Sample Size Re-estimation from Observed Blinded Interim Data

Re-estimates the sample size of an ongoing two-arm trial with a binary endpoint from the blinded data available at an interim analysis, and reports how many patients still have to be enrolled in each group during the second stage. Only the total number of patients and the total number of responders are required, so the treatment allocation remains concealed.

Usage

BinaryBSSR(
  n1,
  n2,
  S,
  Delta.A,
  r,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  N1 = NULL,
  N2 = NULL,
  alternative = c("greater", "less", "two.sided", "equivalence"),
  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,
  margin = 0,
  ref.pvalue = FALSE,
  margin.scale = c("RD", "RR")
)

Arguments

n1: Number of patients of group 1 observed at the interim analysis
n2: Number of patients of group 2 observed at the interim analysis
S: Total number of responders observed at the interim analysis, pooled over both groups
Delta.A: Assumed treatment effect, on the scale given by effect, used to split the blinded pooled proportion into group-specific proportions
r: Allocation ratio to group 1 (i.e., allocation ratio of group 1:group 2 = r:1, r > 0)
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 planned sample size given by N1 and N2. Default is FALSE
N1: Planned sample size of group 1. Required when restricted is TRUE, and when the recovered proportions coincide
N2: Planned sample size of group 2. Required when restricted is TRUE, and when the recovered proportions coincide
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. Options: 'RD' (default), 'RR' or 'OR', as in BinaryPowerBSSR
ss.method: How the sample size is re-estimated. 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', as in BinaryPowerBSSR
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). It can be used to keep the patients who are already enrolled but not yet evaluated
N.max: Upper bound on the final total sample size, or NULL (default)
margin: Non-inferiority margin, on the scale given by margin.scale. On the scale of the risk difference the default of 0 gives a test of superiority, and a value other than 0 tests the null hypothesis p1 - p2 <= -margin against p1 - p2 > -margin when alternative is 'greater', and p1 - p2 >= margin against p1 - p2 < margin when it is 'less'. A negative value tests for superiority by more than its absolute value. On the scale of the risk ratio the margin is a positive ratio R0, and the null hypothesis is p1 / p2 <= R0 against p1 / p2 > R0 when alternative is 'greater', and p1 / p2 >= R0 against p1 / p2 < R0 when it is 'less'. A margin other than 0, and any margin on the scale of the risk ratio, requires Test = 'Blackwelder' or 'Farrington-Manning' and a one-sided alternative or 'equivalence', see BinaryRR. With a margin other than 0 on the scale of the risk difference the assumed and the true effects are risk differences (effect = 'RD'), and with a margin on the scale of the risk ratio they are risk ratios (effect = 'RR'). For alternative = 'equivalence' the margin is the pair c(lower, upper) of margins of a test of equivalence, or a single value used for both. On the scale of the risk difference both margins lie in (0, 1), and the null hypothesis p1 - p2 <= -lower or p1 - p2 >= upper is tested against -lower < p1 - p2 < upper. On the scale of the risk ratio the pair satisfies 0 < lower < 1 < upper, a single ratio R gives the pair of R and 1 / R, the smaller as lower, and the null hypothesis is p1 / p2 <= lower or p1 / p2 >= upper. The null hypothesis is rejected when both one-sided tests reject, each at the level alpha
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
margin.scale: Scale of margin. Options: 'RD' (default) for the risk difference p1 - p2 or 'RR' for the risk ratio p1 / p2

Value

An object of class bbssr_bssr, a data frame with one row containing:

n1: Interim sample size of group 1
n2: Interim sample size of group 2
n: Total interim sample size
S: Total number of interim responders
hat.p: Blinded estimate of the pooled response probability
hat.p1: Recovered response probability of group 1
hat.p2: Recovered response probability of group 2
N1.re: Re-estimated total sample size of group 1
N2.re: Re-estimated total sample size of group 2, before the bounds; with ss.method = 'exact' at most the largest size that N.max allows
N.re: Re-estimated total sample size
n1.stage2: Number of additional patients to enrol in group 1
n2.stage2: Number of additional patients to enrol in group 2
n.stage2: Total number of additional patients to enrol
N1.final: Final sample size of group 1
N2.final: Final sample size of group 2
N.final: Final total sample size
Power: Exact power at the final sample size under the recovered proportions

The attribute search holds the search, and the attribute search.limit the limit of the size of group 2 examined by search = 'stable' (otherwise NA).

Details

The blinded estimate of the pooled response probability is hat.p = S / (n1 + n2). For the risk difference, group-specific proportions are recovered as hat.p1 = hat.p + Delta.A / (1 + r) and hat.p2 = hat.p - r Delta.A / (1 + r), truncated to the unit interval. For the risk ratio and the odds ratio they are the proportions with the pooled value hat.p and the ratio Delta.A. The sample size is then re-estimated as in BinarySampleSize.

Under the unrestricted rule and rounding = 'group', the final size of group 2 is the larger of the re-estimated size and what has already been observed. Under the restricted rule it is raised to the planned size first, so the trial can only grow, and N.min and N.max add a further lower and upper bound on the total. The final size of group 1 is then ceiling(r N2.final), so an imbalance already present at the interim is corrected by the remaining enrolment instead of being carried forward. Neither second-stage size is ever negative. The other rounding rules are described in BinaryPowerBSSR.

While BinaryPowerBSSR evaluates the operating characteristics of a BSSR design at the planning stage, this function is applied once, to the data of a trial that is under way.

Author

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

Examples

# Interim data: 20 patients per group, 11 responders in total
BinaryBSSR(n1 = 20, n2 = 20, S = 11, Delta.A = 0.3, r = 1,
           alpha = 0.025, tar.power = 0.8, Test = 'Chisq')
#> Blinded sample size re-estimation from interim data
#> 
#>   Test             : Chisq
#>   Alternative      : greater
#>   Design rule      : unrestricted
#>   Assumed effect   : 0.3
#>   Alpha            : 0.025, target power 0.8
#> 
#> Interim data
#>   Patients         : n1 = 20, n2 = 20, total n = 40
#>   Responders       : S = 11, blinded pooled rate = 0.275
#>   Recovered rates  : hat.p1 = 0.425, hat.p2 = 0.125
#> 
#> Re-estimation
#>   Required total   : N1 = 33, N2 = 33, total N = 66
#>   Still to enrol   : group 1 = 13, group 2 = 13, total = 26
#>   Final size       : N1 = 33, N2 = 33, total N = 66
#>   Power at final N : 0.8136

# \donttest{
# Restricted rule with a planned sample size of 40 per group
BinaryBSSR(n1 = 20, n2 = 20, S = 11, Delta.A = 0.3, r = 1,
           alpha = 0.025, tar.power = 0.8, Test = 'Boschloo',
           restricted = TRUE, N1 = 40, N2 = 40)
#> Blinded sample size re-estimation from interim data
#> 
#>   Test             : Boschloo
#>   Alternative      : greater
#>   Design rule      : restricted
#>   Assumed effect   : 0.3
#>   Alpha            : 0.025, target power 0.8
#> 
#> Interim data
#>   Patients         : n1 = 20, n2 = 20, total n = 40
#>   Responders       : S = 11, blinded pooled rate = 0.275
#>   Recovered rates  : hat.p1 = 0.425, hat.p2 = 0.125
#> 
#> Re-estimation
#>   Required total   : N1 = 34, N2 = 34, total N = 68
#>   Still to enrol   : group 1 = 20, group 2 = 20, total = 40
#>   Final size       : N1 = 40, N2 = 40, total N = 80
#>   Power at final N : 0.8602
# }