Conditional Rejection Probabilities of Blinded Sample Size Re-Estimation for a Stratified Design

Computes, under the null hypothesis of equal response probabilities in every stratum, the probability that a trial with two strata and blinded sample size re-estimation rejects the null hypothesis given the pooled numbers of responders of the strata at the interim analysis and in the second stage. These conditional rejection probabilities do not depend on the response probabilities of the strata, and the type I error rate is their average over the binomial distributions of the pooled counts. With theta, the type I error rate is also returned for each interim outcome. This is the counterpart of BinaryCondRejectBSSR for a stratified design.

Usage

StratCondRejectBSSR(
  Delta.A,
  N1,
  N2,
  omega = NULL,
  r,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  alternative = c("greater", "less", "two.sided"),
  tsmethod = c("minlike", "central", "blaker"),
  effect = c("RD", "RR", "OR"),
  ss.method = c("exact", "standard"),
  ss.Test = Test,
  ss.alpha = alpha,
  ss.strata = c("stratified", "pooled"),
  search = c("crossing", "smallest", "stable"),
  search.limit = c(2, 50),
  N.min = NULL,
  N.max = NULL,
  n.interim = NULL,
  frac = NULL,
  theta = NULL,
  full = FALSE
)

Arguments

Delta.A: Assumed treatment effect, on the scale given by effect, common to the strata and used to divide the blinded estimate of the pooled response probability of each stratum between the groups at the re-estimation
N1: Planned sizes of group 1 in the strata, which are those of the fixed design
N2: Planned sizes of group 2 in the strata
omega: Fraction of the planned sizes at the interim analysis: in stratum j the interim size of group 2 is ceiling(omega N2[j]) and that of group 1 is ceiling(r[j] n2[j]). Supply either omega or n.interim
r: Allocation ratio of group 1 to group 2 within the strata, a single value or one value per stratum. The blinded estimate of the pooled response probability of a stratum is divided between the groups with this ratio, also when the interim sizes are not in this ratio
alpha: Level of significance of the final analysis
tar.power: Target power of the re-estimation
Test: Statistical test. Options: 'CMH' (Cochran-Mantel-Haenszel test with the hypergeometric variance, without continuity correction), 'CMH-cc' (the same with the continuity correction of mantelhaen.test), 'Cochran' (Cochran's test with the binomial variance, the test of Woolson et al. (1986)), 'Cochran-cc' (the same with the continuity correction, which rejects the same outcomes as that of Nam (1992) at one-sided levels up to 0.078), 'Strat-Fisher' (the stratified exact conditional test of Jung (2014)), or one of the unstratified tests of the table of totals: 'Unstrat-Chisq' (Pearson's chi-squared test, the comparison of two proportions with the pooled variance), 'Unstrat-Chisq-cc' (the same with the continuity correction) and 'Unstrat-Fisher' (Fisher's exact test). See Details
restricted: Logical. If TRUE, the final group sizes are at least the planned sizes. Default is FALSE
alternative: Direction of the alternative hypothesis. Options: 'greater' (default), under which group 1 responds more often than group 2, 'less' or 'two.sided'
tsmethod: Convention of the two-sided p-value of 'Strat-Fisher' and 'Unstrat-Fisher', as for the Fisher exact test of BinaryRR. Options: 'minlike' (default), 'central' or 'blaker'. Ignored by the other tests and for one-sided alternatives
effect: Scale of Delta.A and Delta.T. Options: 'RD' (default) for the risk difference, 'RR' for the risk ratio or 'OR' for the odds ratio
ss.method: Re-estimation rule. 'exact' (default) re-estimates the sample size of StratSampleSize with method = 'exact', and 'standard' that of the normal approximation. With three or more strata only 'standard' is available
ss.Test: Test used in the re-estimation. Default is Test
ss.alpha: Level used in the re-estimation. Default is alpha
ss.strata: Blinded estimate of the re-estimation. 'stratified' (default) estimates the pooled response probability of each stratum from its own interim patients, and 'pooled' uses the proportion of responders among all the interim patients for every stratum, as a re-estimation that ignores the strata would. With 'pooled' the final group sizes depend only on the total number of interim responders
search: Search of the exact re-estimation, see StratSampleSize; it does not go beyond the largest size that N.max allows
search.limit: Limit of the search 'stable', see StratSampleSize
N.min: Smallest final total sample size, or NULL (default)
N.max: Largest final total sample size, or NULL (default)
n.interim: Interim group sizes, as a matrix with two rows (group 1 and group 2) and one column per stratum. Supply either omega or n.interim
frac: Fractions of the strata used to divide the re-estimated total sample size between the strata, see StratSampleSize. Default is NULL, which takes the fractions of the planned design, (N1 + N2) / sum(N1 + N2)
theta: Optional common response probabilities of the strata at which the type I error rate is decomposed by the interim outcome: a vector, whose values form a grid of pairs, or a matrix with two columns that gives the points directly. Default is NULL, which skips the decomposition. The time of the decomposition grows with the number of points
full: Logical. If TRUE, the attribute full lists the conditional rejection probability of every interim and second-stage outcome. Default is FALSE. A listing of more than 1e6 rows is refused

Value

An object of class bbssr_scrp, a data frame with one row for every pair of interim pooled numbers of responders of the strata, the first stratum varying fastest, containing:

s1, s2: Pooled numbers of responders of the two strata at the interim analysis
N1.S1, N1.S2, N2.S1, N2.S2: Final group sizes reached from (s1, s2)
CRP.max, CRP.min: Largest and smallest rejection probability over the pooled numbers of responders (w1, w2) of the strata in the second stage
exceed: Number of pairs (w1, w2) at which the rejection probability exceeds alpha
CRP.total.max: Largest conditional size of the final test given the final numbers of responders of the strata, see Details

The attribute max is a data frame with the largest values of CRP and CRP.total over all outcomes, and the first outcome, in the order of the rows and then of (w1, w2) with w1 varying fastest, at which each is attained up to rounding error. The attribute exceed gives the numbers of outcomes (s1, s2, w1, w2) at which each of the two exceeds alpha, and the attribute reestimation holds the final group sizes for every interim outcome, as in StratPowerBSSR. When theta is supplied, the attribute by.s is a data frame with one row for each point and interim outcome, containing the probability prob.s of (s1, s2), the type I error rate TIE.s given (s1, s2) and their product contribution, and the attribute TIE gives the type I error rate at each point. With full = TRUE the attribute full is a data frame with one row for every outcome (s1, s2, w1, w2), ordered by s2, s1, w2 and w1, with the columns CRP and CRP.total.

Details

Let n1[j] and n2[j] be the interim group sizes of stratum j, and m1[j] and m2[j] the second-stage group sizes reached from the interim outcome s = (s1, s2). Under the null hypothesis both groups of stratum j respond with the probability theta_j. Given s_j, the number of interim responders of group 1 in stratum j is hypergeometric, being the number of group 1 patients among s_j responders drawn from the n1[j] + n2[j] interim patients of the stratum. Given the pooled number w_j of second-stage responders of the stratum, the number of second-stage responders of group 1 is an independent hypergeometric count of the same kind. CRP sums the rejection region of the final design over these four distributions and does not depend on theta. The type I error rate is the sum over s and w = (w1, w2) of b(s1; n1[1] + n2[1], theta_1) b(s2; n1[2] + n2[2], theta_2) b(w1; m1[1] + m2[1], theta_1) b(w2; m1[2] + m2[2], theta_2) CRP(s, w), where b is the binomial probability.

CRP.total is the conditional size of the final test given the final numbers of responders z_j = s_j + w_j of the strata, with the number of responders of group 1 of each stratum hypergeometric among all its final patients. The sum of the contributions at a point equals the type I error rate of StratTypeIErrorBSSR there.

Only designs with two strata are covered, as for the exact computations of StratTypeIErrorBSSR.

Author

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

Examples

crp <- StratCondRejectBSSR(Delta.A = 0.3, N1 = c(8, 8), N2 = c(8, 8), omega = 0.5,
                           r = 1, alpha = 0.025, tar.power = 0.8, Test = 'CMH',
                           ss.method = 'standard', theta = c(0.3, 0.5))
print(crp)
#> Conditional rejection probabilities of blinded sample size re-estimation
#> for a stratified design, under equal response probabilities in every stratum
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Planned sizes    : (8, 8) and (8, 8)
#>   Interim sizes    : (4, 4) and (4, 4)
#>   Assumed effect   : 0.3 (RD)
#>   Nominal level    : 0.025
#>   Interim outcomes : 81 pairs (s1, s2)
#>   Above the level  : 10605 outcomes for CRP, 17037 for CRP.total
#> 
#> Largest conditional rejection probability
#>   Quantity   value s1 s2 w1 w2
#>        CRP 0.06250  1  1  1  1
#>  CRP.total 0.05853  2  2  0  0
#> 
#> Type I error rate over 4 points: largest 0.02467 at theta = (0.5, 0.3)
attr(crp, 'TIE')
#>   theta.S1 theta.S2        TIE
#> 1      0.3      0.3 0.02437500
#> 2      0.5      0.3 0.02467373
#> 3      0.3      0.5 0.02467373
#> 4      0.5      0.5 0.02463391