Sample Size Calculation for Two-Arm Trials with Binary Endpoints

Calculates the required sample size 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. The Blackwelder and Farrington-Manning tests also allow a non-inferiority margin and a test of equivalence. The sample size can be obtained from the exact power of the selected test or from the normal approximation.

Usage

BinarySampleSize(
  p1,
  p2,
  r,
  alpha,
  tar.power,
  Test,
  alternative = c("greater", "less", "two.sided", "equivalence"),
  tsmethod = c("minlike", "central", "blaker"),
  n.grid = 100,
  bb.gamma = 0,
  method = c("exact", "standard", "null.variance", "alternative.variance"),
  rounding = c("group", "friede-kieser", "total", "nearest"),
  search = c("crossing", "smallest", "stable"),
  search.limit = c(2, 50),
  margin = 0,
  ref.pvalue = FALSE,
  margin.scale = c("RD", "RR")
)

Arguments

p1

True probability of responders for group 1

p2

True probability of responders for group 2

r

Allocation ratio to group 1 (i.e., allocation ratio of group 1:group 2 = r:1, r > 0)

alpha

Level of significance for the alternative specified by alternative

tar.power

Target power

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), which requires p1 - p2 > -margin, 'less', which requires p1 - p2 < margin, or 'two.sided', which requires p1 != p2. With margin.scale = 'RR', 'greater' requires p1 > margin p2 and 'less' requires p1 < margin p2. 'equivalence' requires -lower < p1 - p2 < upper, or lower < p1 / p2 < upper on the scale of the risk ratio

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

method

How the sample size is obtained. 'exact' (default) steps from the normal approximation, one patient at a time, to the sample size at which the exact power of Test first attains the target, see Details. 'standard' uses the normal approximation with the variance under the null hypothesis for the significance term and the variance under the alternative for the power term, formula (21.3) of Kieser (2020). 'null.variance' uses the variance under the null hypothesis for both terms, formula (1) of Friede and Kieser (2004). 'alternative.variance' uses the variance under the alternative for both terms, as in Blackwelder (1982). See Details for a non-zero margin and for a margin on the scale of the risk ratio

rounding

How an unrounded sample size from the normal approximation is turned into whole numbers. 'group' (default) rounds up the size of group 2 and gives group 1 ceiling(r N2) patients. 'friede-kieser' rounds up the two group sizes separately, as in Friede and Kieser (2004). 'total' rounds up the total and gives group 2 floor(N / (1 + r)) patients. 'nearest' rounds the two group sizes separately to the nearest whole number, as in Farrington and Manning (1990). Only 'group' is available with method = 'exact'

search

How the search of method = 'exact' chooses the size of group 2. 'crossing' (default) steps from the normal approximation to a size that attains the target power while the size one unit smaller does not. 'smallest' returns the smallest size that attains the target power. 'stable' returns the smallest size from which every size up to a limit attains it. See Details. Ignored by the other methods

search.limit

Two numbers a and b giving the limit max(ceiling(a n0), ceiling(n0 + b)) of the size of group 2 examined by search = 'stable', where n0 is the size of group 2 from the normal approximation, rounded up. a must be at least 1 and b non-negative. Default is c(2, 50)

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_samplesize, a data frame with one row containing:

p1

True probability of responders for group 1

p2

True probability of responders for group 2

r

Allocation ratio to group 1

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 at the selected sample size

N1

Required sample size of group 1

N2

Required sample size of group 2

N

Total required sample size

The attribute search holds the search, and the attribute search.limit the limit of the size of group 2 examined by search = 'stable' (otherwise NA).

Details

With the default search = 'crossing' the calculation uses a three-step approach:

1. Calculate an initial sample size from the normal approximation to the chi-squared test

2. Evaluate the exact power at the initial sample size

3. If the exact power attains the target power, lower the size of group 2 one unit at a time as long as the power still attains it; otherwise raise it one unit at a time until the power attains it

The normal approximation of the first step uses alpha for a one-sided alternative, alpha / 2 for a two-sided alternative, and alpha for each of the two one-sided tests of equivalence. Only the starting value of the search is affected, so the returned sample size is exact in every case.

The search stops where the exact power crosses the target: the returned size of group 2 attains the target power and the size one unit smaller does not, unless the returned size is 1. The exact power is not monotone in the sample size, so a smaller size further from the normal approximation may also attain the target power, and a larger size may fall short of it.

The two other searches do not depend on where the normal approximation starts. search = 'smallest' scans the size of group 2 upwards from 1 and returns the first size that attains the target power. It visits every smaller size, which takes longer for the unconditional tests. search = 'stable' scans downwards from the limit given by search.limit and returns the smallest size from which every size up to the limit attains the target power, so that adding patients within this range does not take the power below the target. The search stops with an error if the limit itself falls short of the target power. The three searches can return different sizes because the exact power is not monotone in the sample size.

Under method = 'standard', 'null.variance' or 'alternative.variance' the steps above are replaced by the closed-form normal approximation and the rounding rule selected by rounding. The Power column still reports the exact power of Test at the resulting sample size.

With a non-zero margin, the difference p1 - p2 in the normal approximation is replaced by its distance p1 - p2 + margin from the boundary of the null hypothesis, and the variance under the null hypothesis is evaluated at the large sample values of the restricted maximum likelihood estimates of Farrington and Manning (1990). The method 'standard' then gives formula (4) of Farrington and Manning (1990), which is formula (2) of Friede et al. (2007), and 'alternative.variance' gives the formula of Blackwelder (1982), which is formula (1) of Friede et al. (2007). The exact search starts from the formula of Farrington and Manning.

With margin.scale = 'RR' and the ratio R0 = margin, the difference in the normal approximation is the distance p1 - R0 p2, or R0 p2 - p1 for alternative = 'less', and the variance of the size of group 2 is p1 (1 - p1) / r + R0^2 p2 (1 - p2), evaluated at the large sample values of the restricted maximum likelihood estimates under p1 = R0 p2 for the significance term. The method 'standard' then gives formula (8) of Farrington and Manning (1990), 'alternative.variance' gives the formula of their Method 1, and the exact search starts from formula (8).

For alternative = 'equivalence' the approximate power of each one-sided test is obtained as above for its own margin, at the level alpha, and the sample size of the normal approximation is the smallest unrounded size of group 2 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 exact search starts from this size.

References

Friede T, Kieser M (2004). Sample size recalculation for binary data in internal pilot study designs. Pharmaceutical Statistics, 3(4), 269-279.

Kieser M (2020). Methods and Applications of Sample Size Calculation and Recalculation in Clinical Trials. Springer, Cham.

Blackwelder WC (1982). "Proving the null hypothesis" in clinical trials. Controlled Clinical Trials, 3(4), 345-353.

Farrington CP, Manning G (1990). Test statistics and sample size formulae for comparative binomial trials with null hypothesis of non-zero risk difference or non-unity relative risk. Statistics in Medicine, 9(12), 1447-1454.

Friede T, Mitchell C, Mueller-Velten G (2007). Blinded sample size reestimation in non-inferiority trials with binary endpoints. Biometrical Journal, 49(6), 903-916.

Author

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

Examples

# One-sided chi-squared test
BinarySampleSize(p1 = 0.4, p2 = 0.2, r = 1, alpha = 0.025,
                 tar.power = 0.8, Test = 'Chisq')
#> Sample size for a two-arm trial with a binary endpoint
#> 
#>   Test             : Chisq
#>   Alternative      : greater
#>   Response rates   : p1 = 0.4, p2 = 0.2
#>   Allocation ratio : 1 to 1
#>   Alpha            : 0.025
#>   Target power     : 0.8
#>   Exact search     : crossing
#> 
#>   Required sample size: N1 = 80, N2 = 80, total N = 160
#>   Attained power      : 0.8009

# \donttest{
# Two-sided Fisher exact test
BinarySampleSize(p1 = 0.5, p2 = 0.2, r = 2, alpha = 0.05,
                 tar.power = 0.9, Test = 'Fisher', alternative = 'two.sided')
#> Sample size for a two-arm trial with a binary endpoint
#> 
#>   Test             : Fisher
#>   Alternative      : two.sided
#>   Response rates   : p1 = 0.5, p2 = 0.2
#>   Allocation ratio : 2 to 1
#>   Alpha            : 0.05
#>   Target power     : 0.9
#>   Exact search     : crossing
#> 
#>   Required sample size: N1 = 78, N2 = 39, total N = 117
#>   Attained power      : 0.9013

# Normal approximation with the variance under the null hypothesis, two-sided test
# at level 0.05, as in Friede and Kieser (2004)
BinarySampleSize(p1 = 0.4, p2 = 0.2, r = 1, alpha = 0.05, tar.power = 0.8,
                 Test = 'Chisq', alternative = 'two.sided',
                 method = 'null.variance')
#> Sample size for a two-arm trial with a binary endpoint
#> 
#>   Test             : Chisq
#>   Alternative      : two.sided
#>   Response rates   : p1 = 0.4, p2 = 0.2
#>   Allocation ratio : 1 to 1
#>   Alpha            : 0.05
#>   Target power     : 0.8
#> 
#>   Required sample size: N1 = 83, N2 = 83, total N = 166
#>   Attained power      : 0.8137

# Non-inferiority with a margin of 0.1, formula (2) of Friede et al. (2007)
BinarySampleSize(p1 = 0.7, p2 = 0.7, r = 1, alpha = 0.025, tar.power = 0.8,
                 Test = 'Farrington-Manning', method = 'standard',
                 rounding = 'friede-kieser', margin = 0.1)
#> Sample size for a two-arm trial with a binary endpoint
#> 
#>   Test             : Farrington-Manning
#>   Alternative      : greater
#>   Response rates   : p1 = 0.7, p2 = 0.7
#>   Allocation ratio : 1 to 1
#>   Margin           : 0.1
#>   Alpha            : 0.025
#>   Target power     : 0.8
#> 
#>   Required sample size: N1 = 329, N2 = 329, total N = 658
#>   Attained power      : 0.8006

# Non-inferiority on the scale of the risk ratio: the null hypothesis p1 / p2 <= 0.8
# with equal response probabilities of 0.6
BinarySampleSize(p1 = 0.6, p2 = 0.6, r = 1, alpha = 0.025, tar.power = 0.8,
                 Test = 'Farrington-Manning', margin = 0.8, margin.scale = 'RR')
#> Sample size for a two-arm trial with a binary endpoint
#> 
#>   Test             : Farrington-Manning
#>   Alternative      : greater
#>   Response rates   : p1 = 0.6, p2 = 0.6
#>   Allocation ratio : 1 to 1
#>   Margin           : 0.8 (risk ratio)
#>   Alpha            : 0.025
#>   Target power     : 0.8
#>   Exact search     : crossing
#> 
#>   Required sample size: N1 = 215, N2 = 215, total N = 430
#>   Attained power      : 0.802
# }