Blinded Sample Size Re-Estimation for a Stratified Trial

Re-estimates the final group sizes of the strata of an ongoing two-arm trial with a binary endpoint from the blinded data of an interim analysis: the numbers of patients and the pooled numbers of responders of the strata.

Usage

StratBSSR(
  n1,
  n2,
  S,
  Delta.A,
  r,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  N1 = NULL,
  N2 = NULL,
  frac = NULL,
  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,
  alloc = c("fixed", "stratum", "group", "both"),
  block = 4,
  ss.frac = c("observed", "planned")
)

Arguments

n1: Numbers of patients of group 1 observed in the strata at the interim analysis. With a random allocation a group may be empty in a stratum
n2: Numbers of patients of group 2 observed in the strata at the interim analysis
S: Numbers of responders observed in the strata at the interim analysis, pooled over both groups, one element 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
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
N1: Planned sizes of group 1 in the strata, or NULL (default). Required when restricted is TRUE, when frac is NULL, and when the blinded data show no effect in the direction of the alternative
N2: Planned sizes of group 2 in the strata, or NULL (default)
frac: Fractions of the strata used to divide the re-estimated total sample size, see StratSampleSize. Default is NULL, which takes the fractions of the planned design
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)
alloc: Allocation of the patients to the strata and the groups. 'fixed' (default) uses the group sizes. The others, 'stratum', 'group' and 'both' of StratPower, make the sizes random and need two strata, see Details
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

Value

An object of class bbssr_sbssr, a data frame with one row per stratum containing:

Stratum: Number of the stratum
n1, n2, S: Interim data
hat.p: Blinded estimate of the pooled response probability
hat.p1, hat.p2: Planning probabilities of the two groups, truncated to the unit interval
N1.final, N2.final: Final group sizes, NA with a random allocation
n1.stage2, n2.stage2: Numbers of patients still to be enrolled, NA with a random allocation

The attributes n.raw (unrounded total of the normal approximation), N.final (final total sample size) and Power (exact power of the final design at the planning probabilities) summarize the re-estimation, and the attribute search.limit holds the limit of the nominal total examined by search = 'stable' (otherwise NA).

Details

The re-estimation is that of StratPowerBSSR: in each stratum the blinded estimate of the pooled response probability is divided between the groups with the assumed effect Delta.A, and the total sample size of StratSampleSize for these probabilities is divided between the strata. The final group sizes are at least the interim sizes, at least the planned sizes when restricted = TRUE, and their total lies between N.min and N.max.

With a random allocation (alloc other than 'fixed', two strata only) the re-estimation gives the final total sample size as in StratPowerBSSR, with the fractions of the strata among the interim patients (ss.frac = 'observed') or those of frac, and the remaining patients are allocated at random, so the final group sizes are NA. The attribute Power is then that of StratPower with the final total and the same allocation.

Author

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

Examples

StratBSSR(n1 = c(15, 15), n2 = c(15, 15), S = c(12, 18), Delta.A = 0.2, r = 1,
          alpha = 0.025, tar.power = 0.8, Test = 'CMH', N1 = c(30, 30),
          N2 = c(30, 30), ss.method = 'standard')
#> Blinded sample size re-estimation for a stratified trial
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Re-estimation    : normal approximation (unrestricted)
#>   Effect           : RD, assumed 0.2
#>   Final total      : 188, power 0.7985 at the planning probabilities
#> 
#>  Stratum n1 n2  S hat.p hat.p1 hat.p2 N1.final N2.final n1.stage2 n2.stage2
#>        1 15 15 12   0.4    0.5    0.3       47       47        32        32
#>        2 15 15 18   0.6    0.7    0.5       47       47        32        32