Conditional Rejection Probabilities of a Blinded Sample Size Re-estimation Design

Computes, under the null hypothesis of equal response probabilities, the probability that a two-arm trial with a binary endpoint and blinded sample size re-estimation (BSSR) rejects the null hypothesis given the pooled number of responders at the interim analysis and in the second stage. These conditional rejection probabilities do not depend on the common response probability, and the type I error rate is their average over the binomial distribution of the two pooled counts. With theta, the type I error rate is also returned for each pooled number of interim responders.

Usage

BinaryCondRejectBSSR(
  Delta.A,
  N1,
  N2,
  omega = NULL,
  r,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  alternative = c("greater", "less", "two.sided"),
  tsmethod = c("minlike", "central", "blaker"),
  n.grid = 100,
  bb.gamma = 0,
  effect = c("RD", "RR", "OR"),
  ss.method = c("exact", "standard", "null.variance", "alternative.variance"),
  ss.Test = Test,
  ss.alpha = alpha,
  rounding = c("group", "friede-kieser", "total", "nearest"),
  search = c("crossing", "smallest", "stable"),
  search.limit = c(2, 50),
  N.min = NULL,
  N.max = NULL,
  n.interim = NULL,
  theta = NULL,
  margin = 0,
  ref.pvalue = FALSE
)

Arguments

Delta.A: Assumed treatment effect, on the scale given by effect, used to split the blinded pooled proportion into group-specific proportions
N1: Initial sample size of group 1
N2: Initial sample size of group 2
omega: Fraction of the initial sample size observed at the interim analysis. The interim size of group 2 is ceiling(omega N2) and that of group 1 is ceiling(r ceiling(omega N2)), so the interim analysis keeps the allocation ratio. Either omega or n.interim must be supplied
r: Allocation ratio to group 1
alpha: Level of significance of the final analysis, for the alternative specified by alternative
tar.power: Target power
Test: Type of statistical test of the final analysis. Options: 'Chisq', 'Fisher', 'Fisher-midP', 'Z-pool', 'Boschloo', 'Blackwelder' or 'Farrington-Manning'
restricted: Logical. If TRUE, the re-estimated sample size is not allowed to fall below the initial sample size. Default is FALSE
alternative: Direction of the alternative hypothesis. Options: 'greater' (default), 'less', 'two.sided', or 'equivalence' for the alternative that the effect lies between the margins of a test of equivalence, see margin
tsmethod: Convention used to construct the two-sided version of the conditional tests, see BinaryRR. Options: 'minlike' (default), 'central' or 'blaker'
n.grid: Number of grid points used to search over the nuisance parameter of the unconditional tests. Default is 100
bb.gamma: Confidence level parameter of the Berger-Boos procedure. The default of 0 disables the procedure
effect: Scale of Delta.A and Delta.T. Options: 'RD' (default) for the risk difference p1 - p2, 'RR' for the risk ratio p1 / p2 or 'OR' for the odds ratio
ss.method: How the sample size is re-estimated from the recovered proportions. Options: 'exact' (default), 'standard', 'null.variance' or 'alternative.variance', as in BinarySampleSize
ss.Test: Test whose exact power is used for the re-estimation when ss.method = 'exact'. Default is Test
ss.alpha: Level of significance used for the re-estimation. Default is alpha
rounding: How the re-estimated sample size is turned into whole numbers. Options: 'group' (default), 'friede-kieser', 'total' or 'nearest'. Only 'group' is available with ss.method = 'exact'. See Details
search: How the exact re-estimation chooses the sample size when ss.method = 'exact': 'crossing' (default), 'smallest' or 'stable', as in BinarySampleSize, without going beyond the largest size that N.max allows. Ignored by the other methods
search.limit: Limit of the size of group 2 examined by search = 'stable', as in BinarySampleSize. Default is c(2, 50)
N.min: Lower bound on the final total sample size, or NULL (default) for none beyond the interim total. It can be used to keep the patients who are already enrolled but not yet evaluated at the interim analysis
N.max: Upper bound on the final total sample size, or NULL (default) for none
n.interim: Interim sample sizes of group 1 and group 2, as a vector of length two. An alternative to omega
theta: Optional vector of common response probabilities at which the type I error rate is decomposed by the pooled number of interim responders. Default is NULL, which skips the decomposition
margin: Must be 0, the default, since only tests of superiority are covered (see Details). The argument keeps the arguments the same as those of BinaryTypeIErrorBSSR
ref.pvalue: Logical. If TRUE, the maximization over the nuisance parameter of the unconditional tests is refined between the grid points, see BinaryRR. Default is FALSE. It applies to the final analysis and to the exact re-estimation

Value

An object of class bbssr_crp, a data frame with one row for every pair of pooled responder counts that the design can produce, ordered by s and then by s2, containing:

s: Pooled number of responders at the interim analysis
s2: Pooled number of responders in the second stage
N1: Final sample size of group 1 reached from s
N2: Final sample size of group 2 reached from s
CRP: Rejection probability given s and s2
CRP.total: Rejection probability given only the total number of responders s + s2, when the responder count of group 1 is treated as one hypergeometric count among all N1 + N2 patients

The attribute max is a data frame with the largest value of each of the two probabilities and the first outcome, in the order of the rows, at which it is attained up to rounding error, the attribute exceed gives the number of outcomes at which each of them exceeds alpha, and the attribute reestimation holds the final sample size for every pooled number of interim responders. When theta is supplied, the attribute by.s is a data frame with one row for each value of theta and of s, containing the probability prob.s of s, the type I error rate TIE.s given s and the contribution prob.s * TIE.s of s to the type I error rate, and the attribute TIE gives the type I error rate at each value of theta.

Details

Let n11 and n12 be the interim sample sizes of the two groups, and let n21(s) and n22(s) be the second-stage sizes reached from the pooled number s of interim responders. Under the null hypothesis both groups share the response probability theta. Given s, the number of interim responders in group 1 is hypergeometric, being the number of group 1 patients among s responders drawn from n11 + n12 patients. Given the pooled number s2 of second-stage responders, the number of second-stage responders in group 1 is an independent hypergeometric count of the same kind. CRP sums the rejection region of the final sample size over these two distributions and does not depend on theta. The type I error rate is sum_s sum_s2 b(s; n11 + n12, theta) b(s2; n21(s) + n22(s), theta) CRP(s, s2), where b is the binomial probability.

CRP.total is the conditional size of the final test given the total number of responders, with the responder count of group 1 hypergeometric among all final patients.

Only tests of superiority are covered. With a non-inferiority margin the two response probabilities differ on the boundary of the null hypothesis, and the conditional distributions of the responder counts then depend on them.

Author

Gosuke Homma (my.name.is.gosuke@gmail.com)

Examples

crp <- BinaryCondRejectBSSR(
  Delta.A = 0.3, N1 = 12, N2 = 12, n.interim = c(6, 6), r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Chisq', ss.method = 'standard',
  theta = c(0.3, 0.5)
)
print(crp)
#> Conditional rejection probabilities of blinded sample size re-estimation
#> for a binary endpoint, under equal response probabilities
#> 
#>   Test            : Chisq
#>   Alternative     : greater
#>   Initial size    : N1 = 12, N2 = 12
#>   Interim size    : n1 = 6, n2 = 6
#>   Assumed effect  : 0.3 (RD)
#>   Nominal level   : 0.025
#>   Outcomes        : 567 pairs (s, s2)
#>   Above the level : 198 for CRP, 264 for CRP.total
#> 
#> Largest conditional rejection probability
#>   Quantity   value s s2 N1 N2
#>        CRP 0.05519 2  2 24 24
#>  CRP.total 0.05753 4  0 38 38
#> 
#> Type I error rate over 2 values of theta: largest 0.02762 at theta = 0.5
attr(crp, 'TIE')
#>   theta        TIE
#> 1   0.3 0.02537642
#> 2   0.5 0.02761990

# \donttest{
crp <- BinaryCondRejectBSSR(
  Delta.A = 0.3, N1 = 39, N2 = 39, n.interim = c(20, 20), r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Chisq', ss.method = 'standard'
)
attr(crp, 'max')
#>    Quantity      value s s2 N1 N2
#> 1       CRP 0.05301455 4  0 20 20
#> 2 CRP.total 0.05301455 4  0 20 20
plot(crp)

# }