Package MKinfer

Matthias Kohl

2020-01-07

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.

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)
## [1] "two.sided"
## 
##  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)
## [1] "two.sided"
## 
##  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, Davison and Hinkley (1997)).

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 1.359064 2.987960
## sd   2.393891 3.571155
## 
## sample estimates:
##     mean       sd 
## 2.173512 2.865789 
## 
## additional information:
## SE of mean 
##  0.4052837
meanCI(x)
## 
##  Exact confidence interval(s)
## 
##  percent confidence interval:
##         2.5 %  97.5 %
## mean 1.359064 2.98796
## 
## sample estimates:
##     mean       sd 
## 2.173512 2.865789 
## 
## additional information:
## SE of mean 
##  0.4052837
sdCI(x)
## 
##  Exact confidence interval(s)
## 
##  percent confidence interval:
##       2.5 %   97.5 %
## sd 2.393891 3.571155
## 
## sample estimates:
##     mean       sd 
## 2.173512 2.865789 
## 
## additional information:
## SE of mean 
##  0.4052837
## SD known
normCI(x, sd = 3)
## 
##  Exact confidence interval(s)
## 
## 95 percent confidence interval:
##        2.5 %   97.5 %
## mean 1.34197 3.005054
## 
## sample estimate:
##     mean 
## 2.173512 
## 
## 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 3.443882
## 
## sample estimate:
##       sd 
## 2.865789
## 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%   ( 1.388,  2.946 )   ( 1.367,  2.921 )   ( 1.433,  3.085 )  
## 
## Level     Percentile            BCa          
## 95%   ( 1.426,  2.981 )   ( 1.453,  3.020 )  
## Calculations and Intervals on Original Scale
## 
## sample estimates:
##     mean       sd 
## 2.173512 2.865789

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 -0.5370458 1.11875
## 
## sample estimates:
## mean of differences   sd of differences 
##           0.2908521           1.7689583 
## 
## additional information:
## SE of mean of differences 
##                 0.3955511
## compare
normCI(x-y)
## 
##  Exact confidence interval(s)
## 
## 95 percent confidence intervals:
##           2.5 %  97.5 %
## mean -0.5370458 1.11875
## sd    1.3452758 2.58369
## 
## sample estimates:
##      mean        sd 
## 0.2908521 1.7689583 
## 
## additional information:
## SE of mean 
##  0.3955511
## 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%   (-0.4738,  1.0498 )   (-0.4721,  1.0574 )   (-0.5607,  1.1302 )  
## 
## Level     Percentile            BCa          
## 95%   (-0.4757,  1.0538 )   (-0.4833,  1.0480 )  
## Calculations and Intervals on Original Scale
## 
## sample estimates:
## mean of differences   sd of differences 
##           0.2908521           1.7689583
## 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 -2.877292 -0.6304333
## 
## sample estimate:
## difference in means 
##           -1.753863 
## 
## additional information:
## SE of difference in means           Cohen's d (SMD) 
##                 0.5484406                -1.2385447 
## 
## mean of x   SD of x mean of y   SD of y 
## 0.1195805 0.9493020 1.8734433 2.0823225
## 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 -3.278131 -0.2295949
## 
## sample estimate:
## difference in means 
##           -1.753863 
## 
## additional information:
## SE of difference in means           Cohen's d (SMD) 
##                 0.6918565                -0.7663803 
## 
## mean of x   SD of x mean of y   SD of y 
## 0.1195805 0.9493020 1.8734433 2.0823225
## Hsu
normDiffCI(x, y, method = "hsu")
## 
##  Hsu confidence interval (unpaired)
## 
## 95 percent confidence interval:
##                         2.5 %     97.5 %
## difference in means -3.318951 -0.1887747
## 
## sample estimate:
## difference in means 
##           -1.753863 
## 
## additional information:
## SE of difference in means           Cohen's d (SMD) 
##                 0.6918565                -0.7663803 
## 
## mean of x   SD of x mean of y   SD of y 
## 0.1195805 0.9493020 1.8734433 2.0823225
## 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%   (-2.988, -0.346 )  
## Calculations and Intervals on Original Scale
## 
## sample estimate:
## difference in means 
##           -1.753863 
## 
## additional information:
## SE of difference in means           Cohen's d (SMD) 
##                 0.5484406                -1.2385447 
## 
## mean of x   SD of x mean of y   SD of y 
## 0.1195805 0.9493020 1.8734433 2.0823225
## 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%   (-2.965, -0.386 )  
## Calculations and Intervals on Original Scale
## 
## sample estimate:
## difference in means 
##           -1.753863 
## 
## additional information:
## SE of difference in means           Cohen's d (SMD) 
##                 0.6918565                -0.7663803 
## 
## mean of x   SD of x mean of y   SD of y 
## 0.1195805 0.9493020 1.8734433 2.0823225

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] 0.7
## Welch
sum(CIwelch[,1] < 0 & 0 < CIwelch[,2])/M
## [1] 0.96
## Hsu
sum(CIhsu[,1] < 0 & 0 < CIhsu[,2])/M
## [1] 0.96

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 0.1671466 0.2236147
## 
## sample estimate:
##        CV 
## 0.1953807 
## 
## additional information:
## standard error of CV 
##           0.01440539
## Gulhar et al. (2012)
cvCI(x, method = "gulhar")
## 
##  Gulhar et al (2012) confidence interval
## 
## 95 percent confidence interval:
##        2.5 %    97.5 %
## CV 0.1715456 0.2269689
## 
## sample estimate:
##        CV 
## 0.1953807
## 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%   ( 0.1705,  0.2229 )   ( 0.1710,  0.2236 )  
## 
## Level     Percentile            BCa          
## 95%   ( 0.1672,  0.2197 )   ( 0.1712,  0.2238 )  
## Calculations and Intervals on Original Scale
## 
## sample estimate:
##        CV 
## 0.1953807

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
## 
## 95.14464 percent confidence interval:
##                  lower    upper
## 95 % quantile 4.653803 7.872591
## 
## sample estimate:
## 95 % quantile 
##      5.488418
## asymptotic
quantileCI(x = x, prob = 0.95, method = "asymptotic")
## 
##  asymptotic confidence interval
## 
## 95 percent confidence interval:
##                  2.5 %   97.5 %
## 95 % quantile 4.653803 8.019216
## 
## sample estimate:
## 95 % quantile 
##      5.488418
## 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%   ( 3.285,  7.024 )   ( 3.316,  6.394 )  
## 
## Level     Percentile            BCa          
## 95%   ( 4.583,  7.661 )   ( 4.659,  7.873 )  
## Calculations and Intervals on Original Scale
## 
## sample estimate:
## 95 % quantile 
##      5.488418

Next, we consider the median.

## exact
medianCI(x = x)
## 
##  exact confidence interval
## 
## 95.23684 percent confidence intervals:
##            lower    upper
## median 0.9585896 2.128036
## median 1.1776298 2.442161
## 
## sample estimate:
##   median 
## 1.665435
## asymptotic
medianCI(x = x, method = "asymptotic")
## 
##  asymptotic confidence interval
## 
## 95 percent confidence interval:
##           2.5 %   97.5 %
## median 1.117329 2.271773
## 
## sample estimate:
##   median 
## 1.665435
## 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%   ( 1.060,  2.224 )   ( 1.059,  2.178 )  
## 
## Level     Percentile            BCa          
## 95%   ( 1.153,  2.272 )   ( 1.135,  2.210 )  
## Calculations and Intervals on Original Scale
## 
## sample estimate:
##   median 
## 1.665435

It often happens that quantile confidence intervals are not unique. Here the minimum length interval might be of interest.

medianCI(x = x, minLength = TRUE)
## 
##  minimum length exact confidence interval
## 
## 95.23684 percent confidence interval:
##            lower    upper
## median 0.9585896 2.128036
## 
## sample estimate:
##   median 
## 1.665435

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
## 
## 95.23684 percent confidence intervals:
##        lower    upper
## MAD 1.417652 1.981339
## MAD 1.474332 2.099394
## 
## sample estimate:
##      MAD 
## 1.759466
## aysymptotic
madCI(x = x, method = "asymptotic")
## 
##  asymptotic confidence interval
## 
## 95 percent confidence interval:
##        2.5 %   97.5 %
## MAD 1.465733 2.013086
## 
## sample estimate:
##      MAD 
## 1.759466
## 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%   ( 1.516,  2.057 )   ( 1.506,  2.049 )  
## 
## Level     Percentile            BCa          
## 95%   ( 1.470,  2.013 )   ( 1.468,  1.997 )  
## Calculations and Intervals on Original Scale
## 
## sample estimate:
##      MAD 
## 1.759466
## unstandardized
madCI(x = x, constant = 1)
## 
##  exact confidence interval
## 
## 95.23684 percent confidence intervals:
##         lower    upper
## MAD 0.9561930 1.336395
## MAD 0.9944235 1.416022
## 
## sample estimate:
##      MAD 
## 1.186743

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 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
## 
##  Bootstrapped Welch Two Sample t-test
## 
## data:  1:10 and c(7:20)
## bootstrapped p-value < 2.2e-16 
## 95 percent bootstrap percentile confidence interval:
##  -10.771429  -5.214286
## 
## 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
## 
##  Bootstrapped Welch Two Sample t-test
## 
## data:  1:10 and c(7:20, 200)
## bootstrapped p-value = 0.023 
## 95 percent bootstrap percentile confidence interval:
##  -46.600000  -6.066667
## 
## 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]))
## 
##  Bootstrapped Welch Two Sample t-test
## 
## data:  extra[group == 1] and extra[group == 2]
## bootstrapped p-value = 0.07801 
## 95 percent bootstrap percentile confidence interval:
##  -3.1400 -0.0095
## 
## 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)
## 
##  Bootstrapped Welch Two Sample t-test
## 
## data:  extra by group
## bootstrapped p-value = 0.07781 
## 95 percent bootstrap percentile confidence interval:
##  -3.13  0.01
## 
## 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 < 2.2e-16 
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -12.400000  -3.657143
## 
## 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 = 1e-04 
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -36.86667   1.80000
## 
## 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 = 0.07791 
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -3.34  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 of x mean of y 
##      0.75      2.33
## Formula interface
perm.t.test(extra ~ group, data = sleep)
## 
##  Permutation Welch Two Sample t-test
## 
## data:  extra by group
## (Monte-Carlo) permutation p-value = 0.07801 
## 95 percent (Monte-Carlo) permutation percentile confidence interval:
##  -3.321  0.200
## 
## 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

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)
## Loading required package: Rcpp
## ## 
## ## Amelia II: Multiple Imputation
## ## (Version 1.7.6, built: 2019-11-24)
## ## Copyright (C) 2005-2020 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 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$imputations, x = "variable", y = "group")
## 
##  Multiple Imputation Welch Two Sample t-test
## 
## data:  Variable variable: group no vs group yes
## t = -6.6457, df = 25.142, p-value = 5.63e-07
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -2.668011 -1.405862
## sample estimates:
##  mean (no)    SD (no) mean (yes)   SD (yes) 
## -1.0809584  0.9379070  0.9559786  1.0203219
## Per protocol analysis (Two-sample t-test)
t.test(variable ~ group, data = D, var.equal = TRUE)
## 
##  Two Sample t-test
## 
## data:  variable by group
## t = -6.8168, df = 34, p-value = 7.643e-08
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -2.637703 -1.426171
## sample estimates:
##  mean in group no mean in group yes 
##        -1.0862469         0.9456901
## Intention to treat analysis (Multiple Imputation two-sample t-test)
mi.t.test(res$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 difference in means is not equal to 0
## 95 percent confidence interval:
##  -0.6976921  0.1395993
## sample estimates:
## mean of the differences 
##              -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

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

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, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties

## Warning in wilcox.test.default(xi, xj, paired = paired, ...): cannot compute
## exact p-value with ties
## 
##  Pairwise comparisons using Wilcoxon rank sum test 
## 
## 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

sessionInfo

sessionInfo()
## R version 3.6.2 Patched (2020-01-03 r77636)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Linux Mint 19.3
## 
## 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_0.8-31 Amelia_1.7.6          Rcpp_1.0.3           
## [4] MKinfer_0.4          
## 
## loaded via a namespace (and not attached):
##  [1] gmp_0.5-13.5       pillar_1.4.3       compiler_3.6.2     arrangements_1.1.7
##  [5] tools_3.6.2        boot_1.3-24        digest_0.6.23      evaluate_0.14     
##  [9] lifecycle_0.1.0    tibble_2.1.3       gtable_0.3.0       nlme_3.1-143      
## [13] lattice_0.20-38    pkgconfig_2.0.3    rlang_0.4.2        yaml_2.2.0        
## [17] xfun_0.11          MKdescr_0.5        stringr_1.4.0      dplyr_0.8.3       
## [21] knitr_1.26         grid_3.6.2         tidyselect_0.2.5   glue_1.3.1        
## [25] R6_2.4.1           foreign_0.8-74     rmarkdown_2.0      farver_2.0.1      
## [29] ggplot2_3.2.1      purrr_0.3.3        magrittr_1.5       scales_1.1.0      
## [33] htmltools_0.4.0    assertthat_0.2.1   colorspace_1.4-1   labeling_0.3      
## [37] stringi_1.4.3      lazyeval_0.2.2     munsell_0.5.0      crayon_1.3.4

References

Altman, Douglas G., David Machin, Trevor N. Bryant, and Martin J. Gardner, eds. 2000. Statistics with Confidence: Confidence Intervals and Statistical Guidelines. BMJ Books.

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

DasGupta, Anirban, T. Tony Cai, and Lawrence D. Brown. 2001. “Interval Estimation for a Binomial Proportion.” Statistical Science 16 (2). Institute of Mathematical Statistics: 101–33. doi: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. doi: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. doi: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). Elsevier BV: 9–21. doi: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). Wiley: 873–90. doi: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.