Power of Blinded Sample Size Re-Estimation for a Stratified Design

Calculates the power and the distribution of the final total sample size of a two-arm trial with a binary endpoint, stratified into two or more strata, with blinded sample size re-estimation, together with the power of the fixed design with the planned group sizes. The sample size is re-estimated at an interim analysis from the pooled numbers of responders of the strata, without the treatment groups, and the final analysis is a test of StratRR with all patients. With two strata the power is computed exactly, and with three or more strata by simulation.

Usage

StratPowerBSSR(
  p,
  Delta.A,
  Delta.T,
  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,
  nsim = 1e+05,
  alloc = c("fixed", "stratum", "group", "both"),
  block = 4,
  ss.frac = c("observed", "planned"),
  nseq = 100
)

Arguments

p: True pooled response probabilities of the strata: a vector with one element per stratum, or a matrix with one row per scenario and one column per stratum
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
Delta.T: True treatment effect, on the scale given by effect: a single value common to the strata, or one value per stratum
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)
nsim: Number of simulated trials for each scenario with three or more strata. Default is 1e5. Ignored with two strata
alloc: Allocation of the patients to the strata and the groups. 'fixed' (default) uses the planned and interim group sizes. The others make the sizes random, see Details: 'stratum' (stratum sizes random, permuted blocks within the strata) and 'both' (stratum and group sizes random), with any number of strata, and 'group' (stratum sizes in the fixed fractions, group sizes random), with two strata
block: Size of the permuted blocks of alloc = 'stratum'. Each block holds block r / (1 + r) patients of group 1, which must be a whole number. Default is 4
ss.frac: Fractions of the strata used by the re-estimation with a random allocation: 'observed' (default), those of the interim patients, or 'planned', those given by frac
nseq: Number of allocation sequences drawn with a random allocation and two strata. Default is 100

Value

An object of class bbssr_spowerbssr, a data frame with one row for each scenario in which both response probabilities of every stratum lie in the unit interval, containing:

p.S1, p.S2, ...: True pooled response probabilities of the strata
p1.S1, ..., p2.S1, ...: True response probabilities of the two groups
power.BSSR: Power of the design with re-estimation
power.TRAD: Power of the fixed design with the planned group sizes
E.N: Expected final total sample size
se.BSSR: Monte Carlo standard error of power.BSSR, with three or more strata or with a random allocation
se.TRAD: Monte Carlo standard error of power.TRAD, with three or more strata and a random allocation

The attribute reestimation holds, with two strata, the final group sizes for every pair of interim pooled numbers of responders, N.dist the distribution of the final total sample size in each scenario, scenario the rows of p kept as scenarios, and the other attributes the design settings. summary.bbssr_spowerbssr summarizes the distribution of the final total.

Details

In stratum j the blinded estimate of the pooled response probability is the number s_j of interim responders divided by the interim size of the stratum. It is divided between the groups with the assumed effect Delta.A and the allocation ratio of the stratum, as in BinaryPowerBSSR, and the total sample size is re-estimated from these probabilities by the rule ss.method. The total is divided between the strata and the groups as in StratSampleSize, and the final group sizes are at least the interim sizes, at least the planned sizes when restricted = TRUE, with a total of at least N.min and at most N.max. When the probabilities of every stratum show no effect in the direction of the alternative, which happens with effect = 'RR' when no interim patient responds and with effect = 'OR' when in each stratum no interim patient or every interim patient responds, the planned group sizes are kept. A stratum whose pooled estimate is close to 0 with effect = 'RR', or close to 0 or 1 with effect = 'OR', has planning probabilities close to each other, so the re-estimated size can be very large, and the time of the exact computation grows with it. An upper bound N.max keeps both in check: the final total does not exceed it, and the exact search of ss.method = 'exact' does not go beyond it.

With two strata the final group sizes depend on the interim data only through (s_1, s_2). Given s_j, the number of interim responders of group 1 in stratum j follows from the two binomial distributions of the stratum, and the second-stage numbers of responders are binomial given the final sizes. The power is the sum over (s_1, s_2) of the probability of the interim outcome times the conditional probability of rejection, computed exactly by enumeration in compiled code.

With three or more strata the number of interim outcomes grows as a product over the strata, and the power is estimated from nsim simulated trials, with the random numbers of R, so that set.seed() makes the results reproducible. The power of the fixed design with the fixed allocation is exact for any number of strata.

With a random allocation (alloc other than 'fixed') and two strata the patients enter the strata and the groups as in StratPower, the interim analysis has the first sum(n1 + n2) patients of the interim sizes above, and the final analysis the first N patients, where N is the re-estimated total. The re-estimation computes the total as above, for the fixed allocation, with the fractions of the strata among the interim patients (ss.frac = 'observed') or those of frac, and takes the total of the group sizes it gives, within the same bounds; the allocation of the remaining patients is random. Given an allocation sequence, that is the strata and the groups of the patients in the order of enrolment, the design has fixed group sizes and its power is computed exactly as above. The power is the average over nseq sequences drawn with the random numbers of R, so that set.seed() makes the results reproducible, and se.BSSR is the standard deviation of the powers of the sequences divided by sqrt(nseq). The power of the fixed design with the planned total and the same allocation is exact. A sequence without interim patients in a stratum is drawn again.

With a random allocation and three or more strata (alloc = 'stratum' or 'both') the allocation is simulated with the responses, in nsim trials. The strata of the first sum(n1 + n2) patients are multinomial with the probabilities frac, drawn again until every stratum has one of them, and their groups follow the allocation within the strata. The total is re-estimated as with two strata, and the strata and the groups of the remaining patients continue the allocation, the permuted blocks of alloc = 'stratum' included. The power of the fixed design with the planned total and the same allocation is estimated from nsim other trials, with the standard error se.TRAD. A group without patients is allowed in both designs, and the fixed design is analysed without a stratum that has no patients.

References

Woolson RF, Bean JA, Rojas PB (1986). Sample size for case-control studies using Cochran's statistic. Biometrics, 42(4), 927-932.

Nam J (1992). Sample size determination for case-control studies and the comparison of stratified and unstratified analyses. Biometrics, 48(2), 389-395.

Jung SH (2014). Stratified Fisher's exact test and its sample size calculation. Biometrical Journal, 56(1), 129-140.

Author

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

Examples

res <- StratPowerBSSR(p = rbind(c(0.3, 0.5), c(0.4, 0.6)), Delta.A = 0.2,
                      Delta.T = 0.2, N1 = c(30, 30), N2 = c(30, 30), omega = 0.5,
                      r = 1, alpha = 0.025, tar.power = 0.8, Test = 'CMH',
                      ss.method = 'standard')
print(res)
#> Blinded sample size re-estimation for a stratified design
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Re-estimation    : normal approximation (unrestricted)
#>   Planned sizes    : (30, 30) and (30, 30)
#>   Interim sizes    : (15, 15) and (15, 15)
#>   Effect           : RD, assumed 0.2, true 0.2
#>   Alpha            : 0.025, target power 0.8
#> 
#>  p.S1 p.S2 p1.S1 p1.S2 p2.S1 p2.S2 power.BSSR power.TRAD    E.N
#>   0.3  0.5   0.4   0.6   0.2   0.4     0.7892     0.6274 174.55
#>   0.4  0.6   0.5   0.7   0.3   0.5     0.7888     0.6162 181.99