Sample Size for Paired Binary Data — PairedSampleSize • bbssr

Calculates the number of pairs required for a test comparing two treatments on a binary endpoint in a matched-pairs design to attain a target power. The number can be obtained from the exact power of the selected test or from the normal approximation.

Usage

PairedSampleSize(
  p10,
  p01,
  alpha,
  tar.power,
  Test,
  alternative = c("greater", "less", "two.sided", "equivalence"),
  n.grid = 100,
  bb.gamma = 0,
  method = c("exact", "standard"),
  search = c("crossing", "smallest", "stable"),
  search.limit = c(2, 50),
  margin = 0,
  ref.pvalue = FALSE
)

Arguments

p10: Probability that only treatment 1 responds in a pair
p01: Probability that only treatment 2 responds in a pair
alpha: Level of significance for the alternative specified by alternative
tar.power: Target power
Test: Type of statistical test. Options: 'McNemar', 'McNemar-cc', 'McNemar-exact', 'McNemar-midP', 'McNemar-uncond' or 'Paired-Score'
alternative: Direction of the alternative hypothesis. Options: 'greater' (default) for p10 - p01 > -margin, 'less' for p10 - p01 < margin, 'two.sided' for p10 != p01, or 'equivalence' for -lower < p10 - p01 < upper
n.grid: Number of grid points used to search over the probability of a discordant pair in the test 'McNemar-uncond'. Default is 100
bb.gamma: Confidence level parameter of the Berger-Boos procedure for the test 'McNemar-uncond'. The default of 0 disables the procedure
method: How the number of pairs is obtained. 'exact' (default) steps from the normal approximation to the number of pairs at which the exact power of Test attains the target, see Details. 'standard' rounds up the normal approximation, formula (9) of Nam (1997), which is formula (3) of Connor (1987) when the margin is 0
search: How the search of method = 'exact' chooses the number of pairs. 'crossing' (default) steps from the normal approximation to a number that attains the target power while the number one smaller does not. 'smallest' returns the smallest number that attains the target power. 'stable' returns the smallest number from which every number up to a limit attains it. Ignored by method = 'standard'
search.limit: Two numbers a and b giving the limit max(ceiling(a n0), ceiling(n0 + b)) of the number of pairs examined by search = 'stable', where n0 is the normal approximation rounded up. a must be at least 1 and b non-negative. Default is c(2, 50)
margin: Margin on the scale of the difference p10 - p01. The default of 0 gives a test of superiority, and a value other than 0 tests the null hypothesis p10 - p01 <= -margin against p10 - p01 > -margin when alternative is 'greater', and p10 - p01 >= margin against p10 - p01 < margin when it is 'less'. A negative value tests for superiority by more than its absolute value. A margin other than 0 requires Test = 'Paired-Score' and a one-sided alternative. For alternative = 'equivalence' the margin is the pair c(lower, upper), both in (0, 1), or a single value used for both, and the null hypothesis p10 - p01 <= -lower or p10 - p01 >= upper is tested
ref.pvalue: Logical. If TRUE, the maximization over the probability of a discordant pair in the test 'McNemar-uncond' is refined between the grid points, see Details. Default is FALSE

Value

An object of class bbssr_psamplesize, a data frame with one row containing:

p10, p01: Probabilities of the two kinds of discordant pairs
alpha: Level of significance
tar.power: Target power
Test: Name of the statistical test
alternative: Direction of the alternative hypothesis
Power: Exact power of Test with the required number of pairs
N: Required number of pairs

The attributes hold the remaining settings, among them the unrounded normal approximation n.raw and the limit search.limit of the search 'stable' (otherwise NA).

Details

The normal approximation is formula (9) of Nam (1997), n = (z.a sqrt(v0) + z.b sqrt(v1))^2 / (d1 - d0)^2, where d1 = p10 - p01 is the true difference, d0 = -margin the difference under the null hypothesis, v1 = p10 + p01 - d1^2 the variance under the alternative, and v0 = 2 p01.bar + d0 - d0^2 the variance under the null hypothesis evaluated at the large sample value p01.bar of the restricted maximum likelihood estimate of p01, formula (5) of that article. z.a is the upper alpha quantile of the standard normal distribution, the upper alpha / 2 quantile for a two-sided alternative, and z.b the tar.power quantile. With margin = 0 the formula is formula (3) of Connor (1987). For alternative = 'less' the probabilities p10 and p01 are exchanged. For alternative = 'equivalence' the approximate power of each one-sided test is obtained in the same way for its own margin, at the level alpha, and the normal approximation is the smallest number of pairs at which the sum of the two approximate powers minus 1 attains the target power. The sum minus 1 is a lower bound of the approximate probability that both tests reject.

The normal approximation does not depend on the test, and for 'McNemar-cc' it leaves out the continuity correction. It is the starting point of the exact search for every test. The exact power is not monotone in the number of pairs, so the three searches can return different numbers, see BinarySampleSize.

References

Connor RJ (1987). Sample size for testing differences in proportions for the paired-sample design. Biometrics, 43(1), 207-211.

Nam J (1997). Establishing equivalence of two treatments and sample size requirements in matched-pairs design. Biometrics, 53(4), 1422-1430.

Examples

# Example of Connor (1987, p. 210): theta1 = 0.8, theta2 = 0.7 and psi = 0.3
PairedSampleSize(p10 = 0.2, p01 = 0.1, alpha = 0.05, tar.power = 0.9,
                 Test = 'McNemar', method = 'standard')
#> Number of pairs for paired binary data
#> 
#>   Test           : McNemar
#>   Alternative    : greater
#>   Discordance    : p10 = 0.2, p01 = 0.1
#>   Alpha          : 0.05
#>   Target power   : 0.9
#>   Method         : normal approximation
#> 
#>   Required number of pairs: N = 254
#>   Attained power          : 0.9062

# \donttest{
# Exact number of pairs of the exact conditional test
PairedSampleSize(p10 = 0.2, p01 = 0.1, alpha = 0.05, tar.power = 0.9,
                 Test = 'McNemar-exact')
#> Number of pairs for paired binary data
#> 
#>   Test           : McNemar-exact
#>   Alternative    : greater
#>   Discordance    : p10 = 0.2, p01 = 0.1
#>   Alpha          : 0.05
#>   Target power   : 0.9
#>   Exact search   : crossing
#> 
#>   Required number of pairs: N = 269
#>   Attained power          : 0.9
# }