is the larger of the maxima on the two.

On a boundary the rejection probability is a polynomial in psi: given the numbers of discordant pairs, the number of pairs in which only treatment 1 responds is binomial with probability p10 / psi, which is fixed on the boundary up to the linear reparametrization of paired_boundary, and the re-estimation depends on the interim data only through the number of discordant pairs. The probability is computed exactly. With maximize = 'certified' the Bernstein coefficients of the polynomial over the interval spanned by the values of psi on the boundary are bounded by subdivision, as in BinaryTypeIErrorBSSR, which gives an upper bound of the largest rate over the interval within 1e-12 of the maximum found.

The exact conditional McNemar test keeps its level whatever blinded rule is used (Schepers et al., 2026), so its largest rate does not exceed alpha.

References

Berger RL, Sidik K (2003). Exact unconditional tests for a 2 x 2 matched-pairs design. Statistical Methods in Medical Research, 12(2), 91-108.

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.

Examples

tie <- PairedTypeIErrorBSSR(Delta.A = 0.15, N = 72, n.interim = 30, alpha = 0.05,
                            tar.power = 0.8, Test = 'McNemar',
                            alternative = 'two.sided', ss.method = 'standard')
print(tie)
#> Type I error rate of blinded sample size re-estimation for paired binary data
#> 
#>   Test            : McNemar
#>   Alternative     : two.sided
#>   Pairs           : N = 72 planned, 30 at the interim analysis
#>   Assumed effect  : 0.15
#>   Nominal level   : 0.05
#>   Grid            : 101 values of psi in [0, 1]
#>   Maximum         : certified over psi in [0, 1]
#> 
#> Largest type I error rate
#>        Design      psi     TIE
#>          BSSR 1.000000 0.05313
#>  Fixed sample 0.335197 0.05348

# \donttest{
plot(tie)

# }