# Package MKinfer

## Introduction

Package MKinfer includes a collection of functions for the computation of various confidence intervals (Altman et al. 2000; Hedderich and Sachs 2018) including bootstrapped versions (Davison and Hinkley 1997) as well as Hsu (Hedderich and Sachs 2018), permutation (Janssen 1997), bootstrap (Davison and Hinkley 1997) and multiple imputation (Barnard and Rubin 1999) t-test. Graphical visualization by volcano plots.

We first load the package.

library(MKinfer)

## Confidence Intervals

### Binomial Proportion

There are several functions for computing confidence intervals. We can compute 12 different confidence intervals for binomial proportions (DasGupta, Cai, and Brown 2001); e.g.

## default: "wilson"
binomCI(x = 12, n = 50)
##
##  wilson confidence interval
##
## 95 percent confidence interval:
##          2.5 %    97.5 %
## prob 0.1429739 0.3741268
##
## sample estimate:
## prob
## 0.24
##
## additional information:
## standard error of prob
##             0.05896867
## Clopper-Pearson interval
binomCI(x = 12, n = 50, method = "clopper-pearson")
##
##  clopper-pearson confidence interval
##
## 95 percent confidence interval:
##          2.5 %    97.5 %
## prob 0.1306099 0.3816907
##
## sample estimate:
## prob
## 0.24
## identical to
binom.test(x = 12, n = 50)$conf.int ## [1] 0.1306099 0.3816907 ## attr(,"conf.level") ## [1] 0.95 For all intervals implemented see the help page of function binomCI. One can also compute bootstrap intervals via function boot.ci of package boot (Davison and Hinkley 1997) as well as one-sided intervals. ### Difference of Two Binomial Proportions There are several functions for computing confidence intervals. We can compute different confidence intervals for the difference of two binomial proportions (independent (Newcombe 1998a) and paired case (Newcombe 1998b)); e.g. ## default: wilson binomDiffCI(a = 5, b = 0, c = 51, d = 29) ## ## wilson confidence interval (independent proportions) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## difference of independent proportions -0.03813715 0.19256 ## ## sample estimate: ## difference of proportions ## 0.08928571 ## ## additional information: ## proportion of group 1 proportion of group 2 ## 0.08928571 0.00000000 ## default: wilson with continuity correction binomDiffCI(a = 212, b = 144, c = 256, d = 707, paired = TRUE) ## ## wilson-cc confidence interval (paired data) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## difference of proportions (paired data) -0.1141915 -0.05543347 ## ## sample estimate: ## difference of proportions ## -0.08491281 ## ## additional information: ## proportion of group 1 proportion of group 2 ## 0.2699014 0.3548143 For all intervals implemented see the help page of function binomDiffCI. One can also compute boostrap intervals via function boot.ci of package boot (Davison and Hinkley 1997) as well as one-sided intervals. ### Mean and SD We can compute confidence intervals for mean and SD (Altman et al. 2000, @Davison1997). x <- rnorm(50, mean = 2, sd = 3) ## mean and SD unknown normCI(x) ## ## Exact confidence interval(s) ## ## 95 percent confidence intervals: ## 2.5 % 97.5 % ## mean 1.537619 2.932503 ## sd 2.049976 3.058111 ## ## sample estimates: ## mean sd ## 2.235061 2.454080 ## ## additional information: ## SE of mean ## 0.3470593 meanCI(x) ## ## Exact confidence interval(s) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## mean 1.537619 2.932503 ## ## sample estimates: ## mean sd ## 2.235061 2.454080 ## ## additional information: ## SE of mean ## 0.3470593 sdCI(x) ## ## Exact confidence interval(s) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## sd 2.049976 3.058111 ## ## sample estimates: ## mean sd ## 2.235061 2.454080 ## ## additional information: ## SE of mean ## 0.3470593 ## SD known normCI(x, sd = 3) ## ## Exact confidence interval(s) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## mean 1.403519 3.066603 ## ## sample estimate: ## mean ## 2.235061 ## ## additional information: ## SE of mean ## 0.4242641 ## mean known normCI(x, mean = 2, alternative = "less") ## ## Exact confidence interval(s) ## ## 95 percent confidence interval: ## 0 % 95 % ## sd 0 2.949122 ## ## sample estimate: ## sd ## 2.45408 ## bootstrap meanCI(x, boot = TRUE) ## ## Bootstrap confidence interval(s) ## ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS ## Based on 9999 bootstrap replicates ## ## CALL : ## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type) ## ## Intervals : ## Level Normal Basic Studentized ## 95% ( 1.566, 2.914 ) ( 1.555, 2.902 ) ( 1.545, 2.942 ) ## ## Level Percentile BCa ## 95% ( 1.569, 2.915 ) ( 1.594, 2.942 ) ## Calculations and Intervals on Original Scale ## ## sample estimates: ## mean sd ## 2.235061 2.454080 ### Difference in Means We can compute confidence interval for the difference of means (Altman et al. 2000; Hedderich and Sachs 2018; Davison and Hinkley 1997). x <- rnorm(20) y <- rnorm(20, sd = 2) ## paired normDiffCI(x, y, paired = TRUE) ## ## Confidence interval (paired) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## mean of differences -1.506324 0.3954244 ## ## sample estimates: ## mean of differences sd of differences ## -0.555450 2.031721 ## ## additional information: ## SE of mean of differences ## 0.4543065 ## compare normCI(x-y) ## ## Exact confidence interval(s) ## ## 95 percent confidence intervals: ## 2.5 % 97.5 % ## mean -1.506324 0.3954244 ## sd 1.545104 2.9674733 ## ## sample estimates: ## mean sd ## -0.555450 2.031721 ## ## additional information: ## SE of mean ## 0.4543065 ## bootstrap normDiffCI(x, y, paired = TRUE, boot = TRUE) ## ## Bootstrap confidence interval (paired) ## ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS ## Based on 9999 bootstrap replicates ## ## CALL : ## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type) ## ## Intervals : ## Level Normal Basic Studentized ## 95% (-1.4285, 0.3026 ) (-1.4043, 0.3257 ) (-1.5659, 0.3622 ) ## ## Level Percentile BCa ## 95% (-1.4366, 0.2934 ) (-1.4917, 0.2478 ) ## Calculations and Intervals on Original Scale ## ## sample estimates: ## mean of differences sd of differences ## -0.555450 2.031721 ## unpaired y <- rnorm(10, mean = 1, sd = 2) ## classical normDiffCI(x, y, method = "classical") ## ## Classical confidence interval (unpaired) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## difference in means -2.126827 0.02445559 ## ## sample estimate: ## difference in means ## -1.051186 ## ## additional information: ## SE of difference in means Cohen's d (SMD) ## 0.5251110 -0.7753073 ## ## mean of x SD of x mean of y SD of y ## 0.1083886 0.8620075 1.1595742 2.0372553 ## Welch (default as in case of function t.test) normDiffCI(x, y, method = "welch") ## ## Welch confidence interval (unpaired) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## difference in means -2.537325 0.4349542 ## ## sample estimate: ## difference in means ## -1.051186 ## ## additional information: ## SE of difference in means Cohen's d (SMD) ## 0.6724535 -0.4751942 ## ## mean of x SD of x mean of y SD of y ## 0.1083886 0.8620075 1.1595742 2.0372553 ## Hsu normDiffCI(x, y, method = "hsu") ## ## Hsu confidence interval (unpaired) ## ## 95 percent confidence interval: ## 2.5 % 97.5 % ## difference in means -2.572381 0.47001 ## ## sample estimate: ## difference in means ## -1.051186 ## ## additional information: ## SE of difference in means Cohen's d (SMD) ## 0.6724535 -0.4751942 ## ## mean of x SD of x mean of y SD of y ## 0.1083886 0.8620075 1.1595742 2.0372553 ## bootstrap: assuming equal variances normDiffCI(x, y, method = "classical", boot = TRUE, bootci.type = "bca") ## ## Bootstrap confidence interval (equal variances, unpaired) ## ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS ## Based on 9999 bootstrap replicates ## ## CALL : ## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type) ## ## Intervals : ## Level BCa ## 95% (-2.247, 0.327 ) ## Calculations and Intervals on Original Scale ## ## sample estimate: ## difference in means ## -1.051186 ## ## additional information: ## SE of difference in means Cohen's d (SMD) ## 0.5251110 -0.7753073 ## ## mean of x SD of x mean of y SD of y ## 0.1083886 0.8620075 1.1595742 2.0372553 ## bootstrap: assuming unequal variances normDiffCI(x, y, method = "welch", boot = TRUE, bootci.type = "bca") ## ## Bootstrap confidence interval (unequal variances, unpaired) ## ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS ## Based on 9999 bootstrap replicates ## ## CALL : ## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type) ## ## Intervals : ## Level BCa ## 95% (-2.245, 0.279 ) ## Calculations and Intervals on Original Scale ## ## sample estimate: ## difference in means ## -1.051186 ## ## additional information: ## SE of difference in means Cohen's d (SMD) ## 0.6724535 -0.4751942 ## ## mean of x SD of x mean of y SD of y ## 0.1083886 0.8620075 1.1595742 2.0372553 In case of unequal variances and unequal sample sizes per group the classical confidence interval may have a bad coverage (too long or too short), as is indicated by the small Monte-Carlo simulation study below. M <- 100 CIhsu <- CIwelch <- CIclass <- matrix(NA, nrow = M, ncol = 2) for(i in 1:M){ x <- rnorm(10) y <- rnorm(30, sd = 0.1) CIclass[i,] <- normDiffCI(x, y, method = "classical")$conf.int
CIwelch[i,] <- normDiffCI(x, y, method = "welch")$conf.int CIhsu[i,] <- normDiffCI(x, y, method = "hsu")$conf.int
}
## coverage probabilies
## classical
sum(CIclass[,1] < 0 & 0 < CIclass[,2])/M
## [1] 0.72
## Welch
sum(CIwelch[,1] < 0 & 0 < CIwelch[,2])/M
## [1] 0.98
## Hsu
sum(CIhsu[,1] < 0 & 0 < CIhsu[,2])/M
## [1] 0.98

### Coefficient of Variation

We provide 12 different confidence intervals for the (classical) coefficient of variation (Gulhar, Kibria, and Ahmed 2012; Davison and Hinkley 1997); e.g.

x <- rnorm(100, mean = 10, sd = 2) # CV = 0.2
## default: "miller"
cvCI(x)
##
##  Miller (1991) confidence interval
##
## 95 percent confidence interval:
##        2.5 %   97.5 %
## CV 0.1649112 0.220562
##
## sample estimate:
##        CV
## 0.1927366
##
## additional information:
## standard error of CV
##           0.01419689
## Gulhar et al. (2012)
cvCI(x, method = "gulhar")
##
##  Gulhar et al (2012) confidence interval
##
## 95 percent confidence interval:
##       2.5 %    97.5 %
## CV 0.169224 0.2238973
##
## sample estimate:
##        CV
## 0.1927366
## bootstrap
cvCI(x, method = "boot")
##
##  Bootstrap confidence interval
##
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 9999 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type)
##
## Intervals :
## Level      Normal              Basic
## 95%   ( 0.1707,  0.2179 )   ( 0.1701,  0.2175 )
##
## Level     Percentile            BCa
## 95%   ( 0.1680,  0.2154 )   ( 0.1722,  0.2205 )
## Calculations and Intervals on Original Scale
##
## sample estimate:
##        CV
## 0.1927366

For all intervals implemented see the help page of function cvCI.

### Quantiles, Median and MAD

We start with the computation of confidence intervals for quantiles (Hedderich and Sachs 2018; Davison and Hinkley 1997).

x <- rexp(100, rate = 0.5)
## exact
quantileCI(x = x, prob = 0.95)
##
##  exact confidence interval
##
## 95 percent confidence interval:
##                 2.5 %   97.5 %
## 95 % quantile 4.43398 9.137467
##
## sample estimate:
## 95 % quantile
##      5.087252
## asymptotic
quantileCI(x = x, prob = 0.95, method = "asymptotic")
##
##  asymptotic confidence interval
##
## 95 percent confidence interval:
##                  2.5 %   97.5 %
## 95 % quantile 4.408007 9.137467
##
## sample estimate:
## 95 % quantile
##      5.087252
## boot
quantileCI(x = x, prob = 0.95, method = "boot")
##
##  bootstrap confidence interval
##
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 9999 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type)
##
## Intervals :
## Level      Normal              Basic
## 95%   ( 3.948,  6.166 )   ( 4.257,  5.876 )
##
## Level     Percentile            BCa
## 95%   ( 4.299,  5.918 )   ( 4.409,  6.383 )
## Calculations and Intervals on Original Scale
##
## sample estimate:
## 95 % quantile
##      5.087252

Next, we consider the median.

## exact
medianCI(x = x)
##
##  exact confidence interval
##
## 95 percent confidence interval:
##           2.5 %   97.5 %
## median 1.016525 1.861426
##
## sample estimate:
##   median
## 1.354304
## asymptotic
medianCI(x = x, method = "asymptotic")
##
##  asymptotic confidence interval
##
## 95 percent confidence interval:
##           2.5 %   97.5 %
## median 1.016525 1.858088
##
## sample estimate:
##   median
## 1.354304
## boot
medianCI(x = x, method = "boot")
##
##  bootstrap confidence interval
##
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 9999 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type)
##
## Intervals :
## Level      Normal              Basic
## 95%   ( 0.936,  1.654 )   ( 0.851,  1.601 )
##
## Level     Percentile            BCa
## 95%   ( 1.108,  1.858 )   ( 1.089,  1.835 )
## Calculations and Intervals on Original Scale
##
## sample estimate:
##   median
## 1.354304

Finally, we take a look at MAD (median absolute deviation) where by default the standardized MAD is used (see function mad).

## exact
madCI(x = x)
##
##  exact confidence interval
##
## 95 percent confidence interval:
##        2.5 %  97.5 %
## MAD 1.083396 1.64949
##
## sample estimate:
##     MAD
## 1.43339
## aysymptotic
madCI(x = x, method = "asymptotic")
##
##  asymptotic confidence interval
##
## 95 percent confidence interval:
##        2.5 %   97.5 %
## MAD 1.083396 1.589104
##
## sample estimate:
##     MAD
## 1.43339
## boot
madCI(x = x, method = "boot")
##
##  bootstrap confidence interval
##
## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
## Based on 9999 bootstrap replicates
##
## CALL :
## boot.ci(boot.out = boot.out, conf = 1 - alpha, type = bootci.type)
##
## Intervals :
## Level      Normal              Basic
## 95%   ( 1.224,  1.724 )   ( 1.260,  1.781 )
##
## Level     Percentile            BCa
## 95%   ( 1.086,  1.606 )   ( 1.086,  1.605 )
## Calculations and Intervals on Original Scale
##
## sample estimate:
##     MAD
## 1.43339
## unstandardized
madCI(x = x, constant = 1)
##
##  exact confidence interval
##
## 95 percent confidence interval:
##         2.5 %   97.5 %
## MAD 0.7307408 1.112566
##
## sample estimate:
##       MAD
## 0.9668082

## Hsu Two-Sample t-Test

The Hsu two-sample t-test is an alternative to the Welch two-sample t-test using a different formula for computing the degrees of freedom of the respective t-distribution (Hedderich and Sachs 2018). The following code is taken and adapted from the help page of the t.test function.

t.test(1:10, y = c(7:20))      # P = .00001855
##
##  Welch Two Sample t-test
##
## data:  1:10 and c(7:20)
## t = -5.4349, df = 21.982, p-value = 1.855e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.052802  -4.947198
## sample estimates:
## mean of x mean of y
##       5.5      13.5
t.test(1:10, y = c(7:20, 200)) # P = .1245    -- NOT significant anymore
##
##  Welch Two Sample t-test
##
## data:  1:10 and c(7:20, 200)
## t = -1.6329, df = 14.165, p-value = 0.1245
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -47.242900   6.376233
## sample estimates:
## mean of x mean of y
##   5.50000  25.93333
hsu.t.test(1:10, y = c(7:20))
##
##  Hsu Two Sample t-test
##
## data:  1:10 and c(7:20)
## t = -5.4349, df = 9, p-value = 0.0004137
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.329805  -4.670195
## sample estimates:
## mean of x mean of y   SD of x   SD of y
##   5.50000  13.50000   3.02765   4.18330
hsu.t.test(1:10, y = c(7:20, 200))
##
##  Hsu Two Sample t-test
##
## data:  1:10 and c(7:20, 200)
## t = -1.6329, df = 9, p-value = 0.1369
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -48.740846   7.874179
## sample estimates:
## mean of x mean of y   SD of x   SD of y
##   5.50000  25.93333   3.02765  48.32253
## Traditional interface
with(sleep, t.test(extra[group == 1], extra[group == 2]))
##
##  Welch Two Sample t-test
##
## data:  extra[group == 1] and extra[group == 2]
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.3654832  0.2054832
## sample estimates:
## mean of x mean of y
##      0.75      2.33
with(sleep, hsu.t.test(extra[group == 1], extra[group == 2]))
##
##  Hsu Two Sample t-test
##
## data:  extra[group == 1] and extra[group == 2]
## t = -1.8608, df = 9, p-value = 0.09569
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.5007773  0.3407773
## sample estimates:
## mean of x mean of y   SD of x   SD of y
##  0.750000  2.330000  1.789010  2.002249
## Formula interface
t.test(extra ~ group, data = sleep)
##
##  Welch Two Sample t-test
##
## data:  extra by group
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means between group 1 and group 2 is not equal to 0
## 95 percent confidence interval:
##  -3.3654832  0.2054832
## sample estimates:
## mean in group 1 mean in group 2
##            0.75            2.33
hsu.t.test(extra ~ group, data = sleep)
##
##  Hsu Two Sample t-test
##
## data:  extra by group
## t = -1.8608, df = 9, p-value = 0.09569
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.5007773  0.3407773
## sample estimates:
## mean of x mean of y   SD of x   SD of y
##  0.750000  2.330000  1.789010  2.002249

## Bootstrap t-Test

One and two sample bootstrap t-tests with equal or unequal variances in the two sample case (Efron and Tibshirani 1993).

boot.t.test(1:10, y = c(7:20)) # without bootstrap: P = .00001855
##
##  Bootstrap Welch Two Sample t-test
##
## data:  1:10 and c(7:20)
## bootstrap p-value < 2.2e-16
## bootstrap difference of means (SE) = -8.005112 (1.400082)
## 95 percent bootstrap percentile confidence interval:
##  -10.814286  -5.228571
##
## Results without bootstrap:
## t = -5.4349, df = 21.982, p-value = 1.855e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.052802  -4.947198
## sample estimates:
## mean of x mean of y
##       5.5      13.5
boot.t.test(1:10, y = c(7:20, 200)) # without bootstrap: P = .1245
##
##  Bootstrap Welch Two Sample t-test
##
## data:  1:10 and c(7:20, 200)
## bootstrap p-value = 0.0262
## bootstrap difference of means (SE) = -20.39106 (9.975238)
## 95 percent bootstrap percentile confidence interval:
##  -46.635000  -5.966667
##
## Results without bootstrap:
## t = -1.6329, df = 14.165, p-value = 0.1245
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -47.242900   6.376233
## sample estimates:
## mean of x mean of y
##   5.50000  25.93333
## Traditional interface
with(sleep, boot.t.test(extra[group == 1], extra[group == 2]))
##
##  Bootstrap Welch Two Sample t-test
##
## data:  extra[group == 1] and extra[group == 2]
## bootstrap p-value = 0.08401
## bootstrap difference of means (SE) = -1.57145 (0.8004523)
## 95 percent bootstrap percentile confidence interval:
##  -3.17  0.04
##
## Results without bootstrap:
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.3654832  0.2054832
## sample estimates:
## mean of x mean of y
##      0.75      2.33
## Formula interface
boot.t.test(extra ~ group, data = sleep)
##
##  Bootstrap Welch Two Sample t-test
##
## data:  extra by group
## bootstrap p-value = 0.08701
## bootstrap difference of means (SE) = -1.584426 (0.8001336)
## 95 percent bootstrap percentile confidence interval:
##  -3.17  0.02
##
## Results without bootstrap:
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.3654832  0.2054832
## sample estimates:
## mean in group 1 mean in group 2
##            0.75            2.33

## Permutation t-Test

One and two sample permutation t-tests with equal (Efron and Tibshirani 1993) or unequal variances (Janssen 1997) in the two sample case.

perm.t.test(1:10, y = c(7:20)) # without permutation: P = .00001855
##
##  Permutation Welch Two Sample t-test
##
## data:  1:10 and c(7:20)
## (Monte-Carlo) permutation p-value = 1e-04
## permutation difference of means (SE) = -7.959539 (2.253474)
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -12.400000  -3.485714
##
## Results without permutation:
## t = -5.4349, df = 21.982, p-value = 1.855e-05
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.052802  -4.947198
## sample estimates:
## mean of x mean of y
##       5.5      13.5
## permutation confidence interval sensitive to outlier!
perm.t.test(1:10, y = c(7:20, 200)) # without permutation: P = .1245
##
##  Permutation Welch Two Sample t-test
##
## data:  1:10 and c(7:20, 200)
## (Monte-Carlo) permutation p-value < 2.2e-16
## permutation difference of means (SE) = -20.40047 (15.36564)
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -36.866667   1.966667
##
## Results without permutation:
## t = -1.6329, df = 14.165, p-value = 0.1245
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -47.242900   6.376233
## sample estimates:
## mean of x mean of y
##   5.50000  25.93333
## Traditional interface
with(sleep, perm.t.test(extra[group == 1], extra[group == 2]))
##
##  Permutation Welch Two Sample t-test
##
## data:  extra[group == 1] and extra[group == 2]
## (Monte-Carlo) permutation p-value = 0.07991
## permutation difference of means (SE) = -1.571447 (0.9015273)
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -3.32  0.22
##
## Results without permutation:
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.3654832  0.2054832
## sample estimates:
## mean of x mean of y
##      0.75      2.33
## Formula interface
res <- perm.t.test(extra ~ group, data = sleep)
res
##
##  Permutation Welch Two Sample t-test
##
## data:  extra by group
## (Monte-Carlo) permutation p-value = 0.07371
## permutation difference of means (SE) = -1.587719 (0.9025178)
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -3.30  0.16
##
## Results without permutation:
## t = -1.8608, df = 17.776, p-value = 0.07939
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -3.3654832  0.2054832
## sample estimates:
## mean in group 1 mean in group 2
##            0.75            2.33

In case of skewed distributions, one may use function p2ses to compute an alternative standardized effect size (SES) as proposed by Botta-Dukat (Botta-Dukát 2018).

p2ses(res$p.value) ## [1] 1.754212 ## Multiple Imputation t-Test Function mi.t.test can be used to compute a multiple imputation t-test by applying the approch of Rubin (Rubin 1987) in combination with the adjustment of Barnard and Rubin (Barnard and Rubin 1999). ## Generate some data set.seed(123) x <- rnorm(25, mean = 1) x[sample(1:25, 5)] <- NA y <- rnorm(20, mean = -1) y[sample(1:20, 4)] <- NA pair <- c(rnorm(25, mean = 1), rnorm(20, mean = -1)) g <- factor(c(rep("yes", 25), rep("no", 20))) D <- data.frame(ID = 1:45, variable = c(x, y), pair = pair, group = g) ## Use Amelia to impute missing values library(Amelia) ## Lade nötiges Paket: Rcpp ## ## ## ## Amelia II: Multiple Imputation ## ## (Version 1.8.0, built: 2021-05-26) ## ## Copyright (C) 2005-2022 James Honaker, Gary King and Matthew Blackwell ## ## Refer to http://gking.harvard.edu/amelia/ for more information ## ## res <- amelia(D, m = 10, p2s = 0, idvars = "ID", noms = "group") ## Per protocol analysis (Welch two-sample t-test) t.test(variable ~ group, data = D) ## ## Welch Two Sample t-test ## ## data: variable by group ## t = -6.9214, df = 33.69, p-value = 5.903e-08 ## alternative hypothesis: true difference in means between group no and group yes is not equal to 0 ## 95 percent confidence interval: ## -2.628749 -1.435125 ## sample estimates: ## mean in group no mean in group yes ## -1.0862469 0.9456901 ## Intention to treat analysis (Multiple Imputation Welch two-sample t-test) mi.t.test(res$imputations, x = "variable", y = "group")
##
##  Multiple Imputation Welch Two Sample t-test
##
## data:  Variable variable: group no vs group yes
## t = -6.6457, df = 25.142, p-value = 5.63e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -2.668011 -1.405862
## sample estimates:
##  mean (no)    SD (no) mean (yes)   SD (yes)
## -1.0809584  0.9379070  0.9559786  1.0203219
## Per protocol analysis (Two-sample t-test)
t.test(variable ~ group, data = D, var.equal = TRUE)
##
##  Two Sample t-test
##
## data:  variable by group
## t = -6.8168, df = 34, p-value = 7.643e-08
## alternative hypothesis: true difference in means between group no and group yes is not equal to 0
## 95 percent confidence interval:
##  -2.637703 -1.426171
## sample estimates:
##  mean in group no mean in group yes
##        -1.0862469         0.9456901
## Intention to treat analysis (Multiple Imputation two-sample t-test)
mi.t.test(res$imputations, x = "variable", y = "group", var.equal = TRUE) ## ## Multiple Imputation Two Sample t-test ## ## data: Variable variable: group no vs group yes ## t = -6.5736, df = 26.042, p-value = 5.656e-07 ## alternative hypothesis: true difference in means is not equal to 0 ## 95 percent confidence interval: ## -2.673828 -1.400046 ## sample estimates: ## mean (no) SD (no) mean (yes) SD (yes) ## -1.0809584 0.9379070 0.9559786 1.0203219 ## Specifying alternatives mi.t.test(res$imputations, x = "variable", y = "group", alternative = "less")
##
##  Multiple Imputation Welch Two Sample t-test
##
## data:  Variable variable: group no vs group yes
## t = -6.6457, df = 25.142, p-value = 2.815e-07
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##     -Inf -1.5135
## sample estimates:
##  mean (no)    SD (no) mean (yes)   SD (yes)
## -1.0809584  0.9379070  0.9559786  1.0203219
mi.t.test(res$imputations, x = "variable", y = "group", alternative = "greater") ## ## Multiple Imputation Welch Two Sample t-test ## ## data: Variable variable: group no vs group yes ## t = -6.6457, df = 25.142, p-value = 1 ## alternative hypothesis: true difference in means is greater than 0 ## 95 percent confidence interval: ## -2.560374 Inf ## sample estimates: ## mean (no) SD (no) mean (yes) SD (yes) ## -1.0809584 0.9379070 0.9559786 1.0203219 ## One sample test t.test(D$variable[D$group == "yes"]) ## ## One Sample t-test ## ## data: D$variable[D$group == "yes"] ## t = 4.5054, df = 19, p-value = 0.0002422 ## alternative hypothesis: true mean is not equal to 0 ## 95 percent confidence interval: ## 0.5063618 1.3850184 ## sample estimates: ## mean of x ## 0.9456901 mi.t.test(res$imputations, x = "variable", subset = D$group == "yes") ## ## Multiple Imputation One Sample t-test ## ## data: Variable variable ## t = 4.6847, df = 18.494, p-value = 0.0001725 ## alternative hypothesis: true mean is not equal to 0 ## 95 percent confidence interval: ## 0.5280752 1.3838820 ## sample estimates: ## mean SD ## 0.9559786 1.0203219 mi.t.test(res$imputations, x = "variable", mu = -1, subset = D$group == "yes", alternative = "less") ## ## Multiple Imputation One Sample t-test ## ## data: Variable variable ## t = 9.5851, df = 18.494, p-value = 1 ## alternative hypothesis: true mean is less than -1 ## 95 percent confidence interval: ## -Inf 1.309328 ## sample estimates: ## mean SD ## 0.9559786 1.0203219 mi.t.test(res$imputations, x = "variable", mu = -1, subset = D$group == "yes", alternative = "greater") ## ## Multiple Imputation One Sample t-test ## ## data: Variable variable ## t = 9.5851, df = 18.494, p-value = 6.655e-09 ## alternative hypothesis: true mean is greater than -1 ## 95 percent confidence interval: ## 0.6026297 Inf ## sample estimates: ## mean SD ## 0.9559786 1.0203219 ## paired test t.test(D$variable, D$pair, paired = TRUE) ## ## Paired t-test ## ## data: D$variable and D$pair ## t = -1.3532, df = 35, p-value = 0.1847 ## alternative hypothesis: true mean difference is not equal to 0 ## 95 percent confidence interval: ## -0.6976921 0.1395993 ## sample estimates: ## mean difference ## -0.2790464 mi.t.test(res$imputations, x = "variable", y = "pair", paired = TRUE)
##
##  Multiple Imputation Paired t-test
##
## data:  Variables  variable and pair
## t = -1.0455, df = 39.974, p-value = 0.3021
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.5680741  0.1807237
## sample estimates:
## mean of difference   SD of difference
##         -0.1936752          1.2426515

Function mi.t.test also works with package mice by applying function mids2datlist of package miceadds.

library(mice)
##
## Attache Paket: 'mice'
## Das folgende Objekt ist maskiert 'package:stats':
##
##     filter
## Die folgenden Objekte sind maskiert von 'package:base':
##
##     cbind, rbind
library(miceadds)
## * miceadds 3.15-21 (2022-09-22 09:41:02)
res.mice <- mice(D, m = 10, print = FALSE)
res.list <- mids2datlist(res.mice)
mi.t.test(res.list, x = "variable", y = "group")
##
##  Multiple Imputation Welch Two Sample t-test
##
## data:  Variable variable: group no vs group yes
## t = -7.2111, df = 29.735, p-value = 5.289e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -2.6351 -1.4716
## sample estimates:
##  mean (no)    SD (no) mean (yes)   SD (yes)
## -1.0771373  0.8585436  0.9762127  1.0261920

## Repeated Measures One-Way ANOVA

We provide a simple wrapper function that allows to compute a classical repeated measures one-way ANOVA as well as a respective mixed effects model. In addition, the non-parametric Friedman and Quade tests can be computed.

set.seed(123)
outcome <- c(rnorm(10), rnorm(10, mean = 1.5), rnorm(10, mean = 1))
timepoints <- factor(rep(1:3, each = 10))
patients <- factor(rep(1:10, times = 3))
rm.oneway.test(outcome, timepoints, patients)
##
##  Repeated measures 1-way ANOVA
##
## data:  outcome , timepoints and patients
## F = 6.5898, num df = 2, denom df = 18, p-value = 0.007122
rm.oneway.test(outcome, timepoints, patients, method = "lme")
##
##  Mixed-effects 1-way ANOVA
##
## data:  outcome , timepoints and patients
## F = 7.3674, num df = 2, denom df = 18, p-value = 0.004596
rm.oneway.test(outcome, timepoints, patients, method = "friedman")
##
##  Friedman rank sum test
##
## data:  x, g and id
## Friedman chi-squared = 12.8, df = 2, p-value = 0.001662
rm.oneway.test(outcome, timepoints, patients, method = "quade")
##
##  Quade test
##
## data:  x, g and id
## Quade F = 7.2906, num df = 2, denom df = 18, p-value = 0.004795

## Volcano Plots

Volcano plots can be used to visualize the results when many tests have been applied. They show a measure of effect size in combination with (adjusted) p values.

## Generate some data
x <- matrix(rnorm(1000, mean = 10), nrow = 10)
g1 <- rep("control", 10)
y1 <- matrix(rnorm(500, mean = 11.75), nrow = 10)
y2 <- matrix(rnorm(500, mean = 9.75, sd = 3), nrow = 10)
g2 <- rep("treatment", 10)
group <- factor(c(g1, g2))
Data <- rbind(x, cbind(y1, y2))
## compute Hsu t-test
pvals <- apply(Data, 2, function(x, group) hsu.t.test(x ~ group)$p.value, group = group) ## compute log-fold change logfc <- function(x, group){ res <- tapply(x, group, mean) log2(res[1]/res[2]) } lfcs <- apply(Data, 2, logfc, group = group) volcano(lfcs, p.adjust(pvals, method = "fdr"), effect.low = -0.25, effect.high = 0.25, xlab = "log-fold change", ylab = "-log10(adj. p value)") ## Imputation of Standard Deviations for Changes from Baseline The function imputeSD can be used to impute standard deviations for changes from baseline adopting the approach of the Cochrane handbook (Higgins et al. 2019, Section 6.5.2.8). SD1 <- c(0.149, 0.022, 0.036, 0.085, 0.125, NA, 0.139, 0.124, 0.038) SD2 <- c(NA, 0.039, 0.038, 0.087, 0.125, NA, 0.135, 0.126, 0.038) SDchange <- c(NA, NA, NA, 0.026, 0.058, NA, NA, NA, NA) imputeSD(SD1, SD2, SDchange) ## SD1 SD2 SDchange Cor SDchange.min SDchange.mean SDchange.max ## 1 0.149 0.149 NA NA 0.04491608 0.05829769 0.06913600 ## 2 0.022 0.039 NA NA 0.01915642 0.02050235 0.02176520 ## 3 0.036 0.038 NA NA 0.01132754 0.01460887 0.01727787 ## 4 0.085 0.087 0.026 0.9545639 0.02600000 0.02600000 0.02600000 ## 5 0.125 0.125 0.058 0.8923520 0.05800000 0.05800000 0.05800000 ## 6 NA NA NA NA NA NA NA ## 7 0.139 0.135 NA NA 0.04148755 0.05374591 0.06368696 ## 8 0.124 0.126 NA NA 0.03773311 0.04894677 0.05803262 ## 9 0.038 0.038 NA NA 0.01145511 0.01486787 0.01763200 Correlations can also be provided via argument corr. This option may particularly be useful, if no complete data is available. SDchange2 <- rep(NA, 9) imputeSD(SD1, SD2, SDchange2, corr = c(0.85, 0.9, 0.95)) ## SD1 SD2 SDchange Cor SDchange.min SDchange.mean SDchange.max ## 1 0.149 0.149 NA NA 0.04711794 0.06663483 0.08161066 ## 2 0.022 0.039 NA NA 0.01935975 0.02146159 0.02337520 ## 3 0.036 0.038 NA NA 0.01186592 0.01666133 0.02035682 ## 4 0.085 0.087 NA NA 0.02726720 0.03850974 0.04714340 ## 5 0.125 0.125 NA NA 0.03952847 0.05590170 0.06846532 ## 6 NA NA NA NA NA NA NA ## 7 0.139 0.135 NA NA 0.04350287 0.06139218 0.07513654 ## 8 0.124 0.126 NA NA 0.03957777 0.05593568 0.06849234 ## 9 0.038 0.038 NA NA 0.01201666 0.01699412 0.02081346 ## Pairwise Comparisons Function pairwise.fun enables the application of arbitrary functions for pairwise comparisons. pairwise.wilcox.test(airquality$Ozone, airquality$Month, p.adjust.method = "none") ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## Warning in wilcox.test.default(xi, xj, paired = paired, ...): kann bei Bindungen ## keinen exakten p-Wert Berechnen ## ## Pairwise comparisons using Wilcoxon rank sum test with continuity correction ## ## data: airquality$Ozone and airquality$Month ## ## 5 6 7 8 ## 6 0.19250 - - - ## 7 3e-05 0.01414 - - ## 8 0.00012 0.02591 0.86195 - ## 9 0.11859 0.95887 0.00074 0.00325 ## ## P value adjustment method: none ## To avoid the warnings library(exactRankTests) ## Package 'exactRankTests' is no longer under development. ## Please consider using package 'coin' instead. pairwise.fun(airquality$Ozone, airquality$Month, fun = function(x, y) wilcox.exact(x, y)$p.value)
##       5 vs 6       5 vs 7       5 vs 8       5 vs 9       6 vs 7       6 vs 8
## 1.897186e-01 1.087205e-05 6.108735e-05 1.171796e-01 1.183753e-02 2.333564e-02
##       6 vs 9       7 vs 8       7 vs 9       8 vs 9
## 9.528659e-01 8.595683e-01 5.281796e-04 2.714694e-03

The function is also the basis for our functions pairwise.wilcox.exact and pairwise.ext.t.test, which in contrast to the standard functions pairwise.wilcox.test and pairwise.t.test generate a much more detailed output. In addition, pairwise.wilcox.exact is based on wilcox.exact instead of wilcox.test.

pairwise.wilcox.exact(airquality$Ozone, airquality$Month)
##
##  Pairwise Exact Wilcoxon rank sum tests
##
## data:  airquality$Ozone and airquality$Month
## alternative hypothesis: true location shift is not equal to 0
##
##    groups     W diff. in location 2.5% 97.5%      p-value adj. p-value
## 1  5 vs 6 152.0               6.5   -5    18 1.897186e-01 0.5691559000
## 2  5 vs 7 566.5              37.5   21    51 1.087205e-05 0.0001087205
## 3  5 vs 8 548.5              31.5   14    53 6.108735e-05 0.0005497862
## 4  5 vs 9 470.0               5.5   -2    13 1.171796e-01 0.4687184000
## 5  6 vs 7 182.5              28.5    8    51 1.183753e-02 0.0710251900
## 6  6 vs 8 176.5              23.5    2    53 2.333564e-02 0.1166782000
## 7  6 vs 9 128.5              -0.5  -12    10 9.528659e-01 1.0000000000
## 8  7 vs 8 328.0              -2.5  -21    19 8.595683e-01 1.0000000000
## 9  7 vs 9 176.5             -30.5  -44   -13 5.281796e-04 0.0042254370
## 10 8 vs 9 202.0             -23.5  -46    -7 2.714694e-03 0.0190028600

Furthermore, function pairwise.ext.t.test also enables pairwise Student, Welch, Hsu, bootstrap and permutation t tests.

pairwise.t.test(airquality$Ozone, airquality$Month, pool.sd = FALSE)
##
##  Pairwise comparisons using t tests with non-pooled SD
##
## data:  airquality$Ozone and airquality$Month
##
##   5       6       7       8
## 6 1.00000 -       -       -
## 7 0.00026 0.01527 -       -
## 8 0.00195 0.02135 1.00000 -
## 9 0.86321 1.00000 0.00589 0.01721
##
## P value adjustment method: holm
pairwise.ext.t.test(airquality$Ozone, airquality$Month)
##
##  Pairwise Welch Two Sample t-tests
##
## data:  airquality$Ozone and airquality$Month
## alternative hypothesis: true difference in means is not equal to 0
##
##    groups           t       df diff. in means       SE       2.5%     97.5%
## 1  5 vs 6  0.78010090 16.93764      5.8290598 7.472187  -9.940299  21.59842
## 2  5 vs 7  4.68198923 44.84296     35.5000000 7.582247  20.227090  50.77291
## 3  5 vs 8  4.07487966 39.27916     36.3461538 8.919565  18.308730  54.38358
## 4  5 vs 9  1.25274697 52.95697      7.8328912 6.252572  -4.708419  20.37420
## 5  6 vs 7  3.41859846 24.79227     29.6709402 8.679270  11.788050  47.55383
## 6  6 vs 8  3.09220573 29.98945     30.5170940 9.869037  10.361530  50.67265
## 7  6 vs 9  0.26556793 17.61281      2.0038314 7.545457 -13.873600  17.88127
## 8  7 vs 8  0.08501814 47.63564      0.8461538 9.952627 -19.168900  20.86121
## 9  7 vs 9 -3.61450643 46.58247    -27.6671088 7.654464 -43.069550 -12.26467
## 10 8 vs 9 -3.17483051 40.37577    -28.5132626 8.981035 -46.659350 -10.36718
##         p-value adj. p-value
## 1  4.460968e-01 1.0000000000
## 2  2.646679e-05 0.0002646679
## 3  2.168566e-04 0.0019517090
## 4  2.158016e-01 0.8632062000
## 5  2.181685e-03 0.0152717900
## 6  4.269318e-03 0.0213465900
## 7  7.936555e-01 1.0000000000
## 8  9.326033e-01 1.0000000000
## 9  7.361032e-04 0.0058888260
## 10 2.868290e-03 0.0172097400
pairwise.ext.t.test(airquality$Ozone, airquality$Month,
method = "hsu.t.test")
##
##  Pairwise Hsu Two Sample t-tests
##
## data:  airquality$Ozone and airquality$Month
## alternative hypothesis: true difference in means is not equal to 0
##
##    groups           t df diff. in means       SE       2.5%     97.5%
## 1  5 vs 6  0.78010090  8      5.8290598 7.472187 -11.401830  23.05995
## 2  5 vs 7  4.68198923 25     35.5000000 7.582247  19.884070  51.11593
## 3  5 vs 8  4.07487966 25     36.3461538 8.919565  17.975970  54.71634
## 4  5 vs 9  1.25274697 25      7.8328912 6.252572  -5.044523  20.71031
## 5  6 vs 7  3.41859846  8     29.6709402 8.679270   9.656507  49.68537
## 6  6 vs 8  3.09220573  8     30.5170940 9.869037   7.759053  53.27514
## 7  6 vs 9  0.26556793  8      2.0038314 7.545457 -15.396020  19.40369
## 8  7 vs 8  0.08501814 25      0.8461538 9.952627 -19.651670  21.34397
## 9  7 vs 9 -3.61450643 25    -27.6671088 7.654464 -43.431770 -11.90245
## 10 8 vs 9 -3.17483051 25    -28.5132626 8.981035 -47.010050 -10.01648
##         p-value adj. p-value
## 1  0.4577885000  1.000000000
## 2  0.0000849598  0.000849598
## 3  0.0004086781  0.003678103
## 4  0.2218896000  0.887558600
## 5  0.0091067660  0.054640600
## 6  0.0148398900  0.074199430
## 7  0.7972873000  1.000000000
## 8  0.9329242000  1.000000000
## 9  0.0013234440  0.010587550
## 10 0.0039521140  0.027664800
pairwise.ext.t.test(airquality$Ozone, airquality$Month,
method = "boot.t.test")
##
##  Pairwise Bootstrap Welch Two Sample t-tests
##
## data:  airquality$Ozone and airquality$Month
## alternative hypothesis: true difference in means is not equal to 0
##
##    groups diff. in means       SE       2.5%     97.5%    p-value adj. p-value
## 1  5 vs 6      5.8887363 6.955759  -7.807906  20.27479 0.41824180   1.00000000
## 2  5 vs 7     35.5503781 7.364936  20.653850  49.92692 0.00000000   0.00000000
## 3  5 vs 8     36.2165947 8.644354  18.921150  53.19423 0.00000000   0.00000000
## 4  5 vs 9      7.7751623 6.043571  -4.866512  19.52261 0.23222320   0.92889290
## 5  6 vs 7     29.6561703 8.227607  12.901500  45.82970 0.00400040   0.02280228
## 6  6 vs 8     30.4678318 9.450129  12.173930  48.93611 0.00380038   0.02280228
## 7  6 vs 9      2.0987570 7.044058 -12.905750  15.59425 0.83628360   1.00000000
## 8  7 vs 8      0.8187165 9.716243 -17.771150  19.88462 0.93569360   1.00000000
## 9  7 vs 9    -27.6598737 7.461887 -42.283890 -12.99257 0.00120012   0.00840084
## 10 8 vs 9    -28.6843232 8.742714 -46.125990 -11.72759 0.00100010   0.00800080
pairwise.ext.t.test(airquality$Ozone, airquality$Month,
method = "perm.t.test")
##
##  Pairwise Permutation Welch Two Sample t-tests
##
## data:  airquality$Ozone and airquality$Month
## alternative hypothesis: true difference in means is not equal to 0
##
##    groups diff. in means        SE        2.5%     97.5%    p-value
## 1  5 vs 6       5.727345  7.842643  -7.4871790  22.87607 0.50675070
## 2  5 vs 7      35.573465  9.001921  18.3076900  53.30769 0.00000000
## 3  5 vs 8      36.363152 10.190620  16.2307700  56.31154 0.00010001
## 4  5 vs 9       7.836545  6.311655  -4.4704910  19.82361 0.22582260
## 5  6 vs 7      29.713691 12.042870   5.6452990  52.76068 0.00520052
## 6  6 vs 8      30.709878 14.345440   0.6068376  56.99573 0.01040104
## 7  6 vs 9       1.881445  8.487179 -16.2298900  16.97280 0.79317930
## 8  7 vs 8       0.786594  9.853771 -18.0807700  20.31154 0.93069310
## 9  7 vs 9     -27.797195  8.367606 -43.8126700 -11.49469 0.00100010
## 10 8 vs 9     -28.583130  9.499669 -46.9602100 -10.34218 0.00270027
##    adj. p-value
## 1    1.00000000
## 2    0.00000000
## 3    0.00090009
## 4    0.90329030
## 5    0.03120312
## 6    0.05200520
## 7    1.00000000
## 8    1.00000000
## 9    0.00800080
## 10   0.01890189

## sessionInfo

sessionInfo()
## R version 4.2.2 Patched (2022-11-12 r83339)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Linux Mint 20.3
##
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/libf77blas.so.3.10.3
## LAPACK: /home/kohlm/RTOP/Rbranch/lib/libRlapack.so
##
## 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
##
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base
##
## other attached packages:
## [1] exactRankTests_0.8-35 miceadds_3.15-21      mice_3.14.0
## [4] Amelia_1.8.0          Rcpp_1.0.9            MKinfer_0.9
##
## loaded via a namespace (and not attached):
##  [1] tidyselect_1.2.0   xfun_0.34          bslib_0.4.1        purrr_0.3.5
##  [5] mitools_2.4        lattice_0.20-45    colorspace_2.0-3   vctrs_0.5.1
##  [9] generics_0.1.3     htmltools_0.5.3    yaml_2.3.6         gmp_0.6-8
## [13] utf8_1.2.2         rlang_1.0.6        jquerylib_0.1.4    pillar_1.8.1
## [17] foreign_0.8-83     glue_1.6.2         withr_2.5.0        DBI_1.1.3
## [21] arrangements_1.1.9 lifecycle_1.0.3    stringr_1.4.1      munsell_0.5.0
## [25] gtable_0.3.1       evaluate_0.18      labeling_0.4.2     knitr_1.40
## [29] fastmap_1.1.0      fansi_1.0.3        highr_0.9          broom_1.0.1
## [33] scales_1.2.1       backports_1.4.1    cachem_1.0.6       jsonlite_1.8.3
## [37] farver_2.1.1       ggplot2_3.4.0      digest_0.6.30      stringi_1.7.8
## [41] dplyr_1.0.10       grid_4.2.2         cli_3.4.1          tools_4.2.2
## [45] magrittr_2.0.3     sass_0.4.2         tibble_3.1.8       MKdescr_0.8
## [49] tidyr_1.2.1        pkgconfig_2.0.3    rmarkdown_2.18     R6_2.5.1
## [53] boot_1.3-28        nlme_3.1-160       compiler_4.2.2

## References

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Barnard, J, and D B Rubin. 1999. “Miscellanea. Small-Sample Degrees of Freedom with Multiple Imputation.” Biometrika 86 (4): 948–55. https://doi.org/10.1093/biomet/86.4.948.

Botta-Dukát, Z. 2018. “Cautionary Note on Calculating Standardized Effect Size (Ses) in Randomization Test.” Community Ecology 19 (1): 77–83. https://doi.org/10.1556/168.2018.19.1.8.

DasGupta, Anirban, T. Tony Cai, and Lawrence D. Brown. 2001. “Interval Estimation for a Binomial Proportion.” Statistical Science 16 (2): 101–33. https://doi.org/10.1214/ss/1009213286.

Davison, A. C., and D. V. Hinkley. 1997. Bootstrap Methods and Their Applications. Cambridge: Cambridge University Press. http://statwww.epfl.ch/davison/BMA/.

Efron, Bradley, and Robert J Tibshirani. 1993. An Introduction to the Bootstrap (Chapman & Hall/Crc Monographs on Statistics and Applied Probability). Chapman; Hall/CRC.

Gulhar, Monika, B M Golam Kibria, and Nasar Ahmed. 2012. “A Comparison of Some Confidence Intervals for Estimating the Population Coefficient of Variation: A Simulation Study.” SORT 36 (January): 45–68.

Hedderich, Jürgen, and Lothar Sachs. 2018. Angewandte Statistik: Methodensammlung Mit R. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-56657-2.

Higgins, Julian P.T., James Thomas, Jacqueline Chandler, Miranda Cumpston, Tianjing Li, Matthew J. Page, and Vivian A. Welch, eds. 2019. Cochrane Handbook for Systematic Reviews of Interventions. Wiley. https://doi.org/10.1002/9781119536604.

Janssen, Arnold. 1997. “Studentized Permutation Tests for Non-I.i.d. Hypotheses and the Generalized Behrens-Fisher Problem.” Statistics & Probability Letters 36 (1): 9–21. https://doi.org/10.1016/s0167-7152(97)00043-6.

Newcombe, R. G. 1998a. “Interval Estimation for the Difference Between Independent Proportions: Comparison of Eleven Methods.” Statistics in Medicine 17 (8): 873–90. https://doi.org/10.1002/(sici)1097-0258(19980430)17:8<873::aid-sim779>3.0.co;2-i.

———. 1998b. “Improved confidence intervals for the difference between binomial proportions based on paired data.” Stat Med 17 (22): 2635–50.

Rubin, Donald B. 1987. Multiple Imputation for Nonresponse in Surveys (Wiley Series in Probability and Statistics). Wiley.