Power Calculation for Two-Arm Trials with Binary Endpoints

Calculates the exact power for two-arm trials with binary endpoints. Seven tests are supported (see BinaryRR), each of which can be applied with a one-sided or a two-sided alternative, and two of them also test non-inferiority with a margin on the scale of the risk difference or the risk ratio. Vectors of response probabilities are accepted.

Usage

BinaryPower(
  p1,
  p2,
  N1,
  N2,
  alpha,
  Test,
  alternative = c("greater", "less", "two.sided", "equivalence"),
  tsmethod = c("minlike", "central", "blaker"),
  n.grid = 100,
  bb.gamma = 0,
  margin = 0,
  ref.pvalue = FALSE,
  margin.scale = c("RD", "RR")
)

Arguments

p1: True probability of responders for group 1 (can be a vector)
p2: True probability of responders for group 2 (can be a vector of the same length as p1)
N1: Sample size for group 1
N2: Sample size for group 2
alpha: Level of significance for the alternative specified by alternative
Test: Type of statistical test. Options: 'Chisq', 'Fisher', 'Fisher-midP', 'Z-pool', 'Boschloo', 'Blackwelder' or 'Farrington-Manning'
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
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. 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
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_power, a data frame with one row per element of p1 containing:

p1: True probability of responders for group 1
p2: True probability of responders for group 2
N1: Sample size for group 1
N2: Sample size for group 2
alpha: Level of significance
Test: Name of the statistical test
alternative: Direction of the alternative hypothesis
Power: Exact power

Details

The power is obtained by summing the joint probability mass function of the two independent binomial counts over the rejection region returned by BinaryRR. The summation covers the whole rejection region rather than a row-wise tail, so it remains valid for two-sided tests, whose rejection regions are not contiguous within a row of the outcome grid.

Author

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

Examples

# Power of the one-sided chi-squared test
BinaryPower(p1 = 0.5, p2 = 0.2, N1 = 5, N2 = 5, alpha = 0.025, Test = 'Chisq')
#> Exact power for a two-arm trial with a binary endpoint
#> 
#>   Test         : Chisq
#>   Alternative  : greater
#>   Sample sizes : N1 = 5, N2 = 5
#>   Alpha        : 0.025
#> 
#>   p1  p2 Power
#>  0.5 0.2 0.183

# \donttest{
# Power over a range of response probabilities for the two-sided Boschloo test
pw <- BinaryPower(p1 = c(0.5, 0.6, 0.7, 0.8), p2 = rep(0.2, 4),
                  N1 = 20, N2 = 20, alpha = 0.05, Test = 'Boschloo',
                  alternative = 'two.sided')
print(pw)
#> Exact power for a two-arm trial with a binary endpoint
#> 
#>   Test         : Boschloo
#>   Alternative  : two.sided
#>   Sample sizes : N1 = 20, N2 = 20
#>   Alpha        : 0.05
#> 
#>   p1  p2  Power
#>  0.5 0.2 0.4923
#>  0.6 0.2 0.7279
#>  0.7 0.2 0.9014
#>  0.8 0.2 0.9808
plot(pw)

# }