Package MKpower
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.
##
## 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##
## 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 sizeNext, we compute the sample size via the respective asymptotic confidence interval.
##
## 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##
## Sample size calculation by method of wald
##
## n = 188.1531
## prop = 0.4
## width = 0.14
## conf.level = 0.95
##
## NOTE: Two-sided confidence intervalWelch 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##
## 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##
## 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##
## 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* groupHsu 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).
##
## 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##
## 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##
## 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##
## 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##
## 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* groupANCOVA
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
## [,1] [,2] [,3]
## [1,] 1 -1 0
## [2,] 1 0 -1## [,1] [,2] [,3] [,4]
## [1,] 1 -1 0 0
## [2,] 1 0 -1 0
## [3,] 1 0 0 -1We 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##
## 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##
## 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##
## 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: 57Based 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.
##
## 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##
## 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##
## 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##
## 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: 54We 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: 48power.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: 48power.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: 43power.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: 43We 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 = 0.175, 0.352, 0.432, 0.535, 0.660, 0.731, 0.825
## alternative = two.sided
##
## NOTE: n is number in *each* groupsim.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 = 0.782, 0.817, 0.763, 0.805, 0.810, 0.821
## alternative = two.sided
##
## NOTE: n is number in *each* group##
## 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##
## Wilcoxon signed rank test
##
## n = 10, 20, 30, 40
## rx = rnorm(n, mean = 0.5, sd = 1)
## sig.level = 0.05
## emp.power = 0.273, 0.525, 0.729, 0.851
## alternative = two.sided
##
## NOTE: n is number in *each* groupsim.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 = 0.781, 0.771, 0.786, 0.816, 0.814, 0.862
## alternative = two.sided
##
## NOTE: n is number in *each* group##
## 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 = 0.354, 0.642, 0.801
## alternative = two.sided
##
## NOTE: n is number in *each* groupsim.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, 0.791, 0.769, 0.783, 0.812, 0.820
## alternative = two.sided
##
## NOTE: n is number in *each* groupFor 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 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 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 groupDiagnostic 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 = 0.5
##
## NOTE: n is number of cases, n1 is number of controls##
## Diagnostic test exact power calculation
##
## sens = 0.99
## n = 43
## n1 = 43
## delta = 0.13
## sig.level = 0.05
## power = 0.95
## prev = 0.5
##
## NOTE: n is number of cases, n1 is number of controls##
## Diagnostic test exact power calculation
##
## spec = 0.99
## n = 47
## n1 = 47
## delta = 0.12
## sig.level = 0.05
## power = 0.95
## prev = 0.5
##
## NOTE: n is number of controls, n1 is number of casesGiven 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.
##
## Sample size calculation for AUC confidence interval
##
## 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 controlsThe 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 controlsReference 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 distributionssize.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 distributionssize.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 methodssize.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 methodWe 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 distributionMultiple 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* groupNext, 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
nis considered as lower bound for the sample size in case of unknown covariance and is used asn.minin functionpower.mpe.unkown.var. Moreover, we specify a reasonablen.max, which must be larger thann.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: -0.004605914
## Current precision: 0.05208719
## Current precision: 0.001290097
## Current precision: -0.0004958886
## Current precision: 0.000345691
## Current precision: -0.0001630763
## Current precision: -0.0002535715
## Current precision: -0.0001517639
## Current precision: -0.0005671254
## Current precision: 6.332566e-05
## Current precision: -0.000290352
## Current precision: -0.0001141519
## Current precision: 0.000431662##
## Power calculation for multiple co-primary endpoints (covariance unknown)
##
## n = 106.0241
## 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: -0.00413574
## Current precision: 0.05247003
## Current precision: 0.0001661942
## Current precision: 0.0007070909
## Current precision: -0.002222654
## Current precision: -0.0008780296
## Current precision: -0.0002820227
## Current precision: -0.0004354646
## Current precision: -0.0006927148
## Current precision: -9.569105e-05
## Current precision: 2.640639e-06
## Current precision: -0.0002752459
## Current precision: -6.987101e-05##
## Power calculation for multiple co-primary endpoints (covariance unknown)
##
## n = 106.0449
## 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* groupWe 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* groupPower 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##
## 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 = 0.8126
## emp.type.I.error = 0.0600
##
## Welch Two-sample t-Test
## emp.power = 0.7842
## emp.type.I.error = 0.0524
##
## Hsu Two-sample t-Test
## emp.power = 0.7688
## emp.type.I.error = 0.0469res.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 = 0.7508
## emp.type.I.error = 0.0567
##
## Asymptotic Wilcoxon-Mann-Whitney Test
## emp.power = 0.7508
## emp.type.I.error = 0.0567For further investigation of the results, we provide some diagnostic plots.
## Warning: Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Computation failed in `stat_binhex()`.
## Caused by error in `compute_group()`:
## ! The package "hexbin" is required for `stat_bin_hex()`.
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 4.4.2 (2024-10-31)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04.1 LTS
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so; LAPACK version 3.12.0
##
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## time zone: Etc/UTC
## tzcode source: system (glibc)
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] MKpower_1.1 rmarkdown_2.29
##
## 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_1.0-9
## [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_4.4.2 rlang_1.1.4
## [19] sass_0.4.9 tools_4.4.2 utf8_1.2.4
## [22] yaml_2.3.10 knitr_1.49 labeling_0.4.3
## [25] multcomp_1.4-26 withr_3.0.2 MKinfer_1.2
## [28] purrr_1.0.2 sys_3.4.3 nnet_7.3-19
## [31] grid_4.4.2 stats4_4.4.2 fansi_1.0.6
## [34] opdisDownsampling_1.0.1 jomo_2.7-6 caTools_1.18.3
## [37] colorspace_2.1-1 mice_3.16.0 ggplot2_3.5.1
## [40] scales_1.3.0 iterators_1.0.14 MASS_7.3-61
## [43] cli_3.6.3 mvtnorm_1.3-2 generics_0.1.3
## [46] exactRankTests_0.8-35 twosamples_2.0.1 robustbase_0.99-4-1
## [49] minqa_1.2.8 DBI_1.2.3 cachem_1.1.0
## [52] modeltools_0.2-23 splines_4.4.2 parallel_4.4.2
## [55] matrixStats_1.4.1 mitools_2.4 vctrs_0.6.5
## [58] boot_1.3-31 glmnet_4.1-8 Matrix_1.7-1
## [61] sandwich_3.1-1 jsonlite_1.8.9 mitml_0.4-5
## [64] pbmcapply_1.5.1 maketools_1.3.1 foreach_1.5.2
## [67] qqplotr_0.0.6 tidyr_1.3.1 jquerylib_0.1.4
## [70] glue_1.8.0 nloptr_2.1.1 pan_1.9
## [73] DEoptimR_1.1-3 codetools_0.2-20 shape_1.4.6.1
## [76] gtable_0.3.6 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_1.0.1 lattice_0.22-6
## [88] backports_1.5.0 broom_1.0.7 qqconf_1.3.2
## [91] bslib_0.8.0 Rcpp_1.0.13-1 nlme_3.1-166
## [94] xfun_0.49 coin_1.4-3 zoo_1.8-12
## [97] buildtools_1.0.0 pkgconfig_2.0.3References
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