Rejection Region of a Stratified Test

Returns the critical values of a test for a two-arm trial with a binary endpoint stratified into two or more strata, for every combination of the numbers of responders of the strata. The tests are based on S, the number of responders of group 1 summed over the strata. Given the numbers of responders z = (z_1, ..., z_J) of the strata, each test rejects the null hypothesis when S is at most a lower or at least an upper critical value. The unstratified tests, which ignore the strata, are included for comparison.

Usage

StratRR(
  N1,
  N2,
  alpha,
  Test,
  alternative = c("greater", "less", "two.sided"),
  tsmethod = c("minlike", "central", "blaker")
)

Arguments

N1: Sizes of group 1 in the strata, one element per stratum
N2: Sizes of group 2 in the strata, one element per stratum
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

Value

An object of class bbssr_srr, a data frame with one row for each combination of the numbers of responders of the strata, the first stratum varying fastest, containing:

z.S1, z.S2, ...: Numbers of responders of the strata
S.min, S.max: Smallest and largest possible value of S
lower: Lower critical value: the null hypothesis is rejected when S <= lower. NA when no value of S is rejected in the lower tail
upper: Upper critical value: the null hypothesis is rejected when S >= upper. NA when no value of S is rejected in the upper tail

Details

In stratum j, with N_j = N1[j] + N2[j] patients and z_j responders, the null expectation of the number of responders of group 1 is N1[j] z_j / N_j, and E is the sum of these over the strata. The Cochran-Mantel-Haenszel statistic is (S - E) / sqrt(V) with the hypergeometric variance V = sum_j N1[j] N2[j] z_j (N_j - z_j) / (N_j^2 (N_j - 1)), and Cochran's statistic uses the binomial variance sum_j N1[j] N2[j] z_j (N_j - z_j) / N_j^3 instead. The continuity correction reduces |S - E| by 0.5 when it is at least 0.5, as in mantelhaen.test; a deviation of exactly 0.5 is recognised although E, a sum over the strata, carries rounding errors. Nam (1992) reduces S - E by 0.5 in every outcome, which makes a difference only where |S - E| < 0.5. There S is the integer next to E, and since the hypergeometric variance is at least f (1 - f), with f the fractional part of E, and the binomial variance at least half of it, the one-sided p-value of 'Cochran-cc' exceeds 0.078. The two corrections therefore reject the same outcomes at one-sided levels up to 0.078. The statistics are referred to the standard normal distribution, with the absolute value for a two-sided alternative, and an outcome whose variance is 0 is not rejected. The stratified exact conditional test uses the distribution of S given z, the convolution of the hypergeometric distributions of the strata. The two-sided p-value 'central' is twice the smaller tail probability (Jung, 2014), and 'minlike' and 'blaker' order the values of S by their probability and by their smaller tail probability, with the ties of BinaryRR. As in the other functions of the package, a p-value is rejected when it is below alpha by more than about 1.49e-8.

The unstratified tests apply the corresponding test of a single 2 x 2 table to the table of totals, with sum_j N1[j] and sum_j N2[j] patients and sum_j z_j responders, so their critical values depend on z only through this total. Pearson's statistic of that table is Cochran's statistic of one stratum, as in 'Chisq' of BinaryRR, and the continuity correction is that of 'Cochran-cc'; with the correction applied to every outcome, as in the \(z_u\) test of Nam (1992, Section 2.2), the same outcomes are rejected at one-sided levels up to 0.078. An unstratified test is valid when the distribution of the strata is the same in the two groups, and its level can be distorted otherwise (Nam, 1992; Jung, 2014); its power and type I error rate are computed here with the response probabilities of each stratum, so these consequences can be evaluated.

The function stores the critical values for reuse during the session, like the rejection regions of the other designs.

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

rr <- StratRR(N1 = c(10, 12), N2 = c(10, 12), alpha = 0.025, Test = 'CMH')
print(rr)
#> Rejection region of a stratified test
#> 
#>   Test             : CMH
#>   Alternative      : greater
#>   Group sizes      : N1 = (10, 12), N2 = (10, 12)
#>   Alpha            : 0.025
#>   Combinations     : 525, of which 485 have a rejected value of S
#> 
#>  z.S1 z.S2 S.min S.max lower upper
#>     4    0     0     4    NA     4
#>     5    0     0     5    NA     5
#>     6    0     0     6    NA     6
#>     7    0     0     7    NA     6
#>     8    0     0     8    NA     7
#>     9    0     0     9    NA     7
#>    10    0     0    10    NA     8
#>    11    0     1    10    NA     8
#>    12    0     2    10    NA     9
#>    13    0     3    10    NA     9
#>    14    0     4    10    NA    10
#>    15    0     5    10    NA    10
#>    16    0     6    10    NA    10
#>     3    1     0     4    NA     4
#>     4    1     0     5    NA     5
#>     5    1     0     6    NA     6
#>     6    1     0     7    NA     6
#>     7    1     0     8    NA     7
#>     8    1     0     9    NA     7
#>     9    1     0    10    NA     8
#>   ... 465 more rows