Type I Error Rate of Blinded Sample Size Re-Estimation for a Stratified Design

Calculates the type I error rate of a two-arm trial with a binary endpoint, stratified into two or more strata, with blinded sample size re-estimation, over the common response probabilities of the strata under the null hypothesis, together with that of the fixed design with the planned group sizes. With two strata the rate is computed exactly and its largest value over a rectangle of probabilities is certified; with three or more strata the rate of the re-estimation design is estimated by simulation.

Usage

StratTypeIErrorBSSR(
  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"),
  nsim = 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: 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)
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 is located. With 'certified' (the default with two strata, and available only with two) it is certified over the rectangle from the smallest to the largest value of theta in each stratum, see Details. With 'grid' (the default with three or more strata) it is the largest value at the points of theta
nsim: Number of simulated trials at each point 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_stie, a data frame with one row per point, containing the probabilities theta.S1, theta.S2, ..., the type I error rates TIE.BSSR and TIE.TRAD of the designs with re-estimation and without it, and, with three or more strata or a random allocation, the Monte Carlo standard error se.BSSR, and with three or more strata and a random allocation se.TRAD for TIE.TRAD. The attribute max is a data frame with the location, the value and, with maximize = 'certified', the upper bound bound of the largest rate of each design, and with a random allocation the standard error se of the rate at the point of theta with the largest rate (NA for the exact rate of the fixed design with two strata).

Details

Under the null hypothesis both groups of stratum j respond with the probability theta_j. Given the interim and the second-stage numbers of responders of a stratum, the numbers of responders of group 1 are hypergeometric whatever theta_j, so with two strata the type I error rate is a polynomial in (theta_1, theta_2). Its coefficients in the products of the Bernstein bases of the two strata are computed exactly, and the rate is evaluated at the points of theta from them. Over a rectangle the polynomial lies between the smallest and the largest of its coefficients there, so with maximize = 'certified' the rectangle is halved along its longer side until the largest coefficient of every piece exceeds the largest value found by at most 1e-12. The largest value is reported with this upper bound, which the type I error rate cannot exceed anywhere in the rectangle. The fixed design is certified in the same way.

With three or more strata the type I error rate of the re-estimation design is estimated at each point from nsim simulated trials, with the random numbers of R, and that of the fixed design with the fixed allocation is computed exactly.

With a random allocation and two strata the allocation sequences are drawn as in StratPowerBSSR. Given a sequence the type I error rate is a polynomial as above, and the rate of the re-estimation design is the average of these polynomials over nseq sequences, itself a polynomial, whose largest value is certified as above. The certificate holds for this average, and se.BSSR gives the Monte Carlo standard error of the average at each point, from the variation between the sequences; the attribute max gives it at the point of theta with the largest rate. The rate of the fixed design with the planned total and the same allocation is exact: the sum over the stratum and group sizes of their probability times the polynomial of the fixed design they give.

With a random allocation and three or more strata (alloc = 'stratum' or 'both') the allocation is simulated with the responses as in StratPowerBSSR, and the rates of both designs are estimated at each point from nsim trials each.

Author

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

Examples

tie <- StratTypeIErrorBSSR(Delta.A = 0.2, N1 = c(20, 20), N2 = c(20, 20), omega = 0.5,
                           r = 1, alpha = 0.025, tar.power = 0.8, Test = 'CMH',
                           ss.method = 'standard', theta = seq(0.1, 0.9, by = 0.2))
attr(tie, 'max')
#>   Design theta.S1 theta.S2        TIE      bound
#> 1   BSSR      0.5      0.5 0.02508517 0.02508517
#> 2   TRAD      0.5      0.5 0.02833680 0.02833680