Adjusted Significance Level of Blinded Sample Size Re-Estimation for a Stratified Design

Lowers the nominal significance level of a two-stratum design with blinded sample size re-estimation until the largest type I error rate over the common response probabilities of the strata does not exceed the target level alpha, and returns the adjusted level together with that of the fixed design with the planned group sizes.

Usage

StratAlphaAdjBSSR(
  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 = seq(0, 1, by = 0.05),
  maximize = c("certified", "grid"),
  adjust = c("test", "both"),
  tol = 1e-08,
  step = 1e-05,
  alloc = c("fixed", "stratum", "group", "both"),
  block = 4,
  ss.frac = c("observed", "planned"),
  nseq = 100
)

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: Target level of significance, which the type I error rate must not exceed
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 of significance used for the re-estimation when adjust = 'test'. Default is alpha. It is ignored when adjust = 'both'
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: Common response probabilities of the strata under the null hypothesis. A vector gives, with two strata, the grid of all pairs of its values, and with three or more strata the points at which every stratum has the same value. A matrix with one column per stratum gives the points directly. Default is seq(0, 1, by = 0.05)
maximize: How the largest type I error rate at a level is located, as in StratTypeIErrorBSSR. With 'certified' (default) the adjusted level controls the type I error rate over the whole rectangle spanned by theta. With 'grid' the control is assessed at the points of theta
adjust: Which part of the design uses the adjusted level. 'test' (default) applies it to the final analysis only, so the re-estimation keeps using ss.alpha. 'both' applies it to the re-estimation as well
tol: Tolerance of the bisection, relative to alpha, a single value in (0, 1). Default is 1e-8
step: Decrement of the level in the search used when adjust = 'both'. Default is 1e-5
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_alphaadj, a data frame with one row for the BSSR design and one for the fixed design containing:

Design: 'BSSR' or 'TRAD'
alpha: Target level
max.TIE: Largest type I error rate at the nominal level alpha
alpha.adj: Adjusted nominal level. It is alpha when the nominal level already controls the type I error rate, and 0, with a warning, when no level examined by the bisection does. The search of adjust = 'both' for the BSSR design stops with an error instead when it reaches 0
max.TIE.adj: Largest type I error rate at the adjusted level
theta.S1.adj, theta.S2.adj: Common response probabilities of the strata at which max.TIE.adj occurs, NA when alpha.adj is 0
max.TIE.bound: Upper bound of the type I error rate at the nominal level with maximize = 'certified', otherwise NA
max.TIE.adj.bound: Upper bound of the type I error rate at the adjusted level with maximize = 'certified', otherwise NA

Details

The searches are those of BinaryAlphaAdjBSSR. With adjust = 'test' the final group sizes do not depend on the level, the largest type I error rate is a non-decreasing step function of the level, and a bisection locates the level above which it exceeds alpha, to within tol * alpha. The reported level is the largest value with at most six significant digits that gives the same critical values. With adjust = 'both' the level is lowered from alpha in steps of step until the type I error rate is controlled. Each level requires the coefficients of the type I error rate of StratTypeIErrorBSSR, so the search takes about as long as that many evaluations of the type I error rate.

With a random allocation (alloc other than 'fixed') the type I error rate of the design with re-estimation at each level is that of StratTypeIErrorBSSR, the average over the same nseq allocation sequences at every level, so each level takes about nseq times as long as with the fixed allocation. The adjusted level controls this average, whose Monte Carlo error it does not account for. The rate of the fixed design with the planned total and the same allocation is exact. Both levels are reported as the search found them, without the rounding to six significant digits, since the critical values differ between the allocation sequences and between the stratum and group sizes.

Author

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

Examples

# \donttest{
StratAlphaAdjBSSR(Delta.A = 0.25, N1 = c(15, 15), N2 = c(15, 15), omega = 0.5, r = 1,
                  alpha = 0.025, tar.power = 0.8, Test = 'CMH', ss.method = 'standard',
                  theta = seq(0.05, 0.95, by = 0.05))
#> Adjusted significance level for blinded sample size re-estimation
#> 
#>   Test            : CMH
#>   Alternative     : greater
#>   Adjusted part   : final analysis only
#>   Maximum         : certified over (theta.S1, theta.S2) in [0.05, 0.95] x [0.05, 0.95]
#>   Target level    : 0.025
#> 
#>        Design   max.TIE alpha.adj max.TIE.adj
#>          BSSR 0.0268795 0.0242744   0.0247366
#>  Fixed sample 0.0277886 0.0209052   0.0235488
# }