Adjusted Significance Level of a Blinded Sample Size Re-estimation Design

Lowers the nominal significance level of a two-arm trial with a binary endpoint and blinded sample size re-estimation (BSSR) until the largest type I error rate over the common response probability, or over the pooled response probability on the boundary of a non-inferiority hypothesis, does not exceed the target level alpha, and returns the adjusted level together with that of the fixed-sample design.

Usage

BinaryAlphaAdjBSSR(
  Delta.A,
  N1,
  N2,
  omega = NULL,
  r,
  alpha,
  tar.power,
  Test,
  restricted = FALSE,
  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,
  n.interim = NULL,
  theta = seq(0, 1, by = 0.005),
  maximize = c("certified", "refined", "grid"),
  adjust = c("test", "both"),
  tol = 1e-08,
  step = 1e-05,
  margin = 0,
  ref.pvalue = FALSE,
  margin.scale = c("RD", "RR")
)

Arguments

Delta.A: Assumed treatment effect, on the scale given by effect, used to split the blinded pooled proportion into group-specific proportions
N1: Initial sample size of group 1
N2: Initial sample size of group 2
omega: Fraction of the initial sample size observed at the interim analysis. The interim size of group 2 is ceiling(omega N2) and that of group 1 is ceiling(r ceiling(omega N2)), so the interim analysis keeps the allocation ratio. Either omega or n.interim must be supplied
r: Allocation ratio to group 1
alpha: Target level of significance, which the type I error rate must not exceed
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 initial sample size. Default is FALSE
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 and Delta.T. Options: 'RD' (default) for the risk difference p1 - p2, 'RR' for the risk ratio p1 / p2 or 'OR' for the odds ratio
ss.method: How the sample size is re-estimated from the recovered proportions. 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 when adjust = 'test'. Default is alpha. It is ignored when adjust = 'both'
rounding: How the re-estimated sample size is turned into whole numbers. Options: 'group' (default), 'friede-kieser', 'total' or 'nearest'. Only 'group' is available with ss.method = 'exact'. See Details
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) for none beyond the interim total. It can be used to keep the patients who are already enrolled but not yet evaluated at the interim analysis
N.max: Upper bound on the final total sample size, or NULL (default) for none
n.interim: Interim sample sizes of group 1 and group 2, as a vector of length two. An alternative to omega
theta: Grid of common response probabilities at which the type I error rate is evaluated. Default is seq(0, 1, by = 0.005). With a non-zero margin, or a margin on the scale of the risk ratio, theta is the pooled response probability (r p1 + p2) / (1 + r) on the boundary of the null hypothesis, see Details
maximize: How the largest type I error rate at a level is located, as in BinaryTypeIErrorBSSR. With 'certified' (default) the adjusted level controls the type I error rate over the whole interval from the smallest to the largest value of theta, restricted with a margin to the values at which both response probabilities on the null boundary lie in the unit interval, see Details. With 'refined' and 'grid' the control is assessed on the grid
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). The bisection is used when adjust = 'test' and, whatever adjust, for the fixed-sample design. Default is 1e-8
step: Decrement of the level in the search used when adjust = 'both'. Default is 1e-5
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. It applies to the final analysis, to the fixed-sample comparator and to the exact re-estimation
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_alphaadj, a data frame with one row for the BSSR design and one for the fixed-sample 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, see Details. 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, 0 when alpha.adj is 0
theta.adj: Common response probability, or pooled response probability on the null boundary, 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

For alternative = 'equivalence' the largest rates are taken over both boundaries of the null hypothesis, see BinaryTypeIErrorBSSR, and the column boundary.adj gives the boundary, 'lower' or 'upper', on which max.TIE.adj occurs.

Details

Following Kieser and Friede (2000) and Friede and Kieser (2004), the nominal level of the final analysis is lowered until the largest type I error rate over the common response probability, evaluated as in BinaryTypeIErrorBSSR, does not exceed alpha.

With adjust = 'test' the final sample sizes do not depend on the nominal level, and the rejection regions shrink as the level decreases, so the largest type I error rate is a non-decreasing step function of the level. The bisection locates, to within tol * alpha, the level above which the largest rate exceeds alpha. The package rejects a p-value at a level when it is below the level by more than a tolerance of about 1.49e-8, so every level between the largest p-value rejected at the level found plus this tolerance and the smallest p-value not rejected gives the same rejection regions, also when a p-value is rejected if it is below the level or at most equal to it. The reported alpha.adj is the largest value with at most six significant digits in this range, or with more digits if there is none, so the printed value can be used as it is. The same applies to the fixed-sample design.

With adjust = 'both' the re-estimated sample sizes change with the level as well, so the type I error rate need not be monotone in the level. The level is then lowered from alpha in steps of step until the type I error rate is controlled. The result is the first level among alpha - step, alpha - 2 * step, and so on, at which the type I error rate is controlled, and it can change with step. The levels that control the type I error rate need not form an interval, so a level below the result can fail, and a level above it that is not on this grid can pass. This search re-estimates the sample size at every step and can take much longer than the bisection. The fixed-sample design always uses the bisection, since its sample size does not depend on the level. The script reproduce-published.R in the folder system.file('reproduce', package = 'bbssr') reproduces the published values of Section 5 of Friede and Kieser (2004) and of Example 21.1 of Kieser (2020) only when the re-estimation also uses the adjusted level, as with adjust = 'both'.

The type I error rate can also be controlled within a confidence interval for the common response probability, or for the pooled response probability on the null boundary, instead of over its whole range (Kieser, 2020, Section 22.1.3). A value gamma between 0 and alpha is fixed in advance, and the final analysis uses a level at which the largest type I error rate over the 1 - gamma confidence interval computed from the data of the completed trial does not exceed alpha - gamma. The type I error rate is then controlled at alpha. This level is found when theta spans the interval and alpha is set to alpha - gamma. The interval is known only at the end of the trial, so the re-estimation keeps the nominal level, with adjust = 'test' and ss.alpha set to the nominal level. Neither search examines levels above alpha - gamma. The script reproduce-published.R reproduces the adjusted level of Example 23.1 of Kieser (2020), which uses the Clopper-Pearson interval with gamma = 1e-4.

With maximize = 'certified' the levels are assessed on the grid with the refinement of maximize = 'refined', and the level found is then certified as in BinaryTypeIErrorBSSR: it is accepted only if the upper bound of the type I error rate over the interval from the smallest to the largest value of theta does not exceed alpha. If the bound exceeds alpha, the grid has missed a value of theta at which the level fails, or the largest rate lies within 1e-12 below alpha, and the search continues below that level. The bisection then certifies every level that passes the assessment on the grid, and the search of adjust = 'both' certifies every such level from the start. The adjusted level therefore controls the type I error rate over the whole interval. If a certification stops at its limit of 1e5 subdivisions, with a warning, the bound can exceed the largest rate by more than 1e-12, and the level found can be lower than necessary.

In both searches the type I error rate at a new level is first evaluated at the single value of theta where the largest rate of the last level that failed was found. If it exceeds alpha there, the level fails without the evaluation over the whole grid. The largest rate over theta is never below the rate at any one value, so such a level also fails on the largest rate. With maximize = 'grid' this value is a grid point, and every decision is that of the evaluation on the grid. With 'refined' and 'certified' it can lie between the grid points, where the grid can miss a rate above alpha, so a level can fail that the evaluation on the grid would accept. This can only lower the level found, and with 'certified' the level found is certified in either case.

References

Kieser M, Friede T (2000). Re-calculating the sample size in internal pilot study designs with control of the type I error rate. Statistics in Medicine, 19(7), 901-911.

Friede T, Kieser M (2004). Sample size recalculation for binary data in internal pilot study designs. Pharmaceutical Statistics, 3(4), 269-279.

Kieser M (2020). Methods and Applications of Sample Size Calculation and Recalculation in Clinical Trials. Springer, Cham.

Author

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

Examples

# \donttest{
BinaryAlphaAdjBSSR(
  Delta.A = 0.3, N1 = 39, N2 = 39, n.interim = c(20, 20), r = 1,
  alpha = 0.025, tar.power = 0.8, Test = 'Chisq', ss.method = 'standard',
  theta = seq(0.01, 0.99, by = 0.01)
)
#> Adjusted significance level for blinded sample size re-estimation
#> 
#>   Test            : Chisq
#>   Alternative     : greater
#>   Adjusted part   : final analysis only
#>   Maximum         : certified over theta in [0.01, 0.99]
#>   Target level    : 0.025
#> 
#>        Design   max.TIE alpha.adj max.TIE.adj
#>          BSSR 0.0272864 0.0204375   0.0237418
#>  Fixed sample 0.0293761 0.0207700   0.0247660
# }