Exact Power of a Stratified Test

Calculates the exact power of a test of StratRR for a two-arm trial with a binary endpoint, with fixed group sizes in each stratum or, with two strata, with stratum and group sizes that are random.

Usage

StratPower(
  p1,
  p2,
  N1 = NULL,
  N2 = NULL,
  alpha,
  Test,
  alternative = c("greater", "less", "two.sided"),
  tsmethod = c("minlike", "central", "blaker"),
  alloc = c("fixed", "stratum", "group", "both"),
  N = NULL,
  frac = NULL,
  r = 1,
  block = 4
)

Arguments

p1: Response probabilities of group 1 in the strata: a vector with one element per stratum, or a matrix with one row per scenario and one column per stratum
p2: Response probabilities of group 2 in the strata, of the same form as p1
N1: Sizes of group 1 in the strata, one element per stratum, with alloc = 'fixed'
N2: Sizes of group 2 in the strata, one element per stratum, with alloc = 'fixed'
alpha: Level of significance, for the alternative specified
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
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
alloc: Allocation of the patients to the strata and the groups. 'fixed' (default) uses the group sizes N1 and N2. The others need two strata and make the sizes random, see Details: 'stratum' (stratum sizes binomial, permuted blocks within the strata), 'group' (stratum sizes fixed, group sizes binomial) and 'both' (stratum and group sizes binomial)
N: Total sample size, with a random allocation
frac: Fractions of the two strata, with a random allocation. They are rescaled to sum to 1
r: Allocation ratio of group 1 to group 2 within the strata, a single value or one value per stratum, with a random allocation. Default is 1
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

Value

An object of class bbssr_spower, a data frame with one row per scenario containing the response probabilities p1.S1, p1.S2, ... and p2.S1, p2.S2, ... and the exact power Power. With a random allocation the attribute skipped is the total probability of the configurations left out of the sum, see Details, and n.strata the stratum sizes of alloc = 'group'.

Details

The numbers of responders of the two groups in each stratum are independent binomial variables. The power is the probability of the rejection region of StratRR, computed exactly by enumerating the numbers of responders of the strata and carrying the distribution of the partial sum of S over the strata. The response probabilities may differ between the strata in any way, so the power is available also when the odds ratio or the risk difference is not common to the strata.

With alloc other than 'fixed' the design has two strata and a total of N patients, and the sizes of the strata and of the groups within them are random. With 'group' the stratum sizes are fixed at n1 = floor(N frac[1]) and n2 = N - n1, and each patient of stratum j is in group 1 with the probability r[j] / (1 + r[j]), so that the size of group 1 is binomial (Jung, 2014, Section 3.2, where the size of stratum 1 is the round-off of N frac[1]; the integer part used here reproduces all but one of the sample sizes of the column 'n fixed' of Table 4 of the article). With 'both' the size of stratum 1 is binomial as well, with the probability frac[1] (Jung, 2014, Section 3.1). With 'stratum' the stratum sizes are binomial and the groups are allocated by permuted blocks of size block within each stratum: every complete block holds block r[j] / (1 + r[j]) patients of group 1, and the number of patients of group 1 among the patients of the last, incomplete block is hypergeometric. The power is the sum over the stratum and group sizes of their probability times the power of the fixed design they give. Configurations with a probability below 1e-14 are left out, and their total probability, which bounds the error of the power, is returned in the attribute skipped. A group without patients in a stratum leaves that stratum without variation of S, and a stratum without patients leaves a design with one stratum. The number of configurations grows with N most quickly for 'both': at N = 100 the power of one scenario takes a fraction of a second with 'stratum', about a second with 'group' and tens of seconds with 'both'.

References

Jung SH (2014). Stratified Fisher's exact test and its sample size calculation. Biometrical Journal, 56(1), 129-140.

Author

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

Examples

StratPower(p1 = c(0.5, 0.7), p2 = c(0.3, 0.5), N1 = c(40, 40), N2 = c(40, 40),
           alpha = 0.025, Test = 'CMH')
#> Exact power of a stratified test
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Group sizes      : N1 = (40, 40), N2 = (40, 40)
#>   Alpha            : 0.025
#> 
#>  p1.S1 p1.S2 p2.S1 p2.S2  Power
#>    0.5   0.7   0.3   0.5 0.7322

# Several scenarios, and the stratified exact test
StratPower(p1 = rbind(c(0.5, 0.7), c(0.45, 0.65)), p2 = rbind(c(0.3, 0.5), c(0.3, 0.5)),
           N1 = c(40, 40), N2 = c(40, 40), alpha = 0.025, Test = 'Strat-Fisher')
#> Exact power of a stratified test
#> 
#>   Test             : Strat-Fisher
#>   Alternative      : greater
#>   Group sizes      : N1 = (40, 40), N2 = (40, 40)
#>   Alpha            : 0.025
#> 
#>  p1.S1 p1.S2 p2.S1 p2.S2  Power
#>   0.50  0.70   0.3   0.5 0.6792
#>   0.45  0.65   0.3   0.5 0.4249

# Random stratum sizes, with permuted blocks of 4 within the strata
StratPower(p1 = c(0.5, 0.7), p2 = c(0.3, 0.5), alpha = 0.025, Test = 'CMH',
           alloc = 'stratum', N = 160, frac = c(0.5, 0.5))
#> Exact power of a stratified test
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Allocation       : random stratum sizes, permuted blocks of 4 within the strata
#>   Total size       : N = 160, fractions (0.5, 0.5), ratio 1
#>   Left out         : configurations of total probability 8.4e-14
#>   Alpha            : 0.025
#> 
#>  p1.S1 p1.S2 p2.S1 p2.S2  Power
#>    0.5   0.7   0.3   0.5 0.7332