Package MKpower
Matthias Kohl
2024-09-23
Introduction
The package includes functions for power analysis and sample size
calculation for Welch and Hsu (Hedderich and
Sachs 2018) t tests including Monte-Carlo simulations of
empirical power and type-I-error. In addition, power and sample size
calculation for Wilcoxon rank sum and signed rank tests via Monte-Carlo
simulations can be performed. Moreover, power and sample size required
for the evaluation of a diagnostic test(-system) (Flahault, Cadilhac, and Thomas 2005; Dobbin and Simon
2007) as well as for a single proportion (Fleiss, Levin, and Paik 2003), comparing two
negative binomial rates (Zhu and Lakkis
2014), ANCOVA (Shieh 2020),
reference ranges (Jennen-Steinmetz and Wellek
2005), and multiple primary endpoints (Sozu et al. 2015).
We first load the package.
Single Proportion
The computation of the required sample size for investigating a
single proportion can be determined via the respective test or
confidence interval (Fleiss, Levin, and Paik
2003). First, we consider the asymptotic test.
## with continuity correction
power.prop1.test(p1 = 0.4, power = 0.8)
##
## Power calculation for testing a given proportion (with continuity correction)
##
## n = 203.7246
## delta = 0.1
## p1 = 0.4
## p0 = 0.5
## sig.level = 0.05
## exact.sig.level = 0.04205139
## power = 0.8
## exact.power = 0.8008049
## alternative = two.sided
##
## NOTE: n = total sample size
## without continuity correction
power.prop1.test(p1 = 0.4, power = 0.8, cont.corr = FALSE)
##
## Power calculation for testing a given proportion
##
## n = 193.8473
## delta = 0.1
## p1 = 0.4
## p0 = 0.5
## sig.level = 0.05
## exact.sig.level = 0.05228312
## power = 0.8
## exact.power = 0.8067065
## alternative = two.sided
##
## NOTE: n = total sample size
Next, we compute the sample size via the respective asymptotic
confidence interval.
## with continuity correction
ssize.prop.ci(prop = 0.4, width = 0.14)
##
## Sample size calculation by method of wald-cc
##
## n = 202.1865
## prop = 0.4
## width = 0.14
## conf.level = 0.95
##
## NOTE: Two-sided confidence interval
## without continuity correction
ssize.prop.ci(prop = 0.4, width = 0.14, method = "wald")
##
## Sample size calculation by method of wald
##
## n = 188.1531
## prop = 0.4
## width = 0.14
## conf.level = 0.95
##
## NOTE: Two-sided confidence interval
Welch Two-Sample t-Test
For computing the sample size of the Welch t-test, we only consider
the situation of equal group size (balanced design).
## identical results as power.t.test, since sd = sd1 = sd2 = 1
power.welch.t.test(n = 20, delta = 1)
##
## Two-sample Welch t test power calculation
##
## n = 20
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.8689528
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.welch.t.test(power = .90, delta = 1)
##
## Two-sample Welch t test power calculation
##
## n = 22.0211
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.welch.t.test(power = .90, delta = 1, alternative = "one.sided")
##
## Two-sample Welch t test power calculation
##
## n = 17.84713
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = one.sided
##
## NOTE: n is number in *each* group
## sd1 = 0.5, sd2 = 1
power.welch.t.test(delta = 1, sd1 = 0.5, sd2 = 1, power = 0.9)
##
## Two-sample Welch t test power calculation
##
## n = 14.53583
## delta = 1
## sd1 = 0.5
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
Hsu Two-Sample t-Test
For computing the sample size of the Hsu t-test (Hedderich and Sachs 2018), we only consider the
situation of equal group size (balanced design).
## slightly more conservative than Welch t-test
power.hsu.t.test(n = 20, delta = 1)
##
## Two-sample Hsu t test power calculation
##
## n = 20
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.8506046
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.hsu.t.test(power = .90, delta = 1)
##
## Two-sample Hsu t test power calculation
##
## n = 23.02186
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.hsu.t.test(power = .90, delta = 1, alternative = "one.sided")
##
## Two-sample Hsu t test power calculation
##
## n = 18.5674
## delta = 1
## sd1 = 1
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = one.sided
##
## NOTE: n is number in *each* group
## sd1 = 0.5, sd2 = 1
power.welch.t.test(delta = 0.5, sd1 = 0.5, sd2 = 1, power = 0.9)
##
## Two-sample Welch t test power calculation
##
## n = 53.86822
## delta = 0.5
## sd1 = 0.5
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.hsu.t.test(delta = 0.5, sd1 = 0.5, sd2 = 1, power = 0.9)
##
## Two-sample Hsu t test power calculation
##
## n = 54.49421
## delta = 0.5
## sd1 = 0.5
## sd2 = 1
## sig.level = 0.05
## power = 0.9
## alternative = two.sided
##
## NOTE: n is number in *each* group
ANCOVA
With function power.ancova one can compute power and sample size in
ANCOVA designs (Shieh 2020). The default
matrix of contrasts used in the function is
## 3 groups
cbind(rep(1,2), -diag(2))
## [,1] [,2] [,3]
## [1,] 1 -1 0
## [2,] 1 0 -1
## 4 groups
cbind(rep(1,3), -diag(3))
## [,1] [,2] [,3] [,4]
## [1,] 1 -1 0 0
## [2,] 1 0 -1 0
## [3,] 1 0 0 -1
We use the example provided in Table 9.7 of Maxwell and Delaney (2004).
## Example from Maxwell and Delaney (2004) according to Shieh (2020)
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.8)
##
## ANCOVA power calculation
##
## ns = 14.29481, 14.29481, 14.29481
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.8
##
## NOTE: Total sample size: 45
power.ancova(n = rep(45/3, 3), mu=c(7.5366, 11.9849, 13.9785), var = 29.0898)
##
## ANCOVA power calculation
##
## ns = 15, 15, 15
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.8219894
##
## NOTE: Total sample size: 45
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.9)
##
## ANCOVA power calculation
##
## ns = 18.32456, 18.32456, 18.32456
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.9
##
## NOTE: Total sample size: 57
power.ancova(n = rep(57/3, 3), mu=c(7.5366, 11.9849, 13.9785), var = 29.0898)
##
## ANCOVA power calculation
##
## ns = 19, 19, 19
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.9115114
##
## NOTE: Total sample size: 57
Based on the reported adjusted group means and error variance, a
(total) sample size of 45 is required to achieve a power of at least
80%. The calculated power is 82.2%. With a sample size of 57 the power
will be at least 90%, where the calculated power is 91.2%.
Introducing additional covariates (random covariate model) will
increase the required sample size.
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.8,
nr.covs = 2)
##
## ANCOVA power calculation
##
## ns = 14.65509, 14.65509, 14.65509
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 2
## sig.level = 0.05
## power = 0.8
##
## NOTE: Total sample size: 45
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.8,
nr.covs = 3)
##
## ANCOVA power calculation
##
## ns = 15.01403, 15.01403, 15.01403
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 3
## sig.level = 0.05
## power = 0.8
##
## NOTE: Total sample size: 48
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.8,
nr.covs = 5)
##
## ANCOVA power calculation
##
## ns = 15.72837, 15.72837, 15.72837
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 5
## sig.level = 0.05
## power = 0.8
##
## NOTE: Total sample size: 48
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.8,
nr.covs = 10)
##
## ANCOVA power calculation
##
## ns = 17.4965, 17.4965, 17.4965
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 10
## sig.level = 0.05
## power = 0.8
##
## NOTE: Total sample size: 54
We can also calculate the per group sample sizes for an imbalanced
design.
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.8,
group.ratio = c(1, 1.25, 1.5))
##
## ANCOVA power calculation
##
## ns = 12.25390, 15.31738, 18.38085
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.8
##
## NOTE: Total sample size: 48
power.ancova(n = c(13, 16, 19), mu=c(7.5366, 11.9849, 13.9785), var = 29.0898,
group.ratio = c(1, 1.25, 1.5))
##
## ANCOVA power calculation
##
## ns = 13, 16, 19
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.8266307
##
## NOTE: Total sample size: 48
power.ancova(mu=c(7.5366, 11.9849, 13.9785), var = 29.0898, power = 0.8,
group.ratio = c(1, 0.8, 2/3))
##
## ANCOVA power calculation
##
## ns = 16.87449, 13.49959, 11.24966
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.8
##
## NOTE: Total sample size: 43
power.ancova(n = c(17, 14, 12), mu=c(7.5366, 11.9849, 13.9785), var = 29.0898,
group.ratio = c(1, 0.8, 2/3))
##
## ANCOVA power calculation
##
## ns = 17, 14, 12
## mu = 7.5366, 11.9849, 13.9785
## var = 29.0898
## nr.covs = 1
## sig.level = 0.05
## power = 0.8034654
##
## NOTE: Total sample size: 43
We get a smaller total sample size with an imbalanced design (43
instead of 45).
Wilcoxon Rank Sum and Signed Rank Tests
For computing the sample size of these tests, we offer a function
that performs Monte-Carlo simulations to determine their (empirical)
power.
rx <- function(n) rnorm(n, mean = 0, sd = 1)
ry <- function(n) rnorm(n, mean = 0.5, sd = 1)
## two-sample
sim.ssize.wilcox.test(rx = rx, ry = ry, n.max = 100, iter = 1000)
##
## Wilcoxon rank sum test
##
## n = 10, 20, 30, 40, 50, 60, 70
## rx = rnorm(n, mean = 0, sd = 1)
## ry = rnorm(n, mean = 0.5, sd = 1)
## sig.level = 0.05
## emp.power = 0.160, 0.319, 0.472, 0.573, 0.661, 0.755, 0.816
## alternative = two.sided
##
## NOTE: n is number in *each* group
sim.ssize.wilcox.test(rx = rx, ry = ry, n.min = 65, n.max = 70,
step.size = 1, iter = 1000, BREAK = FALSE)
##
## Wilcoxon rank sum test
##
## n = 65, 66, 67, 68, 69, 70
## rx = rnorm(n, mean = 0, sd = 1)
## ry = rnorm(n, mean = 0.5, sd = 1)
## sig.level = 0.05
## emp.power = 0.789, 0.775, 0.797, 0.800, 0.806, 0.798
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.t.test(delta = 0.5, power = 0.8)
##
## Two-sample t test power calculation
##
## n = 63.76576
## delta = 0.5
## sd = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n is number in *each* group
## one-sample
sim.ssize.wilcox.test(rx = ry, n.max = 100, iter = 1000, type = "one.sample")
##
## Wilcoxon signed rank test
##
## n = 10, 20, 30, 40
## rx = rnorm(n, mean = 0.5, sd = 1)
## sig.level = 0.05
## emp.power = 0.294, 0.552, 0.719, 0.858
## alternative = two.sided
##
## NOTE: n is number in *each* group
sim.ssize.wilcox.test(rx = ry, type = "one.sample", n.min = 33,
n.max = 38, step.size = 1, iter = 1000, BREAK = FALSE)
##
## Wilcoxon signed rank test
##
## n = 33, 34, 35, 36, 37, 38
## rx = rnorm(n, mean = 0.5, sd = 1)
## sig.level = 0.05
## emp.power = 0.777, 0.794, 0.800, 0.801, 0.819, 0.841
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.t.test(delta = 0.5, power = 0.8, type = "one.sample")
##
## One-sample t test power calculation
##
## n = 33.3672
## delta = 0.5
## sd = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
## two-sample
rx <- function(n) rgamma(n, scale = 2, shape = 1.5)
ry <- function(n) rgamma(n, scale = 4, shape = 1.5) # equal shape
## two-sample
sim.ssize.wilcox.test(rx = rx, ry = ry, n.max = 100, iter = 1000)
##
## Wilcoxon rank sum test
##
## n = 10, 20, 30
## rx = rgamma(n, scale = 2, shape = 1.5)
## ry = rgamma(n, scale = 4, shape = 1.5)
## sig.level = 0.05
## emp.power = 0.322, 0.622, 0.822
## alternative = two.sided
##
## NOTE: n is number in *each* group
sim.ssize.wilcox.test(rx = rx, ry = ry, n.min = 25, n.max = 30,
step.size = 1, iter = 1000, BREAK = FALSE)
##
## Wilcoxon rank sum test
##
## n = 25, 26, 27, 28, 29, 30
## rx = rgamma(n, scale = 2, shape = 1.5)
## ry = rgamma(n, scale = 4, shape = 1.5)
## sig.level = 0.05
## emp.power = 0.754, 0.773, 0.753, 0.782, 0.807, 0.829
## alternative = two.sided
##
## NOTE: n is number in *each* group
For practical applications we recommend to use a higher number of
iterations. For more detailed examples we refer to the help page of the
function.
Two Negative Binomial Rates
When we consider two negative binomial rates (Zhu and Lakkis 2014), we can compute sample
size or power applying function power.nb.test.
## examples from Table III in Zhu and Lakkis (2014)
power.nb.test(mu0 = 5.0, RR = 2.0, theta = 1/0.5, duration = 1, power = 0.8, approach = 1)
##
## Power calculation for comparing two negative binomial rates
##
## n = 22.37386
## n1 = 22.37386
## mu0 = 5
## RR = 2
## theta = 2
## duration = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n = sample size of control group, n1 = sample size of treatment group
power.nb.test(mu0 = 5.0, RR = 2.0, theta = 1/0.5, duration = 1, power = 0.8, approach = 2)
##
## Power calculation for comparing two negative binomial rates
##
## n = 21.23734
## n1 = 21.23734
## mu0 = 5
## RR = 2
## theta = 2
## duration = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n = sample size of control group, n1 = sample size of treatment group
power.nb.test(mu0 = 5.0, RR = 2.0, theta = 1/0.5, duration = 1, power = 0.8, approach = 3)
##
## Power calculation for comparing two negative binomial rates
##
## n = 20.85564
## n1 = 20.85564
## mu0 = 5
## RR = 2
## theta = 2
## duration = 1
## sig.level = 0.05
## power = 0.8
## alternative = two.sided
##
## NOTE: n = sample size of control group, n1 = sample size of treatment group
Diagnostic Test
Given an expected sensitivity and specificity we can compute sample
size, power (assurance probability), delta or significance level (error
probability) of diagnostic test (Flahault,
Cadilhac, and Thomas 2005). The calculation is based on a
one-sided confidence interval.
## see n2 on page 1202 of Chu and Cole (2007)
ssize.sens.ci(sens = 0.99, delta = 0.14, power = 0.95) # 40
##
## Diagnostic test exact power calculation
##
## sens = 0.99
## n = 40
## n1 = 40
## delta = 0.14
## sig.level = 0.05
## power = 0.95
## prev = NULL
##
## NOTE: n is number of cases, n1 is number of controls
ssize.sens.ci(sens = 0.99, delta = 0.13, power = 0.95) # 43
##
## Diagnostic test exact power calculation
##
## sens = 0.99
## n = 43
## n1 = 43
## delta = 0.13
## sig.level = 0.05
## power = 0.95
## prev = NULL
##
## NOTE: n is number of cases, n1 is number of controls
ssize.spec.ci(spec = 0.99, delta = 0.12, power = 0.95) # 47
##
## Diagnostic test exact power calculation
##
## spec = 0.99
## n = 47
## n1 = 47
## delta = 0.12
## sig.level = 0.05
## power = 0.95
## prev = NULL
##
## NOTE: n is number of controls, n1 is number of cases
Given an expected AUC we can compute sample size, power (assurance
probability), delta or significance level (error probability) of the AUC
based on a one-sided confidence interval.
ssize.auc.ci(AUC = 0.9, delta = 0.1, power = 0.95)
##
##
##
## AUC = 0.9
## n = 55.75181
## n1 = 55.75181
## delta = 0.1
## sig.level = 0.05
## power = 0.95
## prev = 0.5
##
## NOTE: n is number of cases, n1 is number of controls
The sample size planning for developing binary classifiers in case of
high dimensional data, we can apply function ssize.pcc, which is based
on the probability of correct classification (PCC) (Dobbin and Simon 2007).
## see Table 2 of Dobbin et al. (2008)
g <- 0.1
fc <- 1.6
ssize.pcc(gamma = g, stdFC = fc, nrFeatures = 22000)
##
## Sample Size Planning for Developing Classifiers Using High Dimensional Data
##
## gamma = 0.1
## prev = 0.5
## nrFeatures = 22000
## n1 = 21
## n2 = 21
##
## NOTE: n1 is number of cases, n2 is number of controls
Reference Range
We can apply function ssize.reference.range to determine the sample
size required for a study planned to establish a reference range. The
parametric approach assumes a normal distribution whereas the
non-parametric approach only assumes a continuous distribution.
ssize.reference.range(delta = 0.01, ref.prob = 0.95, conf.prob = 0.9,
method = "parametric", exact = TRUE)
##
## Reference range sample size calculation
##
## n = 271.6691
## delta = 0.01
## ref.prob = 0.95
## conf.prob = 0.9
## alternative = two.sided
##
## NOTE: Exact method for normal distribution
ssize.reference.range(delta = 0.01, ref.prob = 0.95, conf.prob = 0.9,
method = "parametric", exact = FALSE)
##
## Reference range sample size calculation
##
## n = 269.9241
## delta = 0.01
## ref.prob = 0.95
## conf.prob = 0.9
## alternative = two.sided
##
## NOTE: Approximate method for normal distribution
ssize.reference.range(delta = 0.01, ref.prob = 0.95, conf.prob = 0.9,
method = "nonparametric", exact = TRUE)
##
## Reference range sample size calculation
##
## n = 635.2654
## delta = 0.01
## ref.prob = 0.95
## conf.prob = 0.9
## alternative = two.sided
##
## NOTE: Exact non-parametric method
ssize.reference.range(delta = 0.01, ref.prob = 0.95, conf.prob = 0.9,
method = "nonparametric", exact = FALSE)
##
## Reference range sample size calculation
##
## n = 659.4762
## delta = 0.01
## ref.prob = 0.95
## conf.prob = 0.9
## alternative = two.sided
##
## NOTE: Approximate non-parametric method
We can also calculate the sample size for one-sided reference
ranges.
ssize.reference.range(delta = 0.015, ref.prob = 0.95, conf.prob = 0.9,
method = "parametric", exact = TRUE,
alternative = "one.sided")
##
## Reference range sample size calculation
##
## n = 301.617
## delta = 0.015
## ref.prob = 0.95
## conf.prob = 0.9
## alternative = one.sided
##
## NOTE: Exact method for normal distribution
Multiple Primary Endpoints (MPE)
We demonstrate how to calculate the sample size for a trial with two
co-primary endpoints with known covariance.
Sigma <- matrix(c(1, 0.8, 0.8, 1), nrow = 2)
power.mpe.known.var(K = 2, delta = c(0.25, 0.4), Sigma = Sigma,
sig.level = 0.025, power = 0.8)
##
## Power calculation for multiple co-primary endpoints (covariance known)
##
## n = 251.2079
## delta = 0.25, 0.40
## SD = 1, 1
## rho = 0.8
## sig.level = 0.025
## power = 0.8
##
## Sigma =
## [,1] [,2]
## [1,] 1.0 0.8
## [2,] 0.8 1.0
##
## NOTE: n is number in *each* group
## equivalent: specify SDs and correlation rho
power.mpe.known.var(K = 2, delta = c(0.25, 0.4), SD = c(1,1), rho = 0.8,
sig.level = 0.025, power = 0.8)
##
## Power calculation for multiple co-primary endpoints (covariance known)
##
## n = 251.2079
## delta = 0.25, 0.40
## SD = 1, 1
## rho = 0.8
## sig.level = 0.025
## power = 0.8
##
## Sigma =
## [,1] [,2]
## [1,] 1.0 0.8
## [2,] 0.8 1.0
##
## NOTE: n is number in *each* group
Next, we show how to calculate the sample size for a trial with two
co-primary endpoints with unknown covariance. Here, we follow three
steps to determine the sample size.
Step 1: As we need starting values for our algorithm that
computes the sample size in this case, we first act as if the covariance
would be known and compute the sample size by applying our function
power.mpe.known.var.
Step 2: The resulting value of n is considered as
lower bound for the sample size in case of unknown covariance and is
used as n.min in function
power.mpe.unkown.var. Moreover, we specify a reasonable
n.max, which must be larger than
n.min.
Step 3: Finally, by using the arguments from the step 2, we can
compute the sample size for the situation with unknown
covariance.
## Step 1:
power.mpe.known.var(K = 2, delta = c(0.5, 0.4), Sigma = Sigma,
sig.level = 0.025, power = 0.8)
##
## Power calculation for multiple co-primary endpoints (covariance known)
##
## n = 100.0826
## delta = 0.5, 0.4
## SD = 1, 1
## rho = 0.8
## sig.level = 0.025
## power = 0.8
##
## Sigma =
## [,1] [,2]
## [1,] 1.0 0.8
## [2,] 0.8 1.0
##
## NOTE: n is number in *each* group
## Step 2 + 3:
Sigma <- matrix(c(1, 0.5, 0.5, 1), nrow = 2)
power.mpe.unknown.var(K = 2, delta = c(0.5, 0.4), Sigma = Sigma,
sig.level = 0.025, power = 0.8,
n.min = 105, n.max = 120)
## Current precision: -0.004521862
## Current precision: 0.05232739
## Current precision: 0.0007625384
## Current precision: 5.571479e-05
## Current precision: -0.0002569758
## Current precision: -0.0001545662
## Current precision: 2.350941e-05
## Current precision: -0.00025997
## Current precision: 0.0003175776
## Current precision: -0.0001569394
## Current precision: -0.0001571391
## Current precision: -0.0003360948
## Current precision: 6.854054e-05
##
## Power calculation for multiple co-primary endpoints (covariance unknown)
##
## n = 106.0202
## delta = 0.5, 0.4
## SD = 1, 1
## rho = 0.5
## sig.level = 0.025
## power = 0.8
##
## Sigma =
## [,1] [,2]
## [1,] 1.0 0.5
## [2,] 0.5 1.0
##
## NOTE: n is number in *each* group
## equivalent: specify SDs and correlation rho
power.mpe.unknown.var(K = 2, delta = c(0.5, 0.4), SD = c(1,1), rho = 0.5,
sig.level = 0.025, power = 0.8,
n.min = 105, n.max = 120)
## Current precision: -0.003838119
## Current precision: 0.05231136
## Current precision: 0.0004913149
## Current precision: -0.001156639
## Current precision: -0.0009737845
## Current precision: -0.0005025679
## Current precision: -0.0002141294
## Current precision: 0.0001505083
## Current precision: 0.0006028865
## Current precision: -0.0008063434
## Current precision: 0.000183268
## Current precision: -0.0001798484
## Current precision: -0.0005059941
## Current precision: -3.131932e-05
##
## Power calculation for multiple co-primary endpoints (covariance unknown)
##
## n = 106.0202
## delta = 0.5, 0.4
## SD = 1, 1
## rho = 0.5
## sig.level = 0.025
## power = 0.8
##
## Sigma =
## [,1] [,2]
## [1,] 1.0 0.5
## [2,] 0.5 1.0
##
## NOTE: n is number in *each* group
We finally demonstrate how to calculate the sample size for a trial
with two primary endpoints with known covariance, where the trial is
designed to find a significant difference for at least one endpoint.
Sigma <- matrix(c(1, 0.3, 0.3, 1), nrow = 2)
power.mpe.atleast.one(K = 2, delta = c(0.2, 0.3), Sigma = Sigma,
power = 0.8, sig.level = 0.025)
##
## Power calculation for multiple primary endpoints for at least one endpoint
##
## n = 146.6651
## delta = 0.2, 0.3
## SD = 1, 1
## rho = 0.3
## sig.level = 0.025
## power = 0.8
##
## Sigma =
## [,1] [,2]
## [1,] 1.0 0.3
## [2,] 0.3 1.0
##
## NOTE: n is number in *each* group
## equivalent: specify SDs and correlation rho
power.mpe.atleast.one(K = 2, delta = c(0.2, 0.3), SD = c(1,1), rho = 0.3,
power = 0.8, sig.level = 0.025)
##
## Power calculation for multiple primary endpoints for at least one endpoint
##
## n = 146.6651
## delta = 0.2, 0.3
## SD = 1, 1
## rho = 0.3
## sig.level = 0.025
## power = 0.8
##
## Sigma =
## [,1] [,2]
## [1,] 1.0 0.3
## [2,] 0.3 1.0
##
## NOTE: n is number in *each* group
Power and Type-I-Error Simulations
There are quite some discussions and various proposals, which test is
the best under which circumstances when we want to compare the location
(mean, median) of two groups (Fagerland and
Sandvik 2009; Fagerland 2012; Sezer, Ozkip, and Yazici 2017). In
addition, it is questioned whether pre-testing of assumptions is
appropriate/useful at least from a practical point of view (Zimmerman 2004; Rasch, Kubinger, and Moder 2011;
Rochon, Gondan, and Kieser 2012; Hoekstra, Kiers, and Johnson
2012).
We provide functions to simulate power and type-I-error of classical
(Gosset 1908), Welch (Welch 1947) and Hsu (Hsu
1938) t-tests as well as of Wilcoxon-Mann-Whitney tests (Wilcoxon 1945; Mann and Whitney 1947).
## functions to simulate the data
## group 1
rx <- function(n) rnorm(n, mean = 0, sd = 1)
rx.H0 <- rx
## group 2
ry <- function(n) rnorm(n, mean = 3, sd = 3)
ry.H0 <- function(n) rnorm(n, mean = 0, sd = 3)
## theoretical results
power.welch.t.test(n = 10, delta = 3, sd1 = 1, sd2 = 3)
##
## Two-sample Welch t test power calculation
##
## n = 10
## delta = 3
## sd1 = 1
## sd2 = 3
## sig.level = 0.05
## power = 0.7794139
## alternative = two.sided
##
## NOTE: n is number in *each* group
power.hsu.t.test(n = 10, delta = 3, sd1 = 1, sd2 = 3)
##
## Two-sample Hsu t test power calculation
##
## n = 10
## delta = 3
## sd1 = 1
## sd2 = 3
## sig.level = 0.05
## power = 0.7611553
## alternative = two.sided
##
## NOTE: n is number in *each* group
## simulation
res.t <- sim.power.t.test(nx = 10, rx = rx, rx.H0 = rx.H0,
ny = 10, ry = ry, ry.H0 = ry.H0)
res.t
##
## Simulation Set-up
## nx = 10
## rx = function (n) , rnorm(n, mean = 0, sd = 1)
## rx.H0 = function (n) , rnorm(n, mean = 0, sd = 1)
## ny = 10
## ry = function (n) , rnorm(n, mean = 3, sd = 3)
## ry.H0 = function (n) , rnorm(n, mean = 0, sd = 3)
## sig.level = 0.05
## mu = 0
## alternative = two.sided
## iter = 10000
##
## Classical Two-sample t-Test
## emp.power = 0.8023
## emp.type.I.error = 0.0597
##
## Welch Two-sample t-Test
## emp.power = 0.7741
## emp.type.I.error = 0.0516
##
## Hsu Two-sample t-Test
## emp.power = 0.7564
## emp.type.I.error = 0.0444
res.w <- sim.power.wilcox.test(nx = 10, rx = rx, rx.H0 = rx.H0,
ny = 10, ry = ry, ry.H0 = ry.H0)
res.w
##
## Simulation Set-up
## nx = 10
## rx = function (n) , rnorm(n, mean = 0, sd = 1)
## rx.H0 = function (n) , rnorm(n, mean = 0, sd = 1)
## ny = 10
## ry = function (n) , rnorm(n, mean = 3, sd = 3)
## ry.H0 = function (n) , rnorm(n, mean = 0, sd = 3)
## sig.level = 0.05
## mu = 0
## alternative = two.sided
## iter = 10000
## conf.int = FALSE
## approximate = FALSE
## ties = FALSE
##
## Exact Wilcoxon-Mann-Whitney Test
## emp.power = 0.7542
## emp.type.I.error = 0.0594
##
## Asymptotic Wilcoxon-Mann-Whitney Test
## emp.power = 0.7542
## emp.type.I.error = 0.0594
For further investigation of the results, we provide some diagnostic
plots.
qqunif(res.t, alpha = 0.05)
volcano(res.t, hex = TRUE)
## Wilcoxon-Mann-Whitney tests
hist(res.w)
qqunif(res.w, alpha = 0.05)
We can also generate a volcano plot for the Wilcoxon-Mann-Whitney
test, but this would require setting argument conf.int to
TRUE, which would lead to a much higher computation time, hence
we omitted it here. Furthermore, it is also possible to compute an
approximate version of the test by setting argument approximate
to TRUE (Hothorn et al. 2008)
again by the cost of an increased computation time.
sessionInfo
## R version 4.4.1 Patched (2024-09-18 r87189)
## Platform: x86_64-pc-linux-gnu
## Running under: Linux Mint 22
##
## Matrix products: default
## BLAS: /home/kohlm/RTOP/Rbranch/lib/libRblas.so
## LAPACK: /home/kohlm/RTOP/Rbranch/lib/libRlapack.so; LAPACK version 3.12.0
##
## locale:
## [1] LC_CTYPE=de_DE.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=de_DE.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=de_DE.UTF-8 LC_MESSAGES=de_DE.UTF-8
## [7] LC_PAPER=de_DE.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C
##
## time zone: Europe/Berlin
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] MKpower_1.0
##
## loaded via a namespace (and not attached):
## [1] tidyselect_1.2.1 farver_2.1.2 libcoin_1.0-10
## [4] dplyr_1.1.4 matrixTests_0.2.3 bitops_1.0-8
## [7] fastmap_1.2.0 TH.data_1.1-2 pracma_2.4.4
## [10] MKdescr_0.9 digest_0.6.37 rpart_4.1.23
## [13] lifecycle_1.0.4 arrangements_1.1.9 survival_3.7-0
## [16] magrittr_2.0.3 compiler_4.4.1 rlang_1.1.4
## [19] sass_0.4.9 tools_4.4.1 utf8_1.2.4
## [22] yaml_2.3.10 knitr_1.48 labeling_0.4.3
## [25] multcomp_1.4-26 MKinfer_1.2 withr_3.0.1
## [28] purrr_1.0.2 nnet_7.3-19 grid_4.4.1
## [31] stats4_4.4.1 fansi_1.0.6 opdisDownsampling_1.0.1
## [34] jomo_2.7-6 caTools_1.18.2 colorspace_2.1-1
## [37] mice_3.16.0 ggplot2_3.5.1 scales_1.3.0
## [40] iterators_1.0.14 MASS_7.3-61 cli_3.6.3
## [43] mvtnorm_1.3-0 rmarkdown_2.28 generics_0.1.3
## [46] exactRankTests_0.8-35 twosamples_2.0.1 robustbase_0.99-4
## [49] minqa_1.2.8 DBI_1.2.3 cachem_1.1.0
## [52] modeltools_0.2-23 splines_4.4.1 parallel_4.4.1
## [55] matrixStats_1.3.0 mitools_2.4 vctrs_0.6.5
## [58] boot_1.3-31 glmnet_4.1-8 Matrix_1.7-0
## [61] sandwich_3.1-0 jsonlite_1.8.8 mitml_0.4-5
## [64] pbmcapply_1.5.1 foreach_1.5.2 hexbin_1.28.3
## [67] qqplotr_0.0.6 tidyr_1.3.1 jquerylib_0.1.4
## [70] glue_1.7.0 nloptr_2.1.1 pan_1.9
## [73] DEoptimR_1.1-3 codetools_0.2-20 shape_1.4.6.1
## [76] gtable_0.3.5 lme4_1.1-35.5 gmp_0.7-5
## [79] munsell_0.5.1 tibble_3.2.1 pillar_1.9.0
## [82] miceadds_3.17-44 htmltools_0.5.8.1 R6_2.5.1
## [85] doParallel_1.0.17 evaluate_0.24.0 lattice_0.22-6
## [88] highr_0.11 backports_1.5.0 broom_1.0.6
## [91] qqconf_1.3.2 bslib_0.8.0 Rcpp_1.0.13
## [94] nlme_3.1-166 xfun_0.47 coin_1.4-3
## [97] zoo_1.8-12 pkgconfig_2.0.3
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