Sample Size of a Stratified Test

Calculates the total sample size of a two-arm trial with a binary endpoint stratified into two or more strata, for a test of StratRR, either exactly or by the normal approximation of Woolson et al. (1986) and Nam (1992).

Usage

StratSampleSize(
  p1,
  p2,
  frac,
  r = 1,
  alpha,
  tar.power,
  Test,
  alternative = c("greater", "less", "two.sided"),
  tsmethod = c("minlike", "central", "blaker"),
  method = c("exact", "standard"),
  search = c("crossing", "smallest", "stable"),
  search.limit = c(2, 50),
  alloc = c("fixed", "stratum", "group", "both"),
  block = 4
)

Arguments

p1: Response probabilities of group 1 in the strata, one element per stratum
p2: Response probabilities of group 2 in the strata, one element per stratum
frac: Fractions of the strata, one positive element per stratum. They are rescaled to sum to 1, so they may also be given as numbers of patients
r: Allocation ratio of group 1 to group 2 within the strata, a single value or one value per stratum. Default is 1
alpha: Level of significance, for the alternative specified
tar.power: Target power
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
method: Calculation of the sample size. 'exact' (default) searches for the total sample size at which the exact power of StratPower attains tar.power. 'standard' uses the normal approximation, see Details
search: How the exact calculation chooses among the sizes at which the power fluctuates around tar.power, as in BinarySampleSize. Options: 'crossing' (default), 'smallest' or 'stable'
search.limit: Limit of the search 'stable': the nominal sizes are examined downwards from max(ceiling(search.limit[1] n), ceiling(n + search.limit[2])), where n is the nominal size of the normal approximation. Default is c(2, 50)
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)
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_ssamplesize, a list containing the arguments p1, p2, frac, r, alpha, tar.power, Test and alternative, the group sizes N1 and N2 of the strata (NULL with a random allocation), their total N, and the exact power Power of the design. The attribute n.raw holds the unrounded total of the normal approximation, n the nominal total of the design and search.limit the limit of the nominal total examined by search = 'stable' (otherwise NA). With a random allocation the attributes skipped and n.strata are those of StratPower.

Details

A design is described by the fractions frac of the strata and the allocation ratios r. For a nominal total sample size n, the size of group 2 in stratum j is ceiling(frac[j] n / (1 + r[j])) and that of group 1 is ceiling(r[j] N2[j]), so the total N is at least n. The exact calculation searches over n within this family of designs.

The normal approximation is that of Cochran's test. With s[j] = r[j] / (1 + r[j]), w[j] = frac[j] s[j] (1 - s[j]), D = sum_j w[j] (p1[j] - p2[j]), v0 = sum_j w[j] pbar[j] (1 - pbar[j]) with pbar[j] = s[j] p1[j] + (1 - s[j]) p2[j], and v1 = sum_j w[j] ((1 - s[j]) p1[j] (1 - p1[j]) + s[j] p2[j] (1 - p2[j])), the total sample size is N* = (z.a sqrt(v0) + z.b sqrt(v1))^2 / D^2, formula (2.8) of Woolson et al. (1986), where z.a is the upper alpha quantile (alpha / 2 for a two-sided alternative) and z.b the tar.power quantile of the standard normal distribution. For the tests with a continuity correction it is N = (1 + sqrt(1 + 2 / (D N*)))^2 N* / 4, formula (2.3) of Nam (1992). The same approximation is used for every test, since the Cochran-Mantel-Haenszel statistic differs from Cochran's only by the factors (N_j - 1) / N_j of the variance, and it serves as the starting value of the exact search. The nominal total is ceiling(N*), or ceiling(N); the published tables of Woolson et al. (1986) and Nam (1992) compute N* and N with the quantiles rounded to three decimals, such as 1.645 for the level 0.05 and 0.841 or 0.842 for the power 0.8, and round them to the nearest integer instead.

For the unstratified tests the approximation is that of the comparison of two proportions on the table of totals, formula (3.3) of Woolson et al. (1986), or (2.3) of Nam (1992) with one stratum for 'Unstrat-Chisq-cc': the formula above with one stratum, in which group 1 holds the fraction sum_j frac[j] s[j] of the patients and the response probability of each group is the average of its probabilities over the strata, weighted by its sizes in the strata. The exact power of these tests is that of the stratified design, with the response probabilities of each stratum, so the search accounts for the difference between the strata that the table of totals ignores.

The exact power is not monotone in the sample size, so the result depends on search, as described in BinarySampleSize.

With a random allocation (alloc other than 'fixed', two strata only) the total sample size N is searched directly, the nominal total being the total itself, and the power is that of StratPower with the same allocation. Jung (2014) searches upwards from the sample size of the Mantel-Haenszel test. The power is not monotone, so the result can differ from that of search = 'crossing', which starts from the normal approximation.

References

Woolson RF, Bean JA, Rojas PB (1986). Sample size for case-control studies using Cochran's statistic. Biometrics, 42(4), 927-932.

Nam J (1992). Sample size determination for case-control studies and the comparison of stratified and unstratified analyses. Biometrics, 48(2), 389-395.

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

# Normal approximation and exact sample size with two strata
StratSampleSize(p1 = c(0.5, 0.7), p2 = c(0.3, 0.5), frac = c(0.5, 0.5),
                alpha = 0.025, tar.power = 0.8, Test = 'CMH', method = 'standard')
#> Sample size of a stratified test
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Method           : normal approximation
#>   Alpha            : 0.025, target power 0.8
#>   Normal approx.   : 186 (unrounded total)
#>   Total size       : N = 188, power 0.7985
#> 
#>  Stratum frac r  p1  p2 N1 N2
#>        1  0.5 1 0.5 0.3 47 47
#>        2  0.5 1 0.7 0.5 47 47
StratSampleSize(p1 = c(0.5, 0.7), p2 = c(0.3, 0.5), frac = c(0.5, 0.5),
                alpha = 0.025, tar.power = 0.8, Test = 'CMH')
#> Sample size of a stratified test
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Method           : exact (search crossing)
#>   Alpha            : 0.025, target power 0.8
#>   Normal approx.   : 186 (unrounded total)
#>   Total size       : N = 192, power 0.8085
#> 
#>  Stratum frac r  p1  p2 N1 N2
#>        1  0.5 1 0.5 0.3 48 48
#>        2  0.5 1 0.7 0.5 48 48