Movatterモバイル変換

Type:	Package
Title:	Calculate Pairwise Multiple Comparisons of Mean Rank SumsExtended
Version:	1.9.12
Date:	2024-09-07
Description:	For one-way layout experiments the one-way ANOVA can be performed as an omnibus test. All-pairs multiple comparisons tests (Tukey-Kramer test, Scheffe test, LSD-test) and many-to-one tests (Dunnett test) for normally distributed residuals and equal within variance are available. Furthermore, all-pairs tests (Games-Howell test, Tamhane's T2 test, Dunnett T3 test, Ury-Wiggins-Hochberg test) and many-to-one (Tamhane-Dunnett Test) for normally distributed residuals and heterogeneous variances are provided. Van der Waerden's normal scores test for omnibus, all-pairs and many-to-one tests is provided for non-normally distributed residuals and homogeneous variances. The Kruskal-Wallis, BWS and Anderson-Darling omnibus test and all-pairs tests (Nemenyi test, Dunn test, Conover test, Dwass-Steele-Critchlow- Fligner test) as well as many-to-one (Nemenyi test, Dunn test, U-test) are given for the analysis of variance by ranks. Non-parametric trend tests (Jonckheere test, Cuzick test, Johnson-Mehrotra test, Spearman test) are included. In addition, a Friedman-test for one-way ANOVA with repeated measures on ranks (CRBD) and Skillings-Mack test for unbalanced CRBD is provided with consequent all-pairs tests (Nemenyi test, Siegel test, Miller test, Conover test, Exact test) and many-to-one tests (Nemenyi test, Demsar test, Exact test). A trend can be tested with Pages's test. Durbin's test for a two-way balanced incomplete block design (BIBD) is given in this package as well as Gore's test for CRBD with multiple observations per cell is given. Outlier tests, Mandel's k- and h statistic as well as functions for Type I error and Power analysis as well as generic summary, print and plot methods are provided.
Depends:	R (≥ 3.5.0)
Imports:	mvtnorm (≥ 1.0), multcompView, gmp, Rmpfr, SuppDists,kSamples (≥ 1.2.7), BWStest (≥ 0.2.1), MASS, stats
Suggests:	xtable, graphics, knitr, rmarkdown, car, e1071, multcomp,pwr, NSM3
SystemRequirements:	gmp (>= 4.2.3), mpfr (>= 3.0.0) | file README.md
SystemRequirementsNote:	see >> README.md
SysDataCompression:	gzip
VignetteBuilder:	knitr, rmarkdown
Classification/MSC-2010:	62J15, 62J10, 62G10, 62F03, 62G30
NeedsCompilation:	yes
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	7.3.1
License:	GPL (≥ 3)
Packaged:	2024-09-08 09:18:42 UTC; thorsten
Author:	Thorsten Pohlert [aut, cre]
Maintainer:	Thorsten Pohlert <thorsten.pohlert@gmx.de>
Repository:	CRAN
Date/Publication:	2024-09-08 10:10:03 UTC

Cochran's distribution

Description

Distribution function and quantile functionfor Cochran's distribution.

Usage

qcochran(p, k, n, lower.tail = TRUE, log.p = FALSE)pcochran(q, k, n, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

pcochran gives the distribution function andqcochran gives the quantile function.

References

Cochran, W.G. (1941) The distribution of the largest of a set of estimatedvariances as a fraction of their total.Ann. Eugen.11, 47–52.

Wilrich, P.-T. (2011) Critical values of Mandel's h and k,Grubbs and the Cochran test statistic.Adv. Stat. Anal..doi:10.1007/s10182-011-0185-y.

See Also

FDist

Examples

qcochran(0.05, 7, 3)

Grubbs D* distribution

Description

Distribution function for Grubbs D* distribution.

Usage

pdgrubbs(q, n, m = 10000, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

pgrubbs gives the distribution function

References

Grubbs, F.E. (1950) Sample criteria for testing outlying observations,Ann. Math. Stat.21, 27–58.

Wilrich, P.-T. (2011) Critical values of Mandel's h and k,Grubbs and the Cochran test statistic,Adv. Stat. Anal..doi:10.1007/s10182-011-0185-y.

See Also

Grubbs

Examples

pdgrubbs(0.62, 7, 1E4)

Generalized Siegel-Tukey Test of Homogeneity ofScales

Description

Performs a Siegel-Tukey k-sample rank dispersion test.

Usage

GSTTest(x, ...)## Default S3 method:GSTTest(x, g, dist = c("Chisquare", "KruskalWallis"), ...)## S3 method for class 'formula'GSTTest(  formula,  data,  subset,  na.action,  dist = c("Chisquare", "KruskalWallis"),  ...)

Arguments

Details

Meyer-Bahlburg (1970) has proposed a generalized Siegel-Tukeyrank dispersion test for thek-sample case.Likewise to thefligner.test, this testis a nonparametric test for testing the homogegeneity ofscales in several groups.Let\theta_i, and\lambda_i denotelocation and scale parameter of theith group,then for the two-tailed case, the null hypothesisH:\lambda_i / \lambda_j = 1 | \theta_i = \theta_j, ~ i \ne j istested against the alternative,A:\lambda_i / \lambda_j \ne 1with at least one inequality beeing strict.

The data are combinedly ranked according to Siegel-Tukey.The ranking is done by alternate extremes (rank 1 is lowest,2 and 3 are the two highest, 4 and 5 are the two next lowest, etc.).

Meyer-Bahlburg (1970) showed, that the Kruskal-Wallis H-testcan be employed on the Siegel-Tukey ranks.The H-statistic is assymptoticallychi-squared distributed withv = k - 1 degreeof freedom, the default test distribution is consequentlydist = "Chisquare". Ifdist = "KruskalWallis" is selected,an incomplete beta approximation is used for the calculationof p-values as implemented in the functionpKruskalWallis of the packageSuppDists.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

If ties are present, a tie correction is performed anda warning message is given. The GSTTest is sensitive tomedian differences, likewise to the Siegel-Tukey test.It is thus appropriate to apply this test on the residualsof a one-way ANOVA, rather than on the original data(see example).

References

H.F.L. Meyer-Bahlburg (1970), A nonparametric test for relativespread in k unpaired samples,Metrika15, 23–29.

See Also

fligner.test,pKruskalWallis,Chisquare,fligner.test

Examples

GSTTest(count ~ spray, data = InsectSprays)## as means/medians differ, apply the test to residuals## of one-way ANOVAans <- aov(count ~ spray, data = InsectSprays)GSTTest( residuals( ans) ~ spray, data =InsectSprays)

Grubbs distribution

Description

Distribution function and quantile functionfor Grubbs distribution.

Usage

qgrubbs(p, n)pgrubbs(q, n, lower.tail = TRUE)

Arguments

Value

pgrubbs gives the distribution function andqgrubbs gives the quantile function.

References

Grubbs, F. E. (1950) Sample criteria for testing outlying observations.Ann. Math. Stat.21, 27–58.

Wilrich, P.-T. (2011) Critical values of Mandel's h and k,Grubbs and the Cochran test statistic.Adv. Stat. Anal..doi:10.1007/s10182-011-0185-y.

See Also

TDist

Examples

qgrubbs(0.05, 7)

Extended One-Sided Studentised Range Test

Description

Performs Nashimoto-Wright's extendedone-sided studentised rangetest against an ordered alternative for normal datawith equal variances.

Usage

MTest(x, ...)## Default S3 method:MTest(x, g, alternative = c("greater", "less"), ...)## S3 method for class 'formula'MTest(  formula,  data,  subset,  na.action,  alternative = c("greater", "less"),  ...)## S3 method for class 'aov'MTest(x, alternative = c("greater", "less"), ...)

Arguments

Details

The procedure uses the property of a simple order,\theta_m' - \mu_m \le \mu_j - \mu_i \le \mu_l' - \mu_l\qquad (l \le i \le m~\mathrm{and}~ m' \le j \le l').The null hypothesis H_{ij}: \mu_i = \mu_j is tested againstthe alternative A_{ij}: \mu_i < \mu_j for any1 \le i < j \le k.

The all-pairs comparisons test statistics for a balanced design are

\hat{h}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{x}_{m'} - \bar{x}_m \right)} {s_{\mathrm{in}} / \sqrt{n}},

withn = n_i; ~ N = \sum_i^k n_i ~~ (1 \le i \le k),\bar{x}_i the arithmetic mean of theith group,ands_{\mathrm{in}}^2 the within ANOVA variance. The null hypothesis is rejected,if\hat{h} > h_{k,\alpha,v}, withv = N - kdegree of freedom.

For the unbalanced case with moderate imbalance the test statistic is

\hat{h}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{x}_{m'} - \bar{x}_m \right)} {s_{\mathrm{in}} \left(1/n_m + 1/n_{m'}\right)^{1/2}},

The null hypothesis is rejected, if\hat{h}_{ij} > h_{k,\alpha,v} / \sqrt{2}.

The function does not return p-values. Instead the critical h-valuesas given in the tables of Hayter (1990) for\alpha = 0.05 (one-sided)are looked up according to the number of groups (k) andthe degree of freedoms (v).

Value

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated statistic(s)

crit.value

critical values for\alpha = 0.05.

alternative

a character string describing the alternative hypothesis.

parameter

the parameter(s) of the test distribution.

dist

a string that denotes the test distribution.

There are print and summary methods available.

Note

The function will give a warning for the unbalanced case and returns thecritical valueh_{k,\alpha,\infty} / \sqrt{2}.

References

Hayter, A. J.(1990) A One-Sided Studentised RangeTest for Testing Against a Simple Ordered Alternative,Journal of the American Statistical Association85, 778–785.

Nashimoto, K., Wright, F.T., (2005) Multiple comparison proceduresfor detecting differences in simply ordered means.Comput. Statist. Data Anal.48, 291–306.

See Also

osrtTest,NPMTest

Examples

##md <- aov(weight ~ group, PlantGrowth)anova(md)osrtTest(md)MTest(md)

Mandel's h Distribution

Description

Distribution function and quantile functionfor Mandel's h distribution.

Usage

qmandelh(p, k, lower.tail = TRUE, log.p = FALSE)pmandelh(q, k, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

pmandelh gives the distribution function andqmandelh gives the quantile function.

Source

The code forpmandelh was taken from:
Stephen L R Ellison. (2017). metRology: Support for MetrologicalApplications. R package version 0.9-26-2.https://CRAN.R-project.org/package=metRology

References

Practice E 691 (2005)Standard Practice forConducting an Interlaboratory Study to Determine thePrecision of a Test Method, ASTM International.

See Also

mandelhTest

Examples

## We need a two-sided upper-tail quantileqmandelh(p = 0.005/2, k = 7, lower.tail=FALSE)

Mandel's k Distribution

Description

Distribution function and quantile functionfor Mandel's k distribution.

Usage

qmandelk(p, k, n, lower.tail = TRUE, log.p = FALSE)pmandelk(q, k, n, lower.tail = TRUE, log.p = FALSE)

Arguments

Value

pmandelk gives the distribution function andqmandelk gives the quantile function.

Source

The code forpmandelk was taken from:
Stephen L R Ellison. (2017). metRology: Support for MetrologicalApplications. R package version 0.9-26-2.https://CRAN.R-project.org/package=metRology

Note

The functions are only appropriate for balanced designs.

References

Practice E 691 (2005)Standard Practice forConducting an Interlaboratory Study to Determine thePrecision of a Test Method, ASTM International.

See Also

mandelkTest

pmandelh,qmandelh

Examples

qmandelk(0.005, 7, 3, lower.tail=FALSE)

All-Pairs Comparisons for Simply Ordered Mean Ranksums

Description

Performs Nashimoto and Wright's all-pairs comparison procedurefor simply ordered mean ranksums.

Usage

NPMTest(x, ...)## Default S3 method:NPMTest(  x,  g,  alternative = c("greater", "less"),  method = c("look-up", "boot", "asympt"),  nperm = 10000,  ...)## S3 method for class 'formula'NPMTest(  formula,  data,  subset,  na.action,  alternative = c("greater", "less"),  method = c("look-up", "boot", "asympt"),  nperm = 10000,  ...)

Arguments

Details

The procedure uses the property of a simple order,\theta_m' - \theta_m \le \theta_j - \theta_i \le \theta_l' - \theta_l\qquad (l \le i \le m~\mathrm{and}~ m' \le j \le l').The null hypothesis H_{ij}: \theta_i = \theta_j is tested againstthe alternative A_{ij}: \theta_i < \theta_j for any1 \le i < j \le k.

The all-pairs comparisons test statistics for a balanced design are

\hat{h}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{R}_{m'} - \bar{R}_m \right)}{\sigma_a / \sqrt{n}},

withn = n_i; ~ N = \sum_i^k n_i ~~ (1 \le i \le k),\bar{R}_i the mean rank for theith group,and\sigma_a = \sqrt{N \left(N + 1 \right) / 12}. The null hypothesis is rejected,ifh_{ij} > h_{k,\alpha,\infty}.

For the unbalanced case with moderate imbalance the test statistic is

\hat{h}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{R}_{m'} - \bar{R}_m \right)} {\sigma_a \left(1/n_m + 1/n_{m'}\right)^{1/2}},

The null hypothesis is rejected, if\hat{h}_{ij} > h_{k,\alpha,\infty} / \sqrt{2}.

Ifmethod = "look-up" the function will not returnp-values. Instead the critical h-valuesas given in the tables of Hayter (1990) for\alpha = 0.05 (one-sided)are looked up according to the number of groups (k) andthe degree of freedoms (v = \infty).

Ifmethod = "boot" an asymetric permutation testis conducted andp-values is returned.

Ifmethod = "asympt" is selected the asymptoticp-value is estimated as implemented in thefunctionpHayStonLSA of the packageNSM3.

Value

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated statistic(s)

crit.value

critical values for\alpha = 0.05.

alternative

a character string describing the alternative hypothesis.

parameter

the parameter(s) of the test distribution.

dist

a string that denotes the test distribution.

There are print and summary methods available.

Either a list of class"PMCMR" or alist with class"osrt" that contains the followingcomponents:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated statistic(s)

crit.value

critical values for\alpha = 0.05.

alternative

a character string describing the alternative hypothesis.

parameter

the parameter(s) of the test distribution.

dist

a string that denotes the test distribution.

There are print and summary methods available.

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

The function will give a warning for the unbalanced case and returns thecritical valueh_{k,\alpha,\infty} / \sqrt{2}.

Source

Ifmethod = "asympt" is selected, this function callsan internal probability functionpHS. The GPL-2 code forthis function was taken frompHayStonLSA of thethe packageNSM3:

Grant Schneider, Eric Chicken and Rachel Becvarik (2020) NSM3:Functions and Datasets to Accompany Hollander, Wolfe, andChicken - Nonparametric Statistical Methods, Third Edition. Rpackage version 1.15.https://CRAN.R-project.org/package=NSM3

References

Hayter, A. J.(1990) A One-Sided Studentised RangeTest for Testing Against a Simple Ordered Alternative,Journal of the American Statistical Association85, 778–785.

Nashimoto, K., Wright, F.T. (2007)Nonparametric Multiple-Comparison Methods for SimplyOrdered Medians.Comput Stat Data Anal51, 5068–5076.

See Also

MTest

Examples

## Example from Shirley (1977)## Reaction times of mice to stimuli to their tails.x <- c(2.4, 3, 3, 2.2, 2.2, 2.2, 2.2, 2.8, 2, 3, 2.8, 2.2, 3.8, 9.4, 8.4, 3, 3.2, 4.4, 3.2, 7.4, 9.8, 3.2, 5.8, 7.8, 2.6, 2.2, 6.2, 9.4, 7.8, 3.4, 7, 9.8, 9.4, 8.8, 8.8, 3.4, 9, 8.4, 2.4, 7.8)g <- gl(4, 10)## Shirley's test## one-sided test using look-up tableshirleyWilliamsTest(x ~ g, alternative = "greater")## Chacko's global hypothesis test for 'greater'chackoTest(x , g)## post-hoc test, default is standard normal distribution (NPT'-test)summary(chaAllPairsNashimotoTest(x, g, p.adjust.method = "none"))## same but h-distribution (NPY'-test)chaAllPairsNashimotoTest(x, g, dist = "h")## NPM-testNPMTest(x, g)## Hayter-Stone testhayterStoneTest(x, g)## all-pairs comparisonshsAllPairsTest(x, g)

Pentosan Dataset

Description

A benchmark dataset of an interlaboratory study fordetermining the precision of a test methodon several levels of the material Pentosan.

Format

A data frame with 189 obs. of 3 variables:

value

numeric, test result (no unit specified)

lab

factor, identifier of the lab (1–7)

material

factor, identifier of the level of the material (A–I)

Source

Tab. 8, Practice E 691, 2005,Standard Practice forConducting an Interlaboratory Study to Determine thePrecision of a Test Method, ASTM International.

Anderson-Darling All-Pairs Comparison Test

Description

Performs Anderson-Darling all-pairs comparison test.

Usage

adAllPairsTest(x, ...)## Default S3 method:adAllPairsTest(x, g, p.adjust.method = p.adjust.methods, ...)## S3 method for class 'formula'adAllPairsTest(  formula,  data,  subset,  na.action,  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals Anderson-Darling'sall-pairs comparison test can be used. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: F_i(x) = F_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: F_i(x) \ne F_j(x), ~~ i \ne j.

This function is a wrapper function that sequentiallycallsadKSampleTest for each pair.The calculated p-values forPr(>|T2N|)can be adjusted to account for Type I error multiplicityusing any method as implemented inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Scholz, F.W., Stephens, M.A. (1987) K-Sample Anderson-Darling Tests.Journal of the American Statistical Association82, 918–924.

See Also

adKSampleTest,adManyOneTest,ad.pval.

Examples

adKSampleTest(count ~ spray, InsectSprays)out <- adAllPairsTest(count ~ spray, InsectSprays, p.adjust="holm")summary(out)summaryGroup(out)

Anderson-Darling k-Sample Test

Description

Performs Anderson-Darling k-sample test.

Usage

adKSampleTest(x, ...)## Default S3 method:adKSampleTest(x, g, ...)## S3 method for class 'formula'adKSampleTest(formula, data, subset, na.action, ...)

Arguments

Details

The null hypothesis, H_0: F_1 = F_2 = \ldots = F_kis tested against the alternative,H_\mathrm{A}: F_i \ne F_j ~~(i \ne j), with at leastone unequality beeing strict.

This function only evaluates version 1 of the k-sample Anderson-Darlingtest (i.e. Eq. 6) of Scholz and Stephens (1987).The p-values are estimated with the extended empirical functionas implemented inad.pval ofthe packagekSamples.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Scholz, F.W., Stephens, M.A. (1987) K-Sample Anderson-Darling Tests.Journal of the American Statistical Association82, 918–924.

See Also

adAllPairsTest,adManyOneTest,ad.pval.

Examples

## Hollander & Wolfe (1973), 116.## Mucociliary efficiency from the rate of removal of dust in normal## subjects, subjects with obstructive airway disease, and subjects## with asbestosis.x <- c(2.9, 3.0, 2.5, 2.6, 3.2) # normal subjectsy <- c(3.8, 2.7, 4.0, 2.4)      # with obstructive airway diseasez <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosisg <- factor(x = c(rep(1, length(x)),                   rep(2, length(y)),                   rep(3, length(z))),             labels = c("ns", "oad", "a"))dat <- data.frame(   g = g,   x = c(x, y, z))## AD-TestadKSampleTest(x ~ g, data = dat)## BWS-TestbwsKSampleTest(x ~ g, data = dat)## Kruskal-Test## Using incomplete beta approximationkruskalTest(x ~ g, dat, dist="KruskalWallis")## Using chisquare distributionkruskalTest(x ~ g, dat, dist="Chisquare")## Not run: ## Check with kruskal.test from R statskruskal.test(x ~ g, dat)## End(Not run)## Using Conover's FkruskalTest(x ~ g, dat, dist="FDist")## Not run: ## Check with aov on ranksanova(aov(rank(x) ~ g, dat))## Check with oneway.testoneway.test(rank(x) ~ g, dat, var.equal = TRUE)## End(Not run)## Median Test asymptoticmedianTest(x ~ g, dat)## Median Test with simulated p-valuesset.seed(112)medianTest(x ~ g, dat, simulate.p.value = TRUE)

Anderson-Darling Many-To-One Comparison Test

Description

Performs Anderson-Darling many-to-one comparison test.

Usage

adManyOneTest(x, ...)## Default S3 method:adManyOneTest(x, g, p.adjust.method = p.adjust.methods, ...)## S3 method for class 'formula'adManyOneTest(  formula,  data,  subset,  na.action,  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in an one-factorial layout with non-normally distributedresiduals Anderson-Darling's non-parametric test can be performed.Let there bek groups including the control,then the number of treatment levels ism = k - 1.Thenm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: F_0 = F_i is tested in the two-tailed case againstA_i: F_0 \ne F_i, ~~ (1 \le i \le m).

This function is a wrapper function that sequentiallycallsadKSampleTest for each pair.The calculated p-values forPr(>|T2N|)can be adjusted to account for Type I error inflationusing any method as implemented inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Scholz, F.W., Stephens, M.A. (1987) K-Sample Anderson-Darling Tests.Journal of the American Statistical Association82, 918–924.

See Also

adKSampleTest,adAllPairsTest,ad.pval.

Examples

## Data set PlantGrowth## Global testadKSampleTest(weight ~ group, data = PlantGrowth)##ans <- adManyOneTest(weight ~ group,                             data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)

Algae Growth Inhibition Data Set

Description

A dose-response experiment was conducted using Atrazineat 9 different dose-levels including the zero-dose controland the biomass of algae (Selenastrumcapricornutum) as the response variable. Three replicateswere measured at day 0, 1 and 2. The fluorescence method (Mayer etal. 1997) was applied to measure biomass.

Format

A data frame with 22 observations on the following 10 variables.

concentration

a numeric vector of dose value in mg / L

Day.0

a numeric vector, total biomass

Day.0.1

a numeric vector, total biomass

Day.0.2

a numeric vector, total biomass

Day.1

a numeric vector, total biomass

Day.1.1

a numeric vector, total biomass

Day.1.2

a numeric vector, total biomass

Day.2

a numeric vector, total biomass

Day.2.1

a numeric vector, total biomass

Day.2.2

a numeric vector, total biomass

Source

ENV/JM/MONO(2006)18/ANN, page 24.

References

OECD (ed. 2006)Current approaches in the statistical analysisof ecotoxicity data: A guidance to application - Annexes, OECD Serieson testing and assessment, No. 54, (ENV/JM/MONO(2006)18/ANN).

See Also

demo(algae)

Plotting PMCMR Objects

Description

Plots a bar-plot for objects of class"PMCMR".

Usage

barPlot(x, alpha = 0.05, ...)

Arguments

Value

A barplot where the height of the bars corresponds to the arithmeticmean. The extend of the whiskers are\pm z_{(1-\alpha/2)}\times s_{\mathrm{E},i}, where the latter denotes the standard errorof theith group. Symbolic letters are depicted on top of the bars,whereas different letters indicate significant differences betweengroups for the selected level of alpha.

Note

The barplot is strictly spoken only valid for normal data, asthe depicted significance intervall implies symetry.

Examples

## data set chickwtsans <- tukeyTest(weight ~ feed, data = chickwts)barPlot(ans)

BWS All-Pairs Comparison Test

Description

Performs Baumgartner-Weiß-Schindler all-pairs comparison test.

Usage

bwsAllPairsTest(x, ...)## Default S3 method:bwsAllPairsTest(  x,  g,  method = c("BWS", "Murakami"),  p.adjust.method = p.adjust.methods,  ...)## S3 method for class 'formula'bwsAllPairsTest(  formula,  data,  subset,  na.action,  method = c("BWS", "Murakami"),  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals Baumgartner-Weiß-Schindlerall-pairs comparison test can be used. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: F_i(x) = F_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: F_i(x) \ne F_j(x), ~~ i \ne j.

This function is a wrapper function that sequentiallycallsbws_test for each pair.The default test method ("BWS") is the originalBaumgartner-Weiß-Schindler test statistic B. Formethod == "Murakami" it is the modified BWS statisticdenoted B*. The calculated p-values forPr(>|B|)orPr(>|B*|) can be adjusted to account for Type I errorinflation using any method as implemented inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for thegeneral two-sample problem,Biometrics54, 1129–1135.

Murakami, H. (2006) K-sample rank test based on modified Baumgartner statistic and its powercomparison,J. Jpn. Comp. Statist.19, 1–13.

See Also

bws_test.

Examples

out <- bwsAllPairsTest(count ~ spray, InsectSprays, p.adjust="holm")summary(out)summaryGroup(out)

Murakami's k-Sample BWS Test

Description

Performs Murakami's k-Sample BWS Test.

Usage

bwsKSampleTest(x, ...)## Default S3 method:bwsKSampleTest(x, g, nperm = 1000, ...)## S3 method for class 'formula'bwsKSampleTest(formula, data, subset, na.action, nperm = 1000, ...)

Arguments

Details

LetX_{ij} ~ (1 \le i \le k,~ 1 \le 1 \le n_i) denote anidentically and independently distributed variable that is obtainedfrom an unknown continuous distributionF_i(x). LetR_{ij}be the rank ofX_{ij}, whereX_{ij} is jointly rankedfrom1 toN, ~ N = \sum_{i=1}^k n_i.In thek-sample test the null hypothesis, H:F_i = F_jis tested against the alternative,A:F_i \ne F_j ~~(i \ne j) with at least one inequalitybeeing strict. Murakami (2006) has generalizedthe two-sample Baumgartner-Weiß-Schindler test(Baumgartner et al. 1998) and proposed amodified statisticB_k^* defined by

B_{k}^* = \frac{1}{k}\sum_{i=1}^k\left\{\frac{1}{n_i} \sum_{j=1}^{n_i} \frac{(R_{ij} - \mathsf{E}[R_{ij}])^2} {\mathsf{Var}[R_{ij}]}\right\},

where

\mathsf{E}[R_{ij}] = \frac{N + 1}{n_i + 1} j

and

\mathsf{Var}[R_{ij}] = \frac{j}{n_i + 1} \left(1 - \frac{j}{n_i + 1}\right)\frac{\left(N-n_i\right)\left(N+1\right)}{n_i + 2}.

Thep-values are estimated via an assymptotic boot-strap method.It should be noted that theB_k^* detects both differences in theunknown location parameters and / or differencesin the unknown scale parameters of thek-samples.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

One may increase the number of permutations to e.g.nperm = 10000in order to get more precise p-values. However, this will be onthe expense of computational time.

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for thegeneral two-sample problem,Biometrics54, 1129–1135.

Murakami, H. (2006) K-sample rank test based on modified Baumgartner statistic and its powercomparison,J. Jpn. Comp. Statist.19, 1–13.

See Also

sample,bwsAllPairsTest,bwsManyOneTest.

Examples

## Hollander & Wolfe (1973), 116.## Mucociliary efficiency from the rate of removal of dust in normal## subjects, subjects with obstructive airway disease, and subjects## with asbestosis.x <- c(2.9, 3.0, 2.5, 2.6, 3.2) # normal subjectsy <- c(3.8, 2.7, 4.0, 2.4)      # with obstructive airway diseasez <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosisg <- factor(x = c(rep(1, length(x)),                   rep(2, length(y)),                   rep(3, length(z))),             labels = c("ns", "oad", "a"))dat <- data.frame(   g = g,   x = c(x, y, z))## AD-TestadKSampleTest(x ~ g, data = dat)## BWS-TestbwsKSampleTest(x ~ g, data = dat)## Kruskal-Test## Using incomplete beta approximationkruskalTest(x ~ g, dat, dist="KruskalWallis")## Using chisquare distributionkruskalTest(x ~ g, dat, dist="Chisquare")## Not run: ## Check with kruskal.test from R statskruskal.test(x ~ g, dat)## End(Not run)## Using Conover's FkruskalTest(x ~ g, dat, dist="FDist")## Not run: ## Check with aov on ranksanova(aov(rank(x) ~ g, dat))## Check with oneway.testoneway.test(rank(x) ~ g, dat, var.equal = TRUE)## End(Not run)## Median Test asymptoticmedianTest(x ~ g, dat)## Median Test with simulated p-valuesset.seed(112)medianTest(x ~ g, dat, simulate.p.value = TRUE)

BWS Many-To-One Comparison Test

Description

Performs Baumgartner-Weiß-Schindler many-to-one comparison test.

Usage

bwsManyOneTest(x, ...)## Default S3 method:bwsManyOneTest(  x,  g,  alternative = c("two.sided", "greater", "less"),  method = c("BWS", "Murakami", "Neuhauser"),  p.adjust.method = p.adjust.methods,  ...)## S3 method for class 'formula'bwsManyOneTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  method = c("BWS", "Murakami", "Neuhauser"),  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in an one-factorial layout with non-normally distributedresiduals Baumgartner-Weiß-Schindler's non-parametric test can be performed.Let there bek groups including the control,then the number of treatment levels ism = k - 1.Thenm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: F_0 = F_i is tested in the two-tailed case againstA_i: F_0 \ne F_i, ~~ (1 \le i \le m).

This function is a wrapper function that sequentiallycallsbws_stat andbws_cdffor each pair. For the default test method ("BWS") the originalBaumgartner-Weiß-Schindler test statistic B and its corresponding Pr(>|B|)is calculated. Formethod == "BWS" only a two-sided test is possible.

Formethod == "Murakami" the modified BWS statisticdenoted B* and its corresponding Pr(>|B*|) is computed by sequentially callingmurakami_stat andmurakami_cdf.Formethod == "Murakami" only a two-sided test is possible.

Ifalternative == "greater" then the alternative, if onepopulation is stochastically larger than the other is tested:H_i: F_0 = F_i against A_i: F_0 \ge F_i, ~~ (1 \le i \le m).The modified test-statistic B* according to Neuhäuser (2001) and itscorresponding Pr(>B*) or Pr(<B*) is computed by sequentally callingmurakami_stat andmurakami_cdfwithflavor = 2.

The p-values can be adjusted to account for Type I errorinflation using any method as implemented inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for thegeneral two-sample problem,Biometrics54, 1129–1135.

Murakami, H. (2006) K-sample rank test based on modified Baumgartner statistic and its powercomparison,J Jpn Comp Statist19, 1–13.

Neuhäuser, M. (2001) One-Side Two-Sample and Trend Tests Based on a ModifiedBaumgartner-Weiss-Schindler Statistic.J Nonparametric Stat13, 729–739.

See Also

murakami_stat,murakami_cdf,bws_stat,bws_cdf.

Examples

out <- bwsManyOneTest(weight ~ group, PlantGrowth, p.adjust="holm")summary(out)## A two-sample testset.seed(1245)x <- c(rnorm(20), rnorm(20,0.3))g <- gl(2, 20)summary(bwsManyOneTest(x ~ g, alternative = "less", p.adjust="none"))summary(bwsManyOneTest(x ~ g, alternative = "greater", p.adjust="none"))## Not run: ## Check with the implementation in package BWStestBWStest::bws_test(x=x[g==1], y=x[g==2], alternative = "less")BWStest::bws_test(x=x[g==1], y=x[g==2], alternative = "greater")## End(Not run)

Testing against Ordered Alternatives (Murakami's BWS Trend Test)

Description

Performs Murakami's modified Baumgartner-Weiß-Schindlertest for testing against ordered alternatives.

Usage

bwsTrendTest(x, ...)## Default S3 method:bwsTrendTest(x, g, nperm = 1000, ...)## S3 method for class 'formula'bwsTrendTest(formula, data, subset, na.action, nperm = 1000, ...)

Arguments

Details

The null hypothesis, H_0: F_1(u) = F_2(u) = \ldots = F_k(u) ~~ u \in Ris tested against a simple order hypothesis,H_\mathrm{A}: F_1(u) \le F_2(u) \le \ldots \leF_k(u),~F_1(u) < F_k(u), ~~ u \in R.

The p-values are estimated through an assymptotic boot-strap methodusing the functionsample.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

One may increase the number of permutations to e.g.nperm = 10000in order to get more precise p-values. However, this will be onthe expense of computational time.

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for thegeneral two-sample problem,Biometrics54, 1129–1135.

Murakami, H. (2006) K-sample rank test based on modified Baumgartner statistic and its powercomparison,J Jpn Comp Statist19, 1–13.

Neuhäuser, M. (2001) One-Side Two-Sample and Trend Tests Based on a ModifiedBaumgartner-Weiss-Schindler Statistic.J Nonparametric Stat13, 729–739.

See Also

sample,bwsAllPairsTest,bwsManyOneTest.

kruskalTest andshirleyWilliamsTestof the packagePMCMRplus,kruskal.test of the librarystats.

Examples

## Example from Sachs (1997, p. 402)x <- c(106, 114, 116, 127, 145,       110, 125, 143, 148, 151,       136, 139, 149, 160, 174)g <- gl(3,5)levels(g) <- c("A", "B", "C")## Chacko's testchackoTest(x, g)## Cuzick's testcuzickTest(x, g)## Johnson-Mehrotra testjohnsonTest(x, g)## Jonckheere-Terpstra testjonckheereTest(x, g)## Le's testleTest(x, g)## Spearman type testspearmanTest(x, g)## Murakami's BWS trend testbwsTrendTest(x, g)## Fligner-Wolfe testflignerWolfeTest(x, g)## Shan-Young-Kang testshanTest(x, g)

All-Pairs Comparisons for Simply Ordered Mean Ranksums

Description

Performs Nashimoto and Wright's all-pairs comparison procedurefor simply ordered mean ranksums (NPT'-test and NPY'-test).

According to the authors, the procedure shall only beapplied after Chacko's test (seechackoTest) indicatesglobal significance.

Usage

chaAllPairsNashimotoTest(x, ...)## Default S3 method:chaAllPairsNashimotoTest(  x,  g,  p.adjust.method = c(p.adjust.methods),  alternative = c("greater", "less"),  dist = c("Normal", "h"),  ...)## S3 method for class 'formula'chaAllPairsNashimotoTest(  formula,  data,  subset,  na.action,  p.adjust.method = c(p.adjust.methods),  alternative = c("greater", "less"),  dist = c("Normal", "h"),  ...)

Arguments

Details

The modified procedure uses the property of a simple order,\theta_m' - \theta_m \le \theta_j - \theta_i \le \theta_l' - \theta_l\qquad (l \le i \le m~\mathrm{and}~ m' \le j \le l').The null hypothesis H_{ij}: \theta_i = \theta_j is tested againstthe alternative A_{ij}: \theta_i < \theta_j for any1 \le i < j \le k.

LetR_{ij} be the rank ofX_{ij},whereX_{ij} is jointly rankedfrom\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i,then the test statistics for all-pairs comparisonsand a balanced design is calculated as

\hat{T}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{R}_{m'} - \bar{R}_m \right)} {\sigma_a / \sqrt{n}},

withn = n_i; ~ N = \sum_i^k n_i ~~ (1 \le i \le k),\bar{R}_ithe mean rank for theith group,and the expected variance (without ties)\sigma_a^2 = N \left(N + 1 \right) / 12.

For the NPY'-test (dist = "h"), ifT_{ij} > h_{k-1,\alpha,\infty}.

For the unbalanced case with moderate imbalance the test statistic is

\hat{T}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{R}_{m'} - \bar{R}_m \right)} {\sigma_a \left(1/n_m + 1/n_{m'}\right)^{1/2}},

For the NPY'-test (dist="h") the null hypothesis is rejected in an unbalanced design,if\hat{T}_{ij} > h_{k,\alpha,\infty} / \sqrt{2}.In case of a NPY'-test, the function does not return p-values. Instead the critical h-valuesas given in the tables of Hayter (1990) for\alpha = 0.05 (one-sided)are looked up according to the number of groups (k-1) andthe degree of freedoms (v = \infty).

For the NPT'-test (dist = "Normal"), the null hypothesis is rejected, ifT_{ij} > \sqrt{2} t_{\alpha,\infty} = \sqrt{2} z_\alpha. Although Nashimoto and Wright (2005) originally did not use any p-adjustment,any method as available byp.adjust.methods canbe selected for the adjustment of p-values estimated fromthe standard normal distribution.

Value

Either a list of class"osrt" ifdist = "h" or a listof class"PMCMR" ifdist = "Normal".

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated statistic(s)

crit.value

critical values for\alpha = 0.05.

alternative

a character string describing the alternative hypothesis.

parameter

the parameter(s) of the test distribution.

dist

a string that denotes the test distribution.

There are print and summary methods available.

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

The function will give a warning for the unbalanced case and returns thecritical valueh_{k-1,\alpha,\infty} / \sqrt{2} if applicable.

References

Hayter, A. J.(1990) A One-Sided Studentised RangeTest for Testing Against a Simple Ordered Alternative,J Amer Stat Assoc85, 778–785.

Nashimoto, K., Wright, F.T. (2007)Nonparametric Multiple-Comparison Methods for Simply Ordered Medians.Comput Stat Data Anal51, 5068–5076.

See Also

Normal,chackoTest,NPMTest

Examples

## Example from Shirley (1977)## Reaction times of mice to stimuli to their tails.x <- c(2.4, 3, 3, 2.2, 2.2, 2.2, 2.2, 2.8, 2, 3, 2.8, 2.2, 3.8, 9.4, 8.4, 3, 3.2, 4.4, 3.2, 7.4, 9.8, 3.2, 5.8, 7.8, 2.6, 2.2, 6.2, 9.4, 7.8, 3.4, 7, 9.8, 9.4, 8.8, 8.8, 3.4, 9, 8.4, 2.4, 7.8)g <- gl(4, 10)## Shirley's test## one-sided test using look-up tableshirleyWilliamsTest(x ~ g, alternative = "greater")## Chacko's global hypothesis test for 'greater'chackoTest(x , g)## post-hoc test, default is standard normal distribution (NPT'-test)summary(chaAllPairsNashimotoTest(x, g, p.adjust.method = "none"))## same but h-distribution (NPY'-test)chaAllPairsNashimotoTest(x, g, dist = "h")## NPM-testNPMTest(x, g)## Hayter-Stone testhayterStoneTest(x, g)## all-pairs comparisonshsAllPairsTest(x, g)

Testing against Ordered Alternatives (Chacko's Test)

Description

Performs Chacko's test for testing against ordered alternatives.

Usage

chackoTest(x, ...)## Default S3 method:chackoTest(x, g, alternative = c("greater", "less"), ...)## S3 method for class 'formula'chackoTest(formula, data, subset, na.action, alternative = alternative, ...)

Arguments

Details

The null hypothesis, H_0: \theta_1 = \theta_2 = \ldots = \theta_kis tested against a simple order hypothesis,H_\mathrm{A}: \theta_1 \le \theta_2 \le \ldots \le\theta_k,~\theta_1 < \theta_k.

LetR_{ij} be the rank ofX_{ij},whereX_{ij} is jointly rankedfrom\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i,then the test statistic is calculated as

H = \frac{1}{\sigma_R^2} \sum_{i=1}^k n_i \left(\bar{R^*}_i - \bar{R}\right),

where\bar{R^*}_i is the isotonic mean of thei-th groupand\sigma_R^2 = N \left(N + 1\right) / 12 the expected variance (without ties).H_0 is rejected, ifH > \chi^2_{v,\alpha} withv = k -1 degree of freedom. The p-values are estimatedfrom the chi-square distribution.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Source

The source code for the application of the pool adjacent violatorstheorem to calculate the isotonic meanswas taken from the file"pava.f", which is included in thepackageIso:

Rolf Turner (2015). Iso: Functions to Perform Isotonic Regression.R package version 0.0-17.https://CRAN.R-project.org/package=Iso.

The file"pava.f" is a Ratfor modification of Algorithm AS 206.1:

Bril, G., Dykstra, R., Pillers, C., Robertson, T. (1984)Statistical Algorithms: Algorithm AS 206: IsotonicRegression in Two Independent Variables,Appl Statist34, 352–357.

The Algorith AS 206 is available from StatLibhttps://lib.stat.cmu.edu/apstat/. The Royal Statistical Societyholds the copyright to these routines,but has given its permission for their distribution provided thatno fee is charged.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

The function does neither check nor correct for ties.

References

Chacko, V. J. (1963) Testing homogeneity against ordered alternatives,Ann Math Statist34, 945–956.

See Also

kruskalTest andshirleyWilliamsTestof the packagePMCMRplus,kruskal.test of the librarystats.

Examples

## Example from Sachs (1997, p. 402)x <- c(106, 114, 116, 127, 145,       110, 125, 143, 148, 151,       136, 139, 149, 160, 174)g <- gl(3,5)levels(g) <- c("A", "B", "C")## Chacko's testchackoTest(x, g)## Cuzick's testcuzickTest(x, g)## Johnson-Mehrotra testjohnsonTest(x, g)## Jonckheere-Terpstra testjonckheereTest(x, g)## Le's testleTest(x, g)## Spearman type testspearmanTest(x, g)## Murakami's BWS trend testbwsTrendTest(x, g)## Fligner-Wolfe testflignerWolfeTest(x, g)## Shan-Young-Kang testshanTest(x, g)

Chen and Jan Many-to-One Comparisons Test

Description

Performs Chen and Jan nonparametric test for contrasting increasing(decreasing) dose levels of a treatment in a randomized block design.

Usage

chenJanTest(y, ...)## Default S3 method:chenJanTest(  y,  groups,  blocks,  alternative = c("greater", "less"),  p.adjust.method = c("single-step", "SD1", p.adjust.methods),  ...)

Arguments

Details

Chen's test is a non-parametric step-down trend test fortesting several treatment levels with a zero control. Letthere bek groups including the control and letthe zero dose level be indicated withi = 0 and the highestdose level withi = m, then the followingm = k - 1 hypotheses are tested:

\begin{array}{ll}\mathrm{H}_{m}: \theta_0 = \theta_1 = \ldots = \theta_m, & \mathrm{A}_{m} = \theta_0 \le \theta_1 \le \ldots \theta_m, \theta_0 < \theta_m \\\mathrm{H}_{m-1}: \theta_0 = \theta_1 = \ldots = \theta_{m-1}, & \mathrm{A}_{m-1} = \theta_0 \le \theta_1 \le \ldots \theta_{m-1}, \theta_0 < \theta_{m-1} \\\vdots & \vdots \\\mathrm{H}_{1}: \theta_0 = \theta_1, & \mathrm{A}_{1} = \theta_0 < \theta_1\\\end{array}

LetY_{ij1}, Y_{ij2}, \ldots, Y_{ijn_{ij}}(i = 1, 2, \dots, b, j = 0, 1, \ldots, k ~ \mathrm{and} ~ n_{ij} \geq 1) bea i.i.d. random variable of at least ordinal scale. Further,the zero dosecontrol is indicated withj = 0.

The Mann-Whittney statistic is

T_{ij} = \sum_{u=0}^{j-1} \sum_{s=1}^{n_{ij}}\sum_{r=1}^{n_{iu}} I(Y_{ijs} - Y_{iur}),\qquad i = 1, 2, \ldots, b, ~ j = 1, 2, \ldots, k,

where where the indicator function returnsI(a) = 1, ~ \mathrm{if}~ a > 0, 0.5 ~ \mathrm{if} a = 0otherwise0.

Let

N_{ij} = \sum_{s=0}^j n_{is} \qquad i = 1, 2, \ldots, b, ~ j = 1, 2, \ldots, k,

and

T_j = \sum_{i=1}^b T_{ij} \qquad j = 1, 2, \ldots, k.

The mean and variance ofT_j are

\mu(T_j) = \sum_{i=1}^b n_{ij} ~ N_{ij-1} / 2 \qquad \mathrm{and}

\sigma(T_j) = \sum_{i=1}^b n_{ij} ~ N_{ij-1} \left[ \left(N_{ij} + 1\right) - \sum_{u=1}^{g_i} \left(t_u^3 - t_u \right) / \left\{N_{ij} \left(N_{ij} - 1\right) \right\} \right]/ 2,

withg_i the number of ties in theith block andt_u the size of the tied groupu.

The test statisticT_j^* is asymptotically multivariate normaldistributed.

T_j^* = \frac{T_j - \mu(T_j)}{\sigma(T_j)}

Ifp.adjust.method = "single-step" than the p-valuesare calculated with the probability function of the multivariatenormal distribution with\Sigma = I_k. Otherwisethe standard normal distribution is used to calculatep-values and any method as availablebyp.adjust or by the step-down procedure as proposedby Chen (1999), ifp.adjust.method = "SD1" can be usedto account for\alpha-error inflation.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Chen, Y.I., Jan, S.L., 2002. Nonparametric Identification ofthe Minimum Effective Dose for Randomized Block Designs.Commun Stat-Simul Comput31, 301–312.

See Also

Normalpmvnorm

Examples

## Example from Chen and Jan (2002, p. 306)## MED is at dose level 2 (0.5 ppm SO2)y <- c(0.2, 6.2, 0.3, 0.3, 4.9, 1.8, 3.9, 2, 0.3, 2.5, 5.4, 2.3, 12.7,-0.2, 2.1, 6, 1.8, 3.9, 1.1, 3.8, 2.5, 1.3, -0.8, 13.1, 1.1,12.8, 18.2, 3.4, 13.5, 4.4, 6.1, 2.8, 4, 10.6, 9, 4.2, 6.7, 35,9, 12.9, 2, 7.1, 1.5, 10.6)groups <- gl(4,11, labels = c("0", "0.25", "0.5", "1.0"))blocks <- structure(rep(1:11, 4), class = "factor",levels = c("1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11"))summary(chenJanTest(y, groups, blocks, alternative = "greater"))summary(chenJanTest(y, groups, blocks, alternative = "greater", p.adjust = "SD1"))

Chen's Many-to-One Comparisons Test

Description

Performs Chen's nonparametric test for contrasting increasing(decreasing) dose levels of a treatment.

Usage

chenTest(x, ...)## Default S3 method:chenTest(  x,  g,  alternative = c("greater", "less"),  p.adjust.method = c("SD1", p.adjust.methods),  ...)## S3 method for class 'formula'chenTest(  formula,  data,  subset,  na.action,  alternative = c("greater", "less"),  p.adjust.method = c("SD1", p.adjust.methods),  ...)

Arguments

Details

Chen's test is a non-parametric step-down trend test fortesting several treatment levels with a zero control.LetX_{0j} denote a variable with thej-threalization of the control group (1 \le j \le n_0)andX_{ij} thej-the realizationin thei-th treatment group (1 \le i \le k).The variables are i.i.d. of a least ordinal scale withF(x) = F(x_0) = F(x_i), ~ (1 \le i \le k).A total ofm = k hypotheses can be tested:

\begin{array}{ll}\mathrm{H}_{m}: \theta_0 = \theta_1 = \ldots = \theta_m, & \mathrm{A}_{m} = \theta_0 \le \theta_1 \le \ldots \theta_m, \theta_0 < \theta_m \\\mathrm{H}_{m-1}: \theta_0 = \theta_1 = \ldots = \theta_{m-1}, & \mathrm{A}_{m-1} = \theta_0 \le \theta_1 \le \ldots \theta_{m-1}, \theta_0 < \theta_{m-1} \\\vdots & \vdots \\\mathrm{H}_{1}: \theta_0 = \theta_1, & \mathrm{A}_{1} = \theta_0 < \theta_1\\\end{array}

The statisticsT_i are based on a Wilcoxon-type ranking:

T_i = \sum_{j=0}^{i=1} \sum_{u=1}^{n_i} \sum_{v=1}^{n_j} I(x_{iu} - x_{jv}), \qquad (1 \leq i \leq k),

where the indicator function returnsI(a) = 1, ~ \mathrm{if}~ a > 0, 0.5 ~ \mathrm{if} a = 0otherwise0.

The expectedith mean is

\mu(T_i) = n_i N_{i-1} / 2,

withN_j = \sum_{j =0}^i n_j and theith variance:

\sigma^2(T_i) = n_i N_{i-1} / 12 ~ \left\{N_i + 1 -\sum_{j=1}^g t_j \left(t_j^2 - 1 \right) /\left[N_i \left( N_i - 1 \right)\right]\right\}.

The test statisticT_i^* is asymptotically standard normal

T_i^* = \frac{T_i - \mu(T_i)} {\sqrt{\sigma^2(T_i)}}, \qquad (1 \leq i \leq k).

The p-values are calculated from the standard normal distribution.The p-values can be adjusted with any method as availablebyp.adjust or by the step-down procedure as proposedby Chen (1999), ifp.adjust.method = "SD1".

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Chen, Y.-I., 1999, Nonparametric Identification of theMinimum Effective Dose.Biometrics55, 1236–1240.doi:10.1111/j.0006-341X.1999.01236.x

See Also

wilcox.test,Normal

Examples

## Chen, 1999, p. 1237,## Minimum effective dose (MED)## is at 2nd dose leveldf <- data.frame(x = c(23, 22, 14,27, 23, 21,28, 37, 35,41, 37, 43,28, 21, 30,16, 19, 13),g = gl(6, 3))levels(df$g) <- 0:5ans <- chenTest(x ~ g, data = df, alternative = "greater",                p.adjust.method = "SD1")summary(ans)

Cochran Test

Description

Performs Cochran's test for testing an outlying (or inlying)variance.

Usage

cochranTest(x, ...)## Default S3 method:cochranTest(x, g, alternative = c("greater", "less"), ...)## S3 method for class 'formula'cochranTest(  formula,  data,  subset,  na.action,  alternative = c("greater", "less"),  ...)

Arguments

Details

For normally distributed data the null hypothesis,H_0: \sigma_1^2 = \sigma_2^2 = \ldots = \sigma_k^2is tested against the alternative (greater)H_{\mathrm{A}}: \sigma_p > \sigma_i ~~ (i \le k, i \ne p) withat least one inequality being strict.

The p-value is computed with the functionpcochran.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Cochran, W.G. (1941) The distribution of the largest of a set of estimatedvariances as a fraction of their total.Ann. Eugen.11, 47–52.

Wilrich, P.-T. (2011) Critical values of Mandel's h and k,Grubbs and the Cochran test statistic.Adv. Stat. Anal..doi:10.1007/s10182-011-0185-y.

See Also

bartlett.test,fligner.test.

Examples

data(Pentosan)cochranTest(value ~ lab, data = Pentosan, subset = (material == "A"))

Testing against Ordered Alternatives (Cuzick's Test)

Description

Performs Cuzick's test for testing against ordered alternatives.

Usage

cuzickTest(x, ...)## Default S3 method:cuzickTest(  x,  g,  alternative = c("two.sided", "greater", "less"),  scores = NULL,  continuity = FALSE,  ...)## S3 method for class 'formula'cuzickTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  scores = NULL,  continuity = FALSE,  ...)

Arguments

Details

The null hypothesis, H_0: \theta_1 = \theta_2 = \ldots = \theta_kis tested against a simple order hypothesis,H_\mathrm{A}: \theta_1 \le \theta_2 \le \ldots \le\theta_k,~\theta_1 < \theta_k.

The p-values are estimated from the standard normal distribution.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Cuzick, J. (1995) A Wilcoxon-type test for trend,Statistics in Medicine4, 87–90.

See Also

kruskalTest andshirleyWilliamsTestof the packagePMCMRplus,kruskal.test of the librarystats.

Examples

## Example from Sachs (1997, p. 402)x <- c(106, 114, 116, 127, 145,       110, 125, 143, 148, 151,       136, 139, 149, 160, 174)g <- gl(3,5)levels(g) <- c("A", "B", "C")## Chacko's testchackoTest(x, g)## Cuzick's testcuzickTest(x, g)## Johnson-Mehrotra testjohnsonTest(x, g)## Jonckheere-Terpstra testjonckheereTest(x, g)## Le's testleTest(x, g)## Spearman type testspearmanTest(x, g)## Murakami's BWS trend testbwsTrendTest(x, g)## Fligner-Wolfe testflignerWolfeTest(x, g)## Shan-Young-Kang testshanTest(x, g)

Grubbs Double Outlier Test

Description

Performs Grubbs double outlier test.

Usage

doubleGrubbsTest(x, alternative = c("two.sided", "greater", "less"), m = 10000)

Arguments

Details

LetX denote an identically and independently distributed continuousvariate with realizationsx_i ~~ (1 \le i \le k).Further, let the increasingly ordered realizationsdenotex_{(1)} \le x_{(2)} \le \ldots \le x_{(n)}. Thenthe following model for testing two maximum outliers can be proposed:

x_{(i)} = \left\{ \begin{array}{lcl} \mu + \epsilon_{(i)}, & \qquad & i = 1, \ldots, n - 2 \\ \mu + \Delta + \epsilon_{(j)} & \qquad & j = n-1, n \\ \end{array} \right.

with\epsilon \approx N(0,\sigma). The null hypothesis,H_0: \Delta = 0 is tested against the alternative,H_{\mathrm{A}}: \Delta > 0.

For testing two minimum outliers, the model can be proposedas

x_{(i)} = \left\{ \begin{array}{lcl} \mu + \Delta + \epsilon_{(j)} & \qquad & j = 1, 2 \\ \mu + \epsilon_{(i)}, & \qquad & i = 3, \ldots, n \\ \end{array} \right.

The null hypothesis is tested against the alternative,H_{\mathrm{A}}: \Delta < 0.

The p-value is computed with the functionpdgrubbs.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Grubbs, F. E. (1950) Sample criteria for testing outlying observations.Ann. Math. Stat.21, 27–58.

Wilrich, P.-T. (2011) Critical values of Mandel's h and k,Grubbs and the Cochran test statistic.Adv. Stat. Anal..doi:10.1007/s10182-011-0185-y.

Examples

data(Pentosan)dat <- subset(Pentosan, subset = (material == "A"))labMeans <- tapply(dat$value, dat$lab, mean)doubleGrubbsTest(x = labMeans, alternative = "less")

Multiple Comparisons of Mean Rank Sums

Description

Performs the all-pairs comparison test for different factorlevels according to Dwass, Steel, Critchlow and Fligner.

Usage

dscfAllPairsTest(x, ...)## Default S3 method:dscfAllPairsTest(x, g, ...)## S3 method for class 'formula'dscfAllPairsTest(formula, data, subset, na.action, ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals the DSCFall-pairs comparison test can be used. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: F_i(x) = F_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: F_i(x) \ne F_j(x), ~~ i \ne j.As opposed to the all-pairs comparison procedures that dependon Kruskal ranks, the DSCF test is basically an extension ofthe U-test as re-ranking is conducted for each pairwise test.

The p-values are estimated from the studentized range distriburtion.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Douglas, C. E., Fligner, A. M. (1991) On distribution-free multiplecomparisons in the one-way analysis of variance,Communications inStatistics - Theory and Methods20, 127–139.

Dwass, M. (1960) Some k-sample rank-order tests. InContributions toProbability and Statistics, Edited by: I. Olkin,Stanford: Stanford University Press.

Steel, R. G. D. (1960) A rank sum test for comparing all pairs oftreatments,Technometrics2, 197–207

See Also

Tukey,pairwise.wilcox.test

Duncan's Multiple Range Test

Description

Performs Duncan's all-pairs comparisons test for normally distributeddata with equal group variances.

Usage

duncanTest(x, ...)## Default S3 method:duncanTest(x, g, ...)## S3 method for class 'formula'duncanTest(formula, data, subset, na.action, ...)## S3 method for class 'aov'duncanTest(x, ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith normally distributed residuals and equal variancesDuncan's multiple range test can be performed.LetX_{ij} denote a continuous random variablewith thej-the realization (1 \le j \le n_i)in thei-th group (1 \le i \le k). Furthermore, the totalsample size isN = \sum_{i=1}^k n_i. A total ofm = k(k-1)/2hypotheses can be tested: The null hypothesis isH_{ij}: \mu_i = \mu_j ~~ (i \ne j) is tested against the alternativeA_{ij}: \mu_i \ne \mu_j (two-tailed). Duncan's all-pairs teststatistics are given by

t_{(i)(j)} \frac{\bar{X}_{(i)} - \bar{X}_{(j)}} {s_{\mathrm{in}} \left(r\right)^{1/2}}, ~~ (i < j)

withs^2_{\mathrm{in}} the within-group ANOVA variance,r = k / \sum_{i=1}^k n_i and\bar{X}_{(i)} the increasinglyordered means1 \le i \le k.The null hypothesis is rejected if

\mathrm{Pr} \left\{ |t_{(i)(j)}| \ge q_{vm'\alpha'} | \mathrm{H} \right\}_{(i)(j)} = \alpha' = \min \left\{1,~ 1 - (1 - \alpha)^{(1 / (m' - 1))} \right\},

withv = N - k degree of freedom, the rangem' = 1 + |i - j| and\alpha' the Bonferroni adjustedalpha-error. The p-values are computedfrom theTukey distribution.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Duncan, D. B. (1955) Multiple range and multiple F tests,Biometrics11, 1–42.

See Also

Tukey,TukeyHSDtukeyTest

Examples

fit <- aov(weight ~ feed, chickwts)shapiro.test(residuals(fit))bartlett.test(weight ~ feed, chickwts)anova(fit)## also works with fitted objects of class aovres <- duncanTest(fit)summary(res)summaryGroup(res)

Dunnett's T3 Test

Description

Performs Dunnett's all-pairs comparison test for normally distributeddata with unequal variances.

Usage

dunnettT3Test(x, ...)## Default S3 method:dunnettT3Test(x, g, ...)## S3 method for class 'formula'dunnettT3Test(formula, data, subset, na.action, ...)## S3 method for class 'aov'dunnettT3Test(x, ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith normally distributed residuals but unequal groups variancesthe T3 test of Dunnett can be performed.LetX_{ij} denote a continuous random variablewith thej-the realization (1 \le j \le n_i)in thei-th group (1 \le i \le k). Furthermore, the totalsample size isN = \sum_{i=1}^k n_i. A total ofm = k(k-1)/2hypotheses can be tested: The null hypothesis isH_{ij}: \mu_i = \mu_j ~~ (i \ne j) is tested against the alternativeA_{ij}: \mu_i \ne \mu_j (two-tailed). Dunnett T3 all-pairstest statistics are given by

t_{ij} \frac{\bar{X}_i - \bar{X_j}} {\left( s^2_j / n_j + s^2_i / n_i \right)^{1/2}}, ~~ (i \ne j)

withs^2_i the variance of thei-th group.The null hypothesis is rejected (two-tailed) if

\mathrm{Pr} \left\{ |t_{ij}| \ge T_{v_{ij}\rho_{ij}\alpha'/2} | \mathrm{H} \right\}_{ij} = \alpha,

with Welch's approximate solution for calculating the degree of freedom.

v_{ij} = \frac{\left( s^2_i / n_i + s^2_j / n_j \right)^2} {s^4_i / n^2_i \left(n_i - 1\right) + s^4_j / n^2_j \left(n_j - 1\right)}.

Thep-values are computed from thestudentized maximum modulus distributionthat is the equivalent of the multivariate t distributionwith\rho_{ii} = 1, ~ \rho_{ij} = 0 ~ (i \ne j).The functionpmvt is used tocalculate thep-values.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

C. W. Dunnett (1980) Pair wise multiple comparisons in the unequalvariance case,Journal of the American StatisticalAssociation75, 796–800.

See Also

pmvt

Examples

fit <- aov(weight ~ feed, chickwts)shapiro.test(residuals(fit))bartlett.test(weight ~ feed, chickwts)anova(fit)## also works with fitted objects of class aovres <- dunnettT3Test(fit)summary(res)summaryGroup(res)

Dunnett's Many-to-One Comparisons Test

Description

Performs Dunnett's multiple comparisons test with one control.

Usage

dunnettTest(x, ...)## Default S3 method:dunnettTest(x, g, alternative = c("two.sided", "greater", "less"), ...)## S3 method for class 'formula'dunnettTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  ...)## S3 method for class 'aov'dunnettTest(x, alternative = c("two.sided", "greater", "less"), ...)

Arguments

Details

For many-to-one comparisons in an one-factorial layoutwith normally distributed residuals Dunnett's testcan be used.LetX_{0j} denote a continuous random variablewith thej-the realization of the control group(1 \le j \le n_0) andX_{ij} thej-the realizationin thei-th treatment group (1 \le i \le k).Furthermore, the total sample size isN = n_0 + \sum_{i=1}^k n_i.A total ofm = k hypotheses can be tested: The null hypothesis isH_{i}: \mu_i = \mu_0 is tested against the alternativeA_{i}: \mu_i \ne \mu_0 (two-tailed). Dunnett's teststatistics are given by

t_{i} \frac{\bar{X}_i - \bar{X_0}} {s_{\mathrm{in}} \left(1/n_0 + 1/n_i\right)^{1/2}}, ~~ (1 \le i \le k)

withs^2_{\mathrm{in}} the within-group ANOVA variance.The null hypothesis is rejected if|t_{ij}| > |T_{kv\rho\alpha}| (two-tailed),withv = N - k degree of freedom andrho the correlation:

\rho_{ij} = \sqrt{\frac{n_i n_j} {\left(n_i + n_0\right) \left(n_j+ n_0\right)}} ~~ (i \ne j).

The p-values are computed with the functionpDunnettthat is a wrapper to the the multivariate-t distribution as implemented in the functionpmvt.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Dunnett, C. W. (1955) A multiple comparison procedure for comparing severaltreatments with a control.Journal of the American Statistical Association50, 1096–1121.

OECD (ed. 2006)Current approaches in the statistical analysisof ecotoxicity data: A guidance to application - Annexes. OECD Serieson testing and assessment, No. 54.

See Also

pmvtpDunnett

Examples

fit <- aov(Y ~ DOSE, data = trout)shapiro.test(residuals(fit))bartlett.test(Y ~ DOSE, data = trout)## works with fitted object of class aovsummary(dunnettTest(fit, alternative = "less"))

All-Pairs Comparisons Test for Balanced Incomplete Block Designs

Description

Performs Conover-Iman all-pairs comparison test for a balanced incompleteblock design (BIBD).

Usage

durbinAllPairsTest(y, ...)## Default S3 method:durbinAllPairsTest(y, groups, blocks, p.adjust.method = p.adjust.methods, ...)

Arguments

Details

For all-pairs comparisons in a balanced incomplete block designthe proposed test of Conover and Imam can be applied.A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: \theta_i = \theta_j is tested in the two-tailed testagainst the alternativeA_{ij}: \theta_i \ne \theta_j, ~~ i \ne j.

The p-values are computed from the t distribution. If no p-value adjustmentis performed (p.adjust.method = "none"),than a simple protected test is recommended, i.e.the all-pairs comparisons should only be applied after a significantdurbinTest. However, any method as implemented inp.adjust.methods can be selected by the user.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Conover, W. J., Iman, R. L. (1979)On multiple-comparisonsprocedures, Tech. Rep. LA-7677-MS, Los Alamos Scientific Laboratory.

Conover, W. J. (1999)Practical nonparametric Statistics,3rd. Edition, Wiley.

See Also

durbinTest

Examples

## Example for an incomplete block design:## Data from Conover (1999, p. 391).y <- matrix(c(2,NA,NA,NA,3, NA,  3,  3,  3, NA, NA, NA,  3, NA, NA,  1,  2, NA, NA, NA,  1,  1, NA,  1,  1,NA, NA, NA, NA,  2, NA,  2,  1, NA, NA, NA, NA, 3, NA,  2,  1, NA, NA, NA, NA,  3, NA,  2,  2),ncol=7, nrow=7, byrow=FALSE, dimnames=list(1:7, LETTERS[1:7]))durbinAllPairsTest(y)

Durbin Test

Description

Performs Durbin's tests whether k groups(or treatments) in a two-way balanced incomplete block design (BIBD)have identical effects.

Usage

durbinTest(y, ...)## Default S3 method:durbinTest(y, groups, blocks, ...)

Arguments

Details

For testing a two factorial layout of a balanced incompleteblock design whether thek groups have identical effects,the Durbin test can be performed. The null hypothesis,H_0: \theta_i = \theta_j ~ (1 \le i < j \le k),is tested against the alternative that at leastone\theta_i \ne \theta_j.

The p-values are computed from the chi-square distribution.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

The function does not test, whether it is a true BIBD.This function does not test for ties.

References

Conover,W. J. (1999)Practical nonparametric Statistics,3rd. Edition, Wiley.

Heckert, N. A., Filliben, J. J. (2003)NIST Handbook 148:Dataplot Reference Manual, Volume 2:Let Subcommands and Library Functions.National Institute of Standards and Technology Handbook Series, June 2003.

Examples

## Example for an incomplete block design:## Data from Conover (1999, p. 391).y <- matrix(c(2,NA,NA,NA,3, NA,  3,  3,  3, NA, NA, NA,  3, NA, NA,  1,  2, NA, NA, NA,  1,  1, NA,  1,  1,NA, NA, NA, NA,  2, NA,  2,  1, NA, NA, NA, NA, 3, NA,  2,  1, NA, NA, NA, NA,  3, NA,  2,  2), ncol=7, nrow=7, byrow=FALSE,dimnames=list(1:7, LETTERS[1:7]))durbinTest(y)

Testing Several Treatments With One Control

Description

Performs Fligner-Wolfe non-parametric test forsimultaneous testing of several locations of treatment groupsagainst the location of the control group.

Usage

flignerWolfeTest(x, ...)## Default S3 method:flignerWolfeTest(  x,  g,  alternative = c("greater", "less"),  dist = c("Wilcoxon", "Normal"),  ...)## S3 method for class 'formula'flignerWolfeTest(  formula,  data,  subset,  na.action,  alternative = c("greater", "less"),  dist = c("Wilcoxon", "Normal"),  ...)

Arguments

Details

For a one-factorial layout with non-normally distributed residualsthe Fligner-Wolfe test can be used.

Let there bek-1-treatment groups and one control group, thenthe null hypothesis, H_0: \theta_i - \theta_c = 0 ~ (1 \le i \le k-1)is tested against the alternative (greater),A_1: \theta_i - \theta_c > 0 ~ (1 \le i \le k-1),with at least one inequality being strict.

Letn_c denote the sample size of the control group,N^t = \sum_{i=1}^{k-1} n_i the sum of all treatmentsample sizes andN = N^t + n_c. The test statistic without takenties into account is

W = \sum_{j=1}^{k-1} \sum_{i=1}^{n_i} r_{ij} - \frac{N^t \left(N^t + 1 \right) }{2}

withr_{ij} the rank of variablex_{ij}.The null hypothesis is rejected,ifW > W_{\alpha,m,n} withm = N^t andn = n_c.

In the presence of ties, the statistic is

\hat{z} = \frac{W - n_c N^t / 2}{s_W},

where

s_W = \frac{n_c N^t}{12 N \left(N - 1 \right)} \sum_{j=1}^g t_j \left(t_j^2 - 1\right),

withg the number of tied groups andt_jthe number of tied values in thejth group. The null hypothesisis rejected, if\hat{z} > z_\alpha (as cited in EPA 2006).

Ifdist = Wilcoxon, then thep-values are estimated from theWilcoxondistribution, else theNormal distribution is used. The latter can be used,if ties are present.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

EPA (2006)Data Quality Assessment:Statistical Methods for Practitioners(Guideline No. EPA QA/G-9S), US-EPA.

Fligner, M.A., Wolfe, D.A. (1982)Distribution-free tests for comparing severaltreatments with a control.Stat Neerl36,119–127.

See Also

kruskalTest andshirleyWilliamsTestof the packagePMCMRplus,kruskal.test of the librarystats.

Examples

## Example from Sachs (1997, p. 402)x <- c(106, 114, 116, 127, 145,       110, 125, 143, 148, 151,       136, 139, 149, 160, 174)g <- gl(3,5)levels(g) <- c("A", "B", "C")## Chacko's testchackoTest(x, g)## Cuzick's testcuzickTest(x, g)## Johnson-Mehrotra testjohnsonTest(x, g)## Jonckheere-Terpstra testjonckheereTest(x, g)## Le's testleTest(x, g)## Spearman type testspearmanTest(x, g)## Murakami's BWS trend testbwsTrendTest(x, g)## Fligner-Wolfe testflignerWolfeTest(x, g)## Shan-Young-Kang testshanTest(x, g)

Conover's All-Pairs Comparisons Test for Unreplicated Blocked Data

Description

Performs Conover's all-pairs comparisons tests of Friedman-type ranked data.

Usage

frdAllPairsConoverTest(y, ...)## Default S3 method:frdAllPairsConoverTest(  y,  groups,  blocks,  p.adjust.method = c("single-step", p.adjust.methods),  ...)

Arguments

Details

For all-pairs comparisons in a two factorial unreplicatedcomplete block designwith non-normally distributed residuals, Conover's test can beperformed on Friedman-type ranked data.

A total ofm = k ( k -1 )/2 hypotheses can be tested.The null hypothesis, H_{ij}: \theta_i = \theta_j, is testedin the two-tailed case against the alternative,A_{ij}: \theta_i \ne \theta_j, ~~ i \ne j.

Ifp.adjust.method == "single-step" the p-values are computedfrom the studentized range distribution. Otherwise,the p-values are computed from the t-distribution usingany of the p-adjustment methods as included inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Conover, W. J., Iman, R. L. (1979)On multiple-comparisonsprocedures, Tech. Rep. LA-7677-MS, Los Alamos Scientific Laboratory.

Conover, W. J. (1999)Practical nonparametric Statistics,3rd. Edition, Wiley.

See Also

friedmanTest,friedman.test,frdAllPairsExactTest,frdAllPairsMillerTest,frdAllPairsNemenyiTest,frdAllPairsSiegelTest

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) print(y) friedmanTest(y) ## Eisinga et al. 2017 frdAllPairsExactTest(y=y, p.adjust = "bonferroni") ## Conover's test frdAllPairsConoverTest(y=y, p.adjust = "bonferroni") ## Nemenyi's test frdAllPairsNemenyiTest(y=y) ## Miller et al. frdAllPairsMillerTest(y=y) ## Siegel-Castellan frdAllPairsSiegelTest(y=y, p.adjust = "bonferroni") ## Irrelevant of group order? x <- as.vector(y) g <- rep(colnames(y), each = length(x)/length(colnames(y))) b <- rep(rownames(y), times = length(x)/length(rownames(y))) xDF <- data.frame(x, g, b) # grouped by colnames frdAllPairsNemenyiTest(xDF$x, groups = xDF$g, blocks = xDF$b) o <- order(xDF$b) # order per block increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) o <- order(xDF$x) # order per value increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) ## formula method (only works for Nemenyi) frdAllPairsNemenyiTest(x ~ g | b, data = xDF)

Exact All-Pairs Comparisons Test for Unreplicated Blocked Data

Description

Performs exact all-pairs comparisons tests of Friedman-type ranked dataaccording to Eisinga et al. (2017).

Usage

frdAllPairsExactTest(y, ...)## Default S3 method:frdAllPairsExactTest(  y,  groups,  blocks,  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For all-pairs comparisons in a two factorial unreplicatedcomplete block designwith non-normally distributed residuals, an exact test can beperformed on Friedman-type ranked data.

A total ofm = k ( k -1 )/2 hypotheses can be tested.The null hypothesis, H_{ij}: \theta_i = \theta_j, is testedin the two-tailed case against the alternative,A_{ij}: \theta_i \ne \theta_j, ~~ i \ne j.

The exactp-valuesare computed using the code of"pexactfrsd.R"that was a supplement to the publication of Eisinga et al. (2017).Additionally, any of thep-adjustment methodsas included inp.adjust can be selected, forp-valueadjustment.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Source

The functionfrdAllPairsExactTest uses the codeof the filepexactfrsd.R that was a supplement to:

R. Eisinga, T. Heskes, B. Pelzer, M. Te Grotenhuis (2017),Exact p-values for Pairwise Comparison of Friedman Rank Sums,with Application to Comparing Classifiers,BMC Bioinformatics, 18:68.

References

Eisinga, R., Heskes, T., Pelzer, B., Te Grotenhuis, M. (2017)Exact p-values for Pairwise Comparison of Friedman Rank Sums,with Application to Comparing Classifiers,BMC Bioinformatics, 18:68.

See Also

friedmanTest,friedman.test,frdAllPairsConoverTest,frdAllPairsMillerTest,frdAllPairsNemenyiTest,frdAllPairsSiegelTest

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) print(y) friedmanTest(y) ## Eisinga et al. 2017 frdAllPairsExactTest(y=y, p.adjust = "bonferroni") ## Conover's test frdAllPairsConoverTest(y=y, p.adjust = "bonferroni") ## Nemenyi's test frdAllPairsNemenyiTest(y=y) ## Miller et al. frdAllPairsMillerTest(y=y) ## Siegel-Castellan frdAllPairsSiegelTest(y=y, p.adjust = "bonferroni") ## Irrelevant of group order? x <- as.vector(y) g <- rep(colnames(y), each = length(x)/length(colnames(y))) b <- rep(rownames(y), times = length(x)/length(rownames(y))) xDF <- data.frame(x, g, b) # grouped by colnames frdAllPairsNemenyiTest(xDF$x, groups = xDF$g, blocks = xDF$b) o <- order(xDF$b) # order per block increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) o <- order(xDF$x) # order per value increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) ## formula method (only works for Nemenyi) frdAllPairsNemenyiTest(x ~ g | b, data = xDF)

Millers's All-Pairs Comparisons Test for Unreplicated Blocked Data

Description

Performs Miller's all-pairs comparisons tests of Friedman-type ranked data.

Usage

frdAllPairsMillerTest(y, ...)## Default S3 method:frdAllPairsMillerTest(y, groups, blocks, ...)

Arguments

Details

For all-pairs comparisons in a two factorial unreplicatedcomplete block designwith non-normally distributed residuals, Miller's test can beperformed on Friedman-type ranked data.

A total ofm = k ( k -1 )/2 hypotheses can be tested.The null hypothesis, H_{ij}: \theta_i = \theta_j, is testedin the two-tailed case against the alternative,A_{ij}: \theta_i \ne \theta_j, ~~ i \ne j.

Thep-values are computed from the chi-square distribution.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Bortz J., Lienert, G. A., Boehnke, K. (1990)VerteilungsfreieMethoden in der Biostatistik. Berlin: Springer.

Miller Jr., R. G. (1996)Simultaneous statistical inference.New York: McGraw-Hill.

Wike, E. L. (2006),Data Analysis. A Statistical Primer forPsychology Students. New Brunswick: Aldine Transaction.

See Also

friedmanTest,friedman.test,frdAllPairsExactTest,frdAllPairsConoverTest,frdAllPairsNemenyiTest,frdAllPairsSiegelTest

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) print(y) friedmanTest(y) ## Eisinga et al. 2017 frdAllPairsExactTest(y=y, p.adjust = "bonferroni") ## Conover's test frdAllPairsConoverTest(y=y, p.adjust = "bonferroni") ## Nemenyi's test frdAllPairsNemenyiTest(y=y) ## Miller et al. frdAllPairsMillerTest(y=y) ## Siegel-Castellan frdAllPairsSiegelTest(y=y, p.adjust = "bonferroni") ## Irrelevant of group order? x <- as.vector(y) g <- rep(colnames(y), each = length(x)/length(colnames(y))) b <- rep(rownames(y), times = length(x)/length(rownames(y))) xDF <- data.frame(x, g, b) # grouped by colnames frdAllPairsNemenyiTest(xDF$x, groups = xDF$g, blocks = xDF$b) o <- order(xDF$b) # order per block increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) o <- order(xDF$x) # order per value increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) ## formula method (only works for Nemenyi) frdAllPairsNemenyiTest(x ~ g | b, data = xDF)

Nemenyi's All-Pairs Comparisons Test for Unreplicated Blocked Data

Description

Performs Nemenyi's all-pairs comparisons tests of Friedman-type ranked data.

Usage

frdAllPairsNemenyiTest(y, ...)## Default S3 method:frdAllPairsNemenyiTest(y, groups, blocks, ...)## S3 method for class 'formula'frdAllPairsNemenyiTest(formula, data, subset, na.action, ...)

Arguments

Details

For all-pairs comparisons in a two factorial unreplicatedcomplete block designwith non-normally distributed residuals, Nemenyi's test can beperformed on Friedman-type ranked data.

A total ofm = k ( k -1 )/2 hypotheses can be tested.The null hypothesis, H_{ij}: \theta_i = \theta_j, is testedin the two-tailed case against the alternative,A_{ij}: \theta_i \ne \theta_j, ~~ i \ne j.

Thep-values are computed from the studentized range distribution.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Demsar, J. (2006) Statistical comparisons of classifiers over multipledata sets,Journal of Machine Learning Research7, 1–30.

Miller Jr., R. G. (1996)Simultaneous statistical inference.New York: McGraw-Hill.

Nemenyi, P. (1963),Distribution-free Multiple Comparisons.Ph.D. thesis, Princeton University.

Sachs, L. (1997)Angewandte Statistik. Berlin: Springer.

See Also

friedmanTest,friedman.test,frdAllPairsExactTest,frdAllPairsConoverTest,frdAllPairsMillerTest,frdAllPairsSiegelTest

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) print(y) friedmanTest(y) ## Eisinga et al. 2017 frdAllPairsExactTest(y=y, p.adjust = "bonferroni") ## Conover's test frdAllPairsConoverTest(y=y, p.adjust = "bonferroni") ## Nemenyi's test frdAllPairsNemenyiTest(y=y) ## Miller et al. frdAllPairsMillerTest(y=y) ## Siegel-Castellan frdAllPairsSiegelTest(y=y, p.adjust = "bonferroni") ## Irrelevant of group order? x <- as.vector(y) g <- rep(colnames(y), each = length(x)/length(colnames(y))) b <- rep(rownames(y), times = length(x)/length(rownames(y))) xDF <- data.frame(x, g, b) # grouped by colnames frdAllPairsNemenyiTest(xDF$x, groups = xDF$g, blocks = xDF$b) o <- order(xDF$b) # order per block increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) o <- order(xDF$x) # order per value increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) ## formula method (only works for Nemenyi) frdAllPairsNemenyiTest(x ~ g | b, data = xDF)

Siegel and Castellan's All-Pairs Comparisons Test forUnreplicated Blocked Data

Description

Performs Siegel and Castellan's all-pairs comparisons testsof Friedman-type ranked data.

Usage

frdAllPairsSiegelTest(y, ...)## Default S3 method:frdAllPairsSiegelTest(  y,  groups,  blocks,  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For all-pairs comparisons in a two factorial unreplicatedcomplete block designwith non-normally distributed residuals, Siegel and Castellan's test can beperformed on Friedman-type ranked data.

A total ofm = k ( k -1 )/2 hypotheses can be tested.The null hypothesis, H_{ij}: \theta_i = \theta_j, is testedin the two-tailed case against the alternative,A_{ij}: \theta_i \ne \theta_j, ~~ i \ne j.

Thep-values are computed from the standard normal distribution.Any method as implemented inp.adjust can be used forp-value adjustment.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Siegel, S., Castellan Jr., N. J. (1988)NonparametricStatistics for the Behavioral Sciences. 2nd ed. New York: McGraw-Hill.

See Also

friedmanTest,friedman.test,frdAllPairsExactTest,frdAllPairsConoverTest,frdAllPairsNemenyiTest,frdAllPairsMillerTest

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) print(y) friedmanTest(y) ## Eisinga et al. 2017 frdAllPairsExactTest(y=y, p.adjust = "bonferroni") ## Conover's test frdAllPairsConoverTest(y=y, p.adjust = "bonferroni") ## Nemenyi's test frdAllPairsNemenyiTest(y=y) ## Miller et al. frdAllPairsMillerTest(y=y) ## Siegel-Castellan frdAllPairsSiegelTest(y=y, p.adjust = "bonferroni") ## Irrelevant of group order? x <- as.vector(y) g <- rep(colnames(y), each = length(x)/length(colnames(y))) b <- rep(rownames(y), times = length(x)/length(rownames(y))) xDF <- data.frame(x, g, b) # grouped by colnames frdAllPairsNemenyiTest(xDF$x, groups = xDF$g, blocks = xDF$b) o <- order(xDF$b) # order per block increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) o <- order(xDF$x) # order per value increasingly frdAllPairsNemenyiTest(xDF$x[o], groups = xDF$g[o], blocks = xDF$b[o]) ## formula method (only works for Nemenyi) frdAllPairsNemenyiTest(x ~ g | b, data = xDF)

House Test

Description

Performs House nonparametric equivalent of William's testfor contrasting increasing dose levels of a treatment ina complete randomized block design.

Usage

frdHouseTest(y, ...)## Default S3 method:frdHouseTest(y, groups, blocks, alternative = c("greater", "less"), ...)

Arguments

Details

House test is a non-parametric step-down trend test for testing several treatment levelswith a zero control. Let there bek groups including the control and letthe zero dose level be indicated withi = 0 and the highestdose level withi = m, then the followingm = k - 1 hypotheses are tested:

\begin{array}{ll}\mathrm{H}_{m}: \theta_0 = \theta_1 = \ldots = \theta_m, & \mathrm{A}_{m} = \theta_0 \le \theta_1 \le \ldots \theta_m, \theta_0 < \theta_m \\\mathrm{H}_{m-1}: \theta_0 = \theta_1 = \ldots = \theta_{m-1}, & \mathrm{A}_{m-1} = \theta_0 \le \theta_1 \le \ldots \theta_{m-1}, \theta_0 < \theta_{m-1} \\\vdots & \vdots \\\mathrm{H}_{1}: \theta_0 = \theta_1, & \mathrm{A}_{1} = \theta_0 < \theta_1\\\end{array}

LetY_{ij} ~ (1 \leq i \leq n, 0 \leq j \leq k) be a i.i.d. random variableof at least ordinal scale. Further, let\bar{R}_0,~\bar{R}_1, \ldots,~\bar{R}_kbe Friedman's average ranks and set\bar{R}_0^*, \leq \ldots \leq \bar{R}_k^*to be its isotonic regression estimators under the order restriction\theta_0 \leq \ldots \leq \theta_k.

The statistics is

T_j = \left(\bar{R}_j^* - \bar{R}_0 \right)~ \left[ \left(V_j - H_j \right)\left(2 / n \right) \right]^{-1/2} \qquad (1 \leq j \leq k),

with

V_j = \left(j + 1\right) ~ \left(j + 2 \right) / 12

and

H_j = \left(t^3 - t \right) / \left(12 j n \right),

wheret is the number of tied ranks.

The criticalt'_{i,v,\alpha}-valuesas given in the tables of Williams (1972) for\alpha = 0.05 (one-sided)are looked up according to the degree of freedoms (v = \infty) and the order number of thedose level (j).

For the comparison of the first dose level(j = 1) with the control, the criticalz-value from the standard normal distribution is used (Normal).

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Chen, Y.-I., 1999. Rank-Based Tests for Dose Finding inNonmonotonic Dose–Response Settings.Biometrics55, 1258–1262.doi:10.1111/j.0006-341X.1999.01258.x

House, D.E., 1986. A Nonparametric Version of Williams’ Test forRandomized Block Design.Biometrics42, 187–190.

See Also

friedmanTest,friedman.test,frdManyOneExactTest,frdManyOneDemsarTest

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## Assume A is the control. y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) ## Global Friedman test friedmanTest(y) ## Demsar's many-one test summary(frdManyOneDemsarTest(y=y, p.adjust = "bonferroni",                      alternative = "greater")) ## Exact many-one test summary(frdManyOneExactTest(y=y, p.adjust = "bonferroni",                     alternative = "greater")) ## Nemenyi's many-one test summary(frdManyOneNemenyiTest(y=y, alternative = "greater")) ## House test frdHouseTest(y, alternative = "greater")

Demsar's Many-to-One Testfor Unreplicated Blocked Data

Description

Performs Demsar's non-parametric many-to-one comparison testfor Friedman-type ranked data.

Usage

frdManyOneDemsarTest(y, ...)## Default S3 method:frdManyOneDemsarTest(  y,  groups,  blocks,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in a two factorial unreplicated complete block designwith non-normally distributed residuals, Demsar's test can beperformed on Friedman-type ranked data.

Let there bek groups including the control,then the number of treatment levels ism = k - 1.A total ofm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: \theta_0 = \theta_i is tested in the two-tailed case againstA_i: \theta_0 \ne \theta_i, ~~ (1 \le i \le m).

Thep-values are computed from the standard normal distribution.Any of thep-adjustment methods as included inp.adjustcan be used for the adjustment ofp-values.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Demsar, J. (2006) Statistical comparisons of classifiers over multipledata sets,Journal of Machine Learning Research7, 1–30.

See Also

friedmanTest,friedman.test,frdManyOneExactTest,frdManyOneNemenyiTest.

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## Assume A is the control. y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) ## Global Friedman test friedmanTest(y) ## Demsar's many-one test summary(frdManyOneDemsarTest(y=y, p.adjust = "bonferroni",                      alternative = "greater")) ## Exact many-one test summary(frdManyOneExactTest(y=y, p.adjust = "bonferroni",                     alternative = "greater")) ## Nemenyi's many-one test summary(frdManyOneNemenyiTest(y=y, alternative = "greater")) ## House test frdHouseTest(y, alternative = "greater")

Exact Many-to-One Testfor Unreplicated Blocked Data

Description

Performs an exact non-parametric many-to-one comparison testfor Friedman-type ranked data according to Eisinga et al. (2017).

Usage

frdManyOneExactTest(y, ...)## Default S3 method:frdManyOneExactTest(y, groups, blocks, p.adjust.method = p.adjust.methods, ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in a two factorial unreplicated complete block designwith non-normally distributed residuals, an exact test can beperformed on Friedman-type ranked data.

Let there bek groups including the control,then the number of treatment levels ism = k - 1.A total ofm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: \theta_0 = \theta_i is tested in the two-tailed case againstA_i: \theta_0 \ne \theta_i, ~~ (1 \le i \le m).

The exactp-valuesare computed using the code of"pexactfrsd.R"that was a supplement to the publication of Eisinga et al. (2017).Additionally, any of thep-adjustment methodsas included inp.adjust can be selected, forp-valueadjustment.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Eisinga, R., Heskes, T., Pelzer, B., Te Grotenhuis, M. (2017)Exact p-values for Pairwise Comparison of Friedman Rank Sums,with Application to Comparing Classifiers,BMC Bioinformatics, 18:68.

See Also

friedmanTest,friedman.test,frdManyOneDemsarTest,frdManyOneNemenyiTest.

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## Assume A is the control. y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) ## Global Friedman test friedmanTest(y) ## Demsar's many-one test summary(frdManyOneDemsarTest(y=y, p.adjust = "bonferroni",                      alternative = "greater")) ## Exact many-one test summary(frdManyOneExactTest(y=y, p.adjust = "bonferroni",                     alternative = "greater")) ## Nemenyi's many-one test summary(frdManyOneNemenyiTest(y=y, alternative = "greater")) ## House test frdHouseTest(y, alternative = "greater")

Nemenyi's Many-to-One Testfor Unreplicated Blocked Data

Description

Performs Nemenyi's non-parametric many-to-one comparison testfor Friedman-type ranked data.

Usage

frdManyOneNemenyiTest(y, ...)## Default S3 method:frdManyOneNemenyiTest(  y,  groups,  blocks,  alternative = c("two.sided", "greater", "less"),  ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in a two factorial unreplicated complete block designwith non-normally distributed residuals, Nemenyi's test can beperformed on Friedman-type ranked data.

Let there bek groups including the control,then the number of treatment levels ism = k - 1.A total ofm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: \theta_0 = \theta_i is tested in the two-tailed case againstA_i: \theta_0 \ne \theta_i, ~~ (1 \le i \le m).

Thep-values are computed from the multivariate normal distribution.Aspmvnorm applies a numerical method, the estimatedp-values are seet depended.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Hollander, M., Wolfe, D. A., Chicken, E. (2014),Nonparametric Statistical Methods. 3rd ed. New York: Wiley. 2014.

Miller Jr., R. G. (1996),Simultaneous Statistical Inference.New York: McGraw-Hill.

Nemenyi, P. (1963),Distribution-free Multiple Comparisons.Ph.D. thesis, Princeton University.

Siegel, S., Castellan Jr., N. J. (1988),NonparametricStatistics for the Behavioral Sciences. 2nd ed.New York: McGraw-Hill.

Zarr, J. H. (1999),Biostatistical Analysis. 4th ed.Upper Saddle River: Prentice-Hall.

See Also

friedmanTest,friedman.test,frdManyOneExactTest,frdManyOneDemsarTestpmvnorm,set.seed

Examples

 ## Sachs, 1997, p. 675 ## Six persons (block) received six different diuretics ## (A to F, treatment). ## The responses are the Na-concentration (mval) ## in the urine measured 2 hours after each treatment. ## Assume A is the control. y <- matrix(c( 3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92, 23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45, 26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72, 32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23, 26.65),nrow=6, ncol=6, dimnames=list(1:6, LETTERS[1:6])) ## Global Friedman test friedmanTest(y) ## Demsar's many-one test summary(frdManyOneDemsarTest(y=y, p.adjust = "bonferroni",                      alternative = "greater")) ## Exact many-one test summary(frdManyOneExactTest(y=y, p.adjust = "bonferroni",                     alternative = "greater")) ## Nemenyi's many-one test summary(frdManyOneNemenyiTest(y=y, alternative = "greater")) ## House test frdHouseTest(y, alternative = "greater")

Friedman Rank Sum Test

Description

Performs a Friedman rank sum test. The null hypothesisH_0: \theta_i = \theta_j~~(i \ne j) is tested against thealternative H_{\mathrm{A}}: \theta_i \ne \theta_j, with at leastone inequality beeing strict.

Usage

friedmanTest(y, ...)## Default S3 method:friedmanTest(y, groups, blocks, dist = c("Chisquare", "FDist"), ...)

Arguments

Details

The function has implemented Friedman's test as well asthe extension of Conover anf Iman (1981). Friedman'stest statistic is assymptotically chi-squared distributed.Consequently, the default test distribution isdist = "Chisquare".

Ifdist = "FDist" is selected, than the approach ofConover and Imam (1981) is performed.The Friedman Test using theF-distribution leads tothe same results as doing an two-way Analysis of Variance withoutinteraction on rank transformed data.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Conover, W.J., Iman, R.L. (1981) Rank Transformations as a BridgeBetween Parametric and Nonparametric Statistics.Am Stat35, 124–129.

Sachs, L. (1997)Angewandte Statistik. Berlin: Springer.

See Also

friedman.test

Examples

## Hollander & Wolfe (1973), p. 140ff.## Comparison of three methods ("round out", "narrow angle", and##  "wide angle") for rounding first base.  For each of 18 players##  and the three method, the average time of two runs from a point on##  the first base line 35ft from home plate to a point 15ft short of##  second base is recorded.RoundingTimes <-matrix(c(5.40, 5.50, 5.55,        5.85, 5.70, 5.75,        5.20, 5.60, 5.50,        5.55, 5.50, 5.40,        5.90, 5.85, 5.70,        5.45, 5.55, 5.60,        5.40, 5.40, 5.35,        5.45, 5.50, 5.35,        5.25, 5.15, 5.00,        5.85, 5.80, 5.70,        5.25, 5.20, 5.10,        5.65, 5.55, 5.45,        5.60, 5.35, 5.45,        5.05, 5.00, 4.95,        5.50, 5.50, 5.40,        5.45, 5.55, 5.50,        5.55, 5.55, 5.35,        5.45, 5.50, 5.55,        5.50, 5.45, 5.25,        5.65, 5.60, 5.40,        5.70, 5.65, 5.55,        6.30, 6.30, 6.25),      nrow = 22,      byrow = TRUE,      dimnames = list(1 : 22,                      c("Round Out", "Narrow Angle", "Wide Angle")))## Chisquare distributionfriedmanTest(RoundingTimes)## check with friedman.test from R statsfriedman.test(RoundingTimes)## F-distributionfriedmanTest(RoundingTimes, dist = "FDist")## Check with One-way repeated measure ANOVArmat <- RoundingTimesfor (i in 1:length(RoundingTimes[,1])) rmat[i,] <- rank(rmat[i,])dataf <- data.frame(    y = y <- as.vector(rmat),    g = g <- factor(c(col(RoundingTimes))),    b = b <- factor(c(row(RoundingTimes))))summary(aov(y ~ g + Error(b), data = dataf))

Games-Howell Test

Description

Performs Games-Howell all-pairs comparison test for normally distributeddata with unequal group variances.

Usage

gamesHowellTest(x, ...)## Default S3 method:gamesHowellTest(x, g, ...)## S3 method for class 'formula'gamesHowellTest(formula, data, subset, na.action, ...)## S3 method for class 'aov'gamesHowellTest(x, ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith normally distributed residuals but unequal between-groups variancesthe Games-Howell Test can be performed. LetX_{ij} denote a continuous random variablewith thej-the realization (1 \le j \le n_i)in thei-th group (1 \le i \le k). Furthermore, the totalsample size isN = \sum_{i=1}^k n_i. A total ofm = k(k-1)/2hypotheses can be tested: The null hypothesis isH_{ij}: \mu_i = \mu_j ~~ (i \ne j) is tested against the alternativeA_{ij}: \mu_i \ne \mu_j (two-tailed). Games-Howell Test all-pairstest statistics are given by

t_{ij} \frac{\bar{X}_i - \bar{X_j}} {\left( s^2_j / n_j + s^2_i / n_i \right)^{1/2}}, ~~ (i \ne j)

withs^2_i the variance of thei-th group.The null hypothesis is rejected (two-tailed) if

\mathrm{Pr} \left\{ |t_{ij}| \sqrt{2} \ge q_{m v_{ij} \alpha} | \mathrm{H} \right\}_{ij} = \alpha,

with Welch's approximate solution for calculating the degree of freedom.

v_{ij} = \frac{\left( s^2_i / n_i + s^2_j / n_j \right)^2} {s^4_i / n^2_i \left(n_i - 1\right) + s^4_j / n^2_j \left(n_j - 1\right)}.

Thep-values are computed from theTukey distribution.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

See Also

Tukey

Examples

fit <- aov(weight ~ feed, chickwts)shapiro.test(residuals(fit))bartlett.test(weight ~ feed, chickwts) # var1 = varNanova(fit)## also works with fitted objects of class aovres <- gamesHowellTest(fit)summary(res)summaryGroup(res)

Generalized Extreme Studentized Deviate Many-Outlier Test

Description

Performs Rosner's generalized extreme studentized deviateprocedure to detect up-tomaxr outliers in aunivariate sample that follows an approximately normal distribution.

Usage

gesdTest(x, maxr)

Arguments

References

Rosner, B. (1983) Percentage Points for a Generalized ESDMany-Outlier Procedure,Technometrics25, 165–172.

Examples

## Taken from Rosner (1983):x <- c(-0.25,0.68,0.94,1.15,1.20,1.26,1.26,1.34,1.38,1.43,1.49,1.49,1.55,1.56,1.58,1.65,1.69,1.70,1.76,1.77,1.81,1.91,1.94,1.96,1.99,2.06,2.09,2.10,2.14,2.15,2.23,2.24,2.26,2.35,2.37,2.40,2.47,2.54,2.62,2.64,2.90,2.92,2.92,2.93,3.21,3.26,3.30,3.59,3.68,4.30,4.64,5.34,5.42,6.01)out <- gesdTest(x, 10)## print methodout## summary methodsummary(out)

Gore Test

Description

Performs Gore's test. The null hypothesisH_0: \theta_i = \theta_j~~(i \ne j) is tested against thealternative H_{\mathrm{A}}: \theta_i \ne \theta_j, with at leastone inequality beeing strict.

Usage

goreTest(y, groups, blocks)

Arguments

Details

The function has implemented Gore's test for testingmain effects in unbalanced CRB designs,i.e. there are one ore more observations per cell.The statistic is assymptotically chi-squared distributed.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Gore, A. P. (1975) Some nonparametric tests and selectionprocedures for main effects in two-way layouts.Ann. Inst. Stat. Math.27, 487–500.

See Also

friedmanTest,skillingsMackTest,durbinTest

Examples

## Crop Yield of 3 varieties on two## soil classesX <-c("130,A,Light115,A,Light123,A,Light142,A,Light117,A,Heavy125,A,Heavy139,A,Heavy108,B,Light114,B,Light124,B,Light106,B,Light91,B,Heavy111,B,Heavy110,B,Heavy155,C,Light146,C,Light151,C,Light165,C,Light97,C,Heavy108,C,Heavy")con <- textConnection(X)x <- read.table(con, header=FALSE, sep=",")close(con)colnames(x) <- c("Yield", "Variety", "SoilType")goreTest(y = x$Yield, groups = x$Variety, blocks = x$SoilType)

Grubbs Outlier Test

Description

Performs Grubbs single outlier test.

Usage

grubbsTest(x, alternative = c("two.sided", "greater", "less"))

Arguments

Details

LetX denote an identically and independently distributed continuousvariate with realizationsx_i ~~ (1 \le i \le k).Further, let the increasingly ordered realizationsdenotex_{(1)} \le x_{(2)} \le \ldots \le x_{(n)}. Thenthe following model for a single maximum outlier can be proposed:

x_{(i)} = \left\{ \begin{array}{lcl} \mu + \epsilon_{(i)}, & \qquad & i = 1, \ldots, n - 1 \\ \mu + \Delta + \epsilon_{(n)} & & \\ \end{array} \right.

with\epsilon \approx N(0,\sigma). The null hypothesis,H_0: \Delta = 0 is tested against the alternative,H_{\mathrm{A}}: \Delta > 0.

For testing a single minimum outlier, the model can be proposedas

x_{(i)} = \left\{ \begin{array}{lcl} \mu + \Delta + \epsilon_{(1)} & & \\ \mu + \epsilon_{(i)}, & \qquad & i = 2, \ldots, n \\ \end{array} \right.

The null hypothesis is tested against the alternative,H_{\mathrm{A}}: \Delta < 0.

The p-value is computed with the functionpgrubbs.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Grubbs, F. E. (1950) Sample criteria for testing outlying observations.Ann. Math. Stat.21, 27–58.

Wilrich, P.-T. (2011) Critical values of Mandel's h and k,Grubbs and the Cochran test statistic.Adv. Stat. Anal..doi:10.1007/s10182-011-0185-y.

Examples

data(Pentosan)dat <- subset(Pentosan, subset = (material == "A"))labMeans <- tapply(dat$value, dat$lab, mean)grubbsTest(x = labMeans, alternative = "two.sided")

Hartley's Maximum F-Ratio Test of Homogeneity ofVariances

Description

Performs Hartley's maximum F-ratio test of the null thatvariances in each of the groups (samples) are the same.

Usage

hartleyTest(x, ...)## Default S3 method:hartleyTest(x, g, ...)## S3 method for class 'formula'hartleyTest(formula, data, subset, na.action, ...)

Arguments

Details

Ifx is a list, its elements are taken as the samplesto be compared for homogeneity of variances. In thiscase, the elements must all be numeric data vectors,g is ignored, and one can simply usehartleyTest(x) to perform the test. If the samples are notyet contained in a list, usehartleyTest(list(x, ...)).

Otherwise,x must be a numeric data vector, andg mustbe a vector or factor object of the same length asx giving thegroup for the corresponding elements ofx.

Hartley's parametric test requires normality anda nearly balanced design. The p-value of the testis calculated with the functionpmaxFratioof the packageSuppDists.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Hartley, H.O. (1950) The maximum F-ratioas a short cut test for heterogeneity of variance,Biometrika37, 308–312.

See Also

bartlett.test,pmaxFratio

Examples

hartleyTest(count ~ spray, data = InsectSprays)

Hayter-Stone Test

Description

Performs the non-parametric Hayter-Stone procedureto test against an monotonically increasing alternative.

Usage

hayterStoneTest(x, ...)## Default S3 method:hayterStoneTest(  x,  g,  alternative = c("greater", "less"),  method = c("look-up", "boot", "asympt"),  nperm = 10000,  ...)## S3 method for class 'formula'hayterStoneTest(  formula,  data,  subset,  na.action,  alternative = c("greater", "less"),  method = c("look-up", "boot", "asympt"),  nperm = 10000,  ...)

Arguments

Details

LetX be an identically and idepentendly distributed variablethat wasn times observed atk increasing treatment levels.Hayter and Stone (1991) proposed a non-parametric procedureto test the null hypothesis, H:\theta_i = \theta_j ~~ (i < j \le k)against a simple order alternative, A:\theta_i < \theta_j, with at leastone inequality being strict.

The statistic for a global test is calculated as,

h = \max_{1 \le i < j \le k} \frac{2 \sqrt{6} \left(U_{ij} - n_i n_j / 2 \right)} {\sqrt{n_i n_j \left(n_i + n_j + 1 \right)}},

with the Mann-Whittney counts:

U_{ij} = \sum_{a=1}^{n_i} \sum_{b=1}^{n_j} I\left\{x_{ia} < x_{ja}\right\}.

Under the large sample approximation, the test statistich is distributedash_{k,\alpha,v}. Thus, the null hypothesis is rejected, ifh > h_{k,\alpha,v}, withv = \inftydegree of freedom.

Ifmethod = "look-up" the function will not returnp-values. Instead the critical h-valuesas given in the tables of Hayter (1990) for\alpha = 0.05 (one-sided)are looked up according to the number of groups (k) andthe degree of freedoms (v = \infty).

Ifmethod = "boot" an asymptotic permutation testis conducted and ap-value is returned.

Ifmethod = "asympt" is selected the asymptoticp-value is estimated as implemented in thefunctionpHayStonLSA of the packageNSM3.

Value

Either a list of classhtest or alist with class"osrt" that contains the followingcomponents:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated statistic(s)

crit.value

critical values for\alpha = 0.05.

alternative

a character string describing the alternative hypothesis.

parameter

the parameter(s) of the test distribution.

dist

a string that denotes the test distribution.

There are print and summary methods available.

Source

Ifmethod = "asympt" is selected, this function callsan internal probability functionpHS. The GPL-2 code forthis function was taken frompHayStonLSA of thethe packageNSM3:

Grant Schneider, Eric Chicken and Rachel Becvarik (2020) NSM3:Functions and Datasets to Accompany Hollander, Wolfe, andChicken - Nonparametric Statistical Methods, Third Edition. Rpackage version 1.15.https://CRAN.R-project.org/package=NSM3

References

Hayter, A. J.(1990) A One-Sided Studentised RangeTest for Testing Against a Simple Ordered Alternative,J Amer Stat Assoc85, 778–785.

Hayter, A.J., Stone, G. (1991)Distribution free multiple comparisons for monotonically ordered treatment effects.Austral J Statist33, 335–346.

See Also

osrtTest,hsAllPairsTest,sample,pHayStonLSA

Examples

## Example from Shirley (1977)## Reaction times of mice to stimuli to their tails.x <- c(2.4, 3, 3, 2.2, 2.2, 2.2, 2.2, 2.8, 2, 3, 2.8, 2.2, 3.8, 9.4, 8.4, 3, 3.2, 4.4, 3.2, 7.4, 9.8, 3.2, 5.8, 7.8, 2.6, 2.2, 6.2, 9.4, 7.8, 3.4, 7, 9.8, 9.4, 8.8, 8.8, 3.4, 9, 8.4, 2.4, 7.8)g <- gl(4, 10)## Shirley's test## one-sided test using look-up tableshirleyWilliamsTest(x ~ g, alternative = "greater")## Chacko's global hypothesis test for 'greater'chackoTest(x , g)## post-hoc test, default is standard normal distribution (NPT'-test)summary(chaAllPairsNashimotoTest(x, g, p.adjust.method = "none"))## same but h-distribution (NPY'-test)chaAllPairsNashimotoTest(x, g, dist = "h")## NPM-testNPMTest(x, g)## Hayter-Stone testhayterStoneTest(x, g)## all-pairs comparisonshsAllPairsTest(x, g)

Hayter-Stone All-Pairs Comparison Test

Description

Performs the non-parametric Hayter-Stone all-pairs procedureto test against monotonically increasing alternatives.

Usage

hsAllPairsTest(x, ...)## Default S3 method:hsAllPairsTest(  x,  g,  alternative = c("greater", "less"),  method = c("look-up", "boot", "asympt"),  nperm = 10000,  ...)## S3 method for class 'formula'hsAllPairsTest(  formula,  data,  subset,  na.action,  alternative = c("greater", "less"),  method = c("look-up", "boot", "asympt"),  nperm = 10000,  ...)

Arguments

Details

LetX be an identically and idepentendly distributed variablethat wasn times observed atk increasing treatment levels.Hayter and Stone (1991) proposed a non-parametric procedureto test the null hypothesis, H:\theta_i = \theta_j ~~ (i < j \le k)against a simple order alternative, A:\theta_i < \theta_j.

The statistic for all-pairs comparisons is calculated as,

S_{ij} = \frac{2 \sqrt{6} \left(U_{ij} - n_i n_j / 2 \right)} {\sqrt{n_i n_j \left(n_i + n_j + 1 \right)}},

with the Mann-Whittney counts:

U_{ij} = \sum_{a=1}^{n_i} \sum_{b=1}^{n_j} I\left\{x_{ia} < x_{ja}\right\}.

Under the large sample approximation, the test statisticS_{ij} is distributedash_{k,\alpha,v}. Thus, the null hypothesis is rejected,ifS_{ij} > h_{k,\alpha,v}, withv = \infty degree of freedom.

Ifmethod = "look-up" the function will not returnp-values. Instead the critical h-valuesas given in the tables of Hayter (1990) for\alpha = 0.05 (one-sided)are looked up according to the number of groups (k) andthe degree of freedoms (v = \infty).

Ifmethod = "boot" an asymetric permutation testis conducted andp-values are returned.

Ifmethod = "asympt" is selected the asymptoticp-value is estimated as implemented in thefunctionpHayStonLSA of the packageNSM3.

Value

Either a list of class"PMCMR" or alist with class"osrt" that contains the followingcomponents:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated statistic(s)

crit.value

critical values for\alpha = 0.05.

alternative

a character string describing the alternative hypothesis.

parameter

the parameter(s) of the test distribution.

dist

a string that denotes the test distribution.

There are print and summary methods available.

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Source

Ifmethod = "asympt" is selected, this function callsan internal probability functionpHS. The GPL-2 code forthis function was taken frompHayStonLSA of thethe packageNSM3:

Grant Schneider, Eric Chicken and Rachel Becvarik (2020) NSM3:Functions and Datasets to Accompany Hollander, Wolfe, andChicken - Nonparametric Statistical Methods, Third Edition. Rpackage version 1.15.https://CRAN.R-project.org/package=NSM3

References

Hayter, A. J.(1990) A One-Sided Studentised RangeTest for Testing Against a Simple Ordered Alternative,Journal of the American Statistical Association85, 778–785.

Hayter, A.J., Stone, G. (1991)Distribution free multiple comparisons for monotonically ordered treatment effects.Austral J Statist33, 335–346.

See Also

hayterStoneTestsample

Examples

## Example from Shirley (1977)## Reaction times of mice to stimuli to their tails.x <- c(2.4, 3, 3, 2.2, 2.2, 2.2, 2.2, 2.8, 2, 3, 2.8, 2.2, 3.8, 9.4, 8.4, 3, 3.2, 4.4, 3.2, 7.4, 9.8, 3.2, 5.8, 7.8, 2.6, 2.2, 6.2, 9.4, 7.8, 3.4, 7, 9.8, 9.4, 8.8, 8.8, 3.4, 9, 8.4, 2.4, 7.8)g <- gl(4, 10)## Shirley's test## one-sided test using look-up tableshirleyWilliamsTest(x ~ g, alternative = "greater")## Chacko's global hypothesis test for 'greater'chackoTest(x , g)## post-hoc test, default is standard normal distribution (NPT'-test)summary(chaAllPairsNashimotoTest(x, g, p.adjust.method = "none"))## same but h-distribution (NPY'-test)chaAllPairsNashimotoTest(x, g, dist = "h")## NPM-testNPMTest(x, g)## Hayter-Stone testhayterStoneTest(x, g)## all-pairs comparisonshsAllPairsTest(x, g)

Testing against Ordered Alternatives (Johnson-Mehrotra Test)

Description

Performs the Johnson-Mehrotra test for testing against ordered alternativesin a balanced one-factorial sampling design.

Usage

johnsonTest(x, ...)## Default S3 method:johnsonTest(x, g, alternative = c("two.sided", "greater", "less"), ...)## S3 method for class 'formula'johnsonTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  ...)

Arguments

Details

The null hypothesis, H_0: \theta_1 = \theta_2 = \ldots = \theta_kis tested against a simple order hypothesis,H_\mathrm{A}: \theta_1 \le \theta_2 \le \ldots \le\theta_k,~\theta_1 < \theta_k.

The p-values are estimated from the standard normal distribution.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Bortz, J. (1993).Statistik für Sozialwissenschaftler (4th ed.).Berlin: Springer.

Johnson, R. A., Mehrotra, K. G. (1972) Some c-samplenonparametric tests for ordered alternatives.Journal of the Indian Statistical Association9, 8–23.

See Also

kruskalTest andshirleyWilliamsTestof the packagePMCMRplus,kruskal.test of the librarystats.

Examples

## Example from Sachs (1997, p. 402)x <- c(106, 114, 116, 127, 145,       110, 125, 143, 148, 151,       136, 139, 149, 160, 174)g <- gl(3,5)levels(g) <- c("A", "B", "C")## Chacko's testchackoTest(x, g)## Cuzick's testcuzickTest(x, g)## Johnson-Mehrotra testjohnsonTest(x, g)## Jonckheere-Terpstra testjonckheereTest(x, g)## Le's testleTest(x, g)## Spearman type testspearmanTest(x, g)## Murakami's BWS trend testbwsTrendTest(x, g)## Fligner-Wolfe testflignerWolfeTest(x, g)## Shan-Young-Kang testshanTest(x, g)

Testing against Ordered Alternatives (Jonckheere-Terpstra Test)

Description

Performs the Jonckheere-Terpstra test for testing against ordered alternatives.

Usage

jonckheereTest(x, ...)## Default S3 method:jonckheereTest(  x,  g,  alternative = c("two.sided", "greater", "less"),  continuity = FALSE,  ...)## S3 method for class 'formula'jonckheereTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  continuity = FALSE,  ...)

Arguments

Details

The null hypothesis, H_0: \theta_1 = \theta_2 = \ldots = \theta_kis tested against a simple order hypothesis,H_\mathrm{A}: \theta_1 \le \theta_2 \le \ldots \le\theta_k,~\theta_1 < \theta_k.

The p-values are estimated from the standard normal distribution.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Source

The code for the computation of the standard deviationfor the Jonckheere-Terpstra test in the presence of ties was taken from:

Kloke, J., McKean, J. (2016)npsm: Package for Nonparametric Statistical Methods using R.R package version 0.5.https://CRAN.R-project.org/package=npsm

Note

jonckheereTest(x, g, alternative = "two.sided", continuity = TRUE) isequivalent to

cor.test(x, as.numeric(g), method = "kendall", alternative = "two.sided", continuity = TRUE)

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Jonckheere, A. R. (1954) A distribution-free k-sample testagainst ordered alternatives.Biometrica41, 133–145.

Kloke, J., McKean, J. W. (2015)Nonparametric statistical methods using R.Boca Raton, FL: Chapman & Hall/CRC.

See Also

kruskalTest andshirleyWilliamsTestof the packagePMCMRplus,kruskal.test of the librarystats.

Examples

## Example from Sachs (1997, p. 402)x <- c(106, 114, 116, 127, 145,       110, 125, 143, 148, 151,       136, 139, 149, 160, 174)g <- gl(3,5)levels(g) <- c("A", "B", "C")## Chacko's testchackoTest(x, g)## Cuzick's testcuzickTest(x, g)## Johnson-Mehrotra testjohnsonTest(x, g)## Jonckheere-Terpstra testjonckheereTest(x, g)## Le's testleTest(x, g)## Spearman type testspearmanTest(x, g)## Murakami's BWS trend testbwsTrendTest(x, g)## Fligner-Wolfe testflignerWolfeTest(x, g)## Shan-Young-Kang testshanTest(x, g)

Kruskal-Wallis Rank Sum Test

Description

Performs a Kruskal-Wallis rank sum test.

Usage

kruskalTest(x, ...)## Default S3 method:kruskalTest(x, g, dist = c("Chisquare", "KruskalWallis", "FDist"), ...)## S3 method for class 'formula'kruskalTest(  formula,  data,  subset,  na.action,  dist = c("Chisquare", "KruskalWallis", "FDist"),  ...)

Arguments

Details

For one-factorial designs with non-normally distributedresiduals the Kruskal-Wallis rank sum test can be performed to testthe H_0: F_1(x) = F_2(x) = \ldots = F_k(x) againstthe H_\mathrm{A}: F_i (x) \ne F_j(x)~ (i \ne j) with at leastone strict inequality.

LetR_{ij} be the joint rank ofX_{ij},withR_{(1)(1)} = 1, \ldots, R_{(n)(n)} = N, ~~ N = \sum_{i=1}^k n_i,The test statistic is calculated as

H = \sum_{i=1}^k n_i \left(\bar{R}_i - \bar{R}\right) / \sigma_R,

with the mean rank of thei-th group

\bar{R}_i = \sum_{j = 1}^{n_{i}} R_{ij} / n_i,

the expected value

\bar{R} = \left(N +1\right) / 2

and the expected variance as

\sigma_R^2 = N \left(N + 1\right) / 12.

In case of ties the statisticH is divided by\left(1 - \sum_{i=1}^r t_i^3 - t_i \right) / \left(N^3 - N\right)

According to Conover and Imam (1981), the statisticH is relatedto theF-quantile as

F = \frac{H / \left(k - 1\right)} {\left(N - 1 - H\right) / \left(N - k\right)}

which is equivalent to a one-way ANOVA F-test using rank transformed data(see examples).

The function provides three differentdist forp-value estimation:

Chisquare

p-values are computed from theChisquaredistribution withv = k - 1 degree of freedom.

KruskalWallis

p-values are computed from thepKruskalWallis of the packageSuppDists.

FDist

p-values are computed from theFDist distributionwithv_1 = k-1, ~ v_2 = N -k degree of freedom.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

References

Conover, W.J., Iman, R.L. (1981) Rank Transformations as a BridgeBetween Parametric and Nonparametric Statistics.Am Stat35, 124–129.

Kruskal, W.H., Wallis, W.A. (1952) Use of Ranks in One-Criterion Variance Analysis.J Am Stat Assoc47, 583–621.

Sachs, L. (1997)Angewandte Statistik. Berlin: Springer.

See Also

kruskal.test,pKruskalWallis,Chisquare,FDist

Examples

## Hollander & Wolfe (1973), 116.## Mucociliary efficiency from the rate of removal of dust in normal## subjects, subjects with obstructive airway disease, and subjects## with asbestosis.x <- c(2.9, 3.0, 2.5, 2.6, 3.2) # normal subjectsy <- c(3.8, 2.7, 4.0, 2.4)      # with obstructive airway diseasez <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosisg <- factor(x = c(rep(1, length(x)),                   rep(2, length(y)),                   rep(3, length(z))),             labels = c("ns", "oad", "a"))dat <- data.frame(   g = g,   x = c(x, y, z))## AD-TestadKSampleTest(x ~ g, data = dat)## BWS-TestbwsKSampleTest(x ~ g, data = dat)## Kruskal-Test## Using incomplete beta approximationkruskalTest(x ~ g, dat, dist="KruskalWallis")## Using chisquare distributionkruskalTest(x ~ g, dat, dist="Chisquare")## Not run: ## Check with kruskal.test from R statskruskal.test(x ~ g, dat)## End(Not run)## Using Conover's FkruskalTest(x ~ g, dat, dist="FDist")## Not run: ## Check with aov on ranksanova(aov(rank(x) ~ g, dat))## Check with oneway.testoneway.test(rank(x) ~ g, dat, var.equal = TRUE)## End(Not run)## Median Test asymptoticmedianTest(x ~ g, dat)## Median Test with simulated p-valuesset.seed(112)medianTest(x ~ g, dat, simulate.p.value = TRUE)

Conover's All-Pairs Rank Comparison Test

Description

Performs Conover's non-parametric all-pairs comparison testfor Kruskal-type ranked data.

Usage

kwAllPairsConoverTest(x, ...)## Default S3 method:kwAllPairsConoverTest(  x,  g,  p.adjust.method = c("single-step", p.adjust.methods),  ...)## S3 method for class 'formula'kwAllPairsConoverTest(  formula,  data,  subset,  na.action,  p.adjust.method = c("single-step", p.adjust.methods),  ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals Conover's non-parametric testcan be performed. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: \mu_i(x) = \mu_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: \mu_i(x) \ne \mu_j(x), ~~ i \ne j.

Ifp.adjust.method == "single-step" the p-values are computedfrom the studentized range distribution. Otherwise,the p-values are computed from the t-distribution usingany of the p-adjustment methods as included inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Conover, W. J, Iman, R. L. (1979)On multiple-comparisonsprocedures, Tech. Rep. LA-7677-MS, Los Alamos Scientific Laboratory.

See Also

Tukey,TDist,p.adjust,kruskalTest,kwAllPairsDunnTest,kwAllPairsNemenyiTest

Examples

## Data set InsectSprays## Global testkruskalTest(count ~ spray, data = InsectSprays)## Conover's all-pairs comparison test## single-step means Tukey's p-adjustmentans <- kwAllPairsConoverTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "single-step")summary(ans)## Dunn's all-pairs comparison testans <- kwAllPairsDunnTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "bonferroni")summary(ans)## Nemenyi's all-pairs comparison testans <- kwAllPairsNemenyiTest(count ~ spray, data = InsectSprays)summary(ans)## Brown-Mood all-pairs median testans <- medianAllPairsTest(count ~ spray, data = InsectSprays)summary(ans)

Dunn's All-Pairs Rank Comparison Test

Description

Performs Dunn's non-parametric all-pairs comparison testfor Kruskal-type ranked data.

Usage

kwAllPairsDunnTest(x, ...)## Default S3 method:kwAllPairsDunnTest(x, g, p.adjust.method = p.adjust.methods, ...)## S3 method for class 'formula'kwAllPairsDunnTest(  formula,  data,  subset,  na.action,  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals Dunn's non-parametric testcan be performed. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: \mu_i(x) = \mu_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: \mu_i(x) \ne \mu_j(x), ~~ i \ne j.

The p-values are computed from the standard normal distribution usingany of the p-adjustment methods as included inp.adjust.Originally, Dunn (1964) proposed Bonferroni's p-adjustment method.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Dunn, O. J. (1964) Multiple comparisons using rank sums,Technometrics6, 241–252.

Siegel, S., Castellan Jr., N. J. (1988)Nonparametric Statisticsfor The Behavioral Sciences. New York: McGraw-Hill.

See Also

Normal,p.adjust,kruskalTest,kwAllPairsConoverTest,kwAllPairsNemenyiTest

Examples

## Data set InsectSprays## Global testkruskalTest(count ~ spray, data = InsectSprays)## Conover's all-pairs comparison test## single-step means Tukey's p-adjustmentans <- kwAllPairsConoverTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "single-step")summary(ans)## Dunn's all-pairs comparison testans <- kwAllPairsDunnTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "bonferroni")summary(ans)## Nemenyi's all-pairs comparison testans <- kwAllPairsNemenyiTest(count ~ spray, data = InsectSprays)summary(ans)## Brown-Mood all-pairs median testans <- medianAllPairsTest(count ~ spray, data = InsectSprays)summary(ans)

Nemenyi's All-Pairs Rank Comparison Test

Description

Performs Nemenyi's non-parametric all-pairs comparison testfor Kruskal-type ranked data.

Usage

kwAllPairsNemenyiTest(x, ...)## Default S3 method:kwAllPairsNemenyiTest(x, g, dist = c("Tukey", "Chisquare"), ...)## S3 method for class 'formula'kwAllPairsNemenyiTest(  formula,  data,  subset,  na.action,  dist = c("Tukey", "Chisquare"),  ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals Nemenyi's non-parametric testcan be performed. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: \theta_i(x) = \theta_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: \theta_i(x) \ne \theta_j(x), ~~ i \ne j.

LetR_{ij} be the rank ofX_{ij},whereX_{ij} is jointly rankedfrom\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i,then the test statistic under the absence of ties is calculated as

t_{ij} = \frac{\bar{R}_j - \bar{R}_i}{\sigma_R \left(1/n_i + 1/n_j\right)^{1/2}} \qquad \left(i \ne j\right),

with\bar{R}_j, \bar{R}_i the mean rank of thei-th andj-th group and the expected variance as

\sigma_R^2 = N \left(N + 1\right) / 12.

A pairwise difference is significant, if|t_{ij}|/\sqrt{2} > q_{kv},withk the number of groups andv = \inftythe degree of freedom.

Sachs(1997) has given a modified approach forNemenyi's test in the presence of ties forN > 6, k > 4provided that thekruskalTest indicates significance:In the presence of ties, the test statistic iscorrected according to\hat{t}_{ij} = t_{ij} / C, with

C = 1 - \frac{\sum_{i=1}^r t_i^3 - t_i}{N^3 - N}.

The function provides two differentdistforp-value estimation:

Tukey

Thep-values are computed from the studentizedrange distribution (aliasTukey),\mathrm{Pr} \left\{ t_{ij} \sqrt{2} \ge q_{k\infty\alpha} | mathrm{H} \right\} = \alpha.

Chisquare

Thep-values are computed from theChisquare distribution withv = k - 1 degreeof freedom.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Nemenyi, P. (1963)Distribution-free Multiple Comparisons.Ph.D. thesis, Princeton University.

Sachs, L. (1997)Angewandte Statistik. Berlin: Springer.

Wilcoxon, F., Wilcox, R. A. (1964)Some rapid approximate statistical procedures.Pearl River: Lederle Laboratories.

See Also

Tukey,Chisquare,p.adjust,kruskalTest,kwAllPairsDunnTest,kwAllPairsConoverTest

Examples

## Data set InsectSprays## Global testkruskalTest(count ~ spray, data = InsectSprays)## Conover's all-pairs comparison test## single-step means Tukey's p-adjustmentans <- kwAllPairsConoverTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "single-step")summary(ans)## Dunn's all-pairs comparison testans <- kwAllPairsDunnTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "bonferroni")summary(ans)## Nemenyi's all-pairs comparison testans <- kwAllPairsNemenyiTest(count ~ spray, data = InsectSprays)summary(ans)## Brown-Mood all-pairs median testans <- medianAllPairsTest(count ~ spray, data = InsectSprays)summary(ans)

Conover's Many-to-One Rank Comparison Test

Description

Performs Conover's non-parametric many-to-one comparisontest for Kruskal-type ranked data.

Usage

kwManyOneConoverTest(x, ...)## Default S3 method:kwManyOneConoverTest(  x,  g,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)## S3 method for class 'formula'kwManyOneConoverTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in an one-factorial layout with non-normally distributedresiduals Conover's non-parametric test can be performed.Let there bek groups including the control,then the number of treatment levels ism = k - 1.Thenm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: \theta_0 = \theta_i is tested in the two-tailed case againstA_i: \theta_0 \ne \theta_i, ~~ (1 \le i \le m).

Ifp.adjust.method == "single-step" is selected,thep-values will be computedfrom the multivariatet distribution. Otherwise,thep-values are computed from thet-distribution usingany of thep-adjustment methods as included inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Conover, W. J, Iman, R. L. (1979)On multiple-comparisonsprocedures, Tech. Rep. LA-7677-MS, Los Alamos Scientific Laboratory.

See Also

pmvt,TDist,kruskalTest,kwManyOneDunnTest,kwManyOneNdwTest

Examples

## Data set PlantGrowth## Global testkruskalTest(weight ~ group, data = PlantGrowth)## Conover's many-one comparison test## single-step means p-value from multivariate t distributionans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "single-step")summary(ans)## Conover's many-one comparison testans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Dunn's many-one comparison testans <- kwManyOneDunnTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Nemenyi's many-one comparison testans <- kwManyOneNdwTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Many one U testans <- manyOneUTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Chen Testans <- chenTest(weight ~ group, data = PlantGrowth,                    p.adjust.method = "holm")summary(ans)

Dunn's Many-to-One Rank Comparison Test

Description

Performs Dunn's non-parametric many-to-one comparisontest for Kruskal-type ranked data.

Usage

kwManyOneDunnTest(x, ...)## Default S3 method:kwManyOneDunnTest(  x,  g,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)## S3 method for class 'formula'kwManyOneDunnTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in an one-factorial layout with non-normally distributedresiduals Dunn's non-parametric test can be performed.Let there bek groups including the control,then the number of treatment levels ism = k - 1.Thenm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: \theta_0 = \theta_i is tested in the two-tailed case againstA_i: \theta_0 \ne \theta_i, ~~ (1 \le i \le m).

Ifp.adjust.method == "single-step" is selected,thep-values will be computedfrom the multivariate normal distribution. Otherwise,thep-values are computed from the standard normal distribution usingany of thep-adjustment methods as included inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Dunn, O. J. (1964) Multiple comparisons using rank sums,Technometrics6, 241–252.

Siegel, S., Castellan Jr., N. J. (1988)Nonparametric Statisticsfor The Behavioral Sciences. New York: McGraw-Hill.

See Also

pmvnorm,TDist,kruskalTest,kwManyOneConoverTest,kwManyOneNdwTest

Examples

## Data set PlantGrowth## Global testkruskalTest(weight ~ group, data = PlantGrowth)## Conover's many-one comparison test## single-step means p-value from multivariate t distributionans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "single-step")summary(ans)## Conover's many-one comparison testans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Dunn's many-one comparison testans <- kwManyOneDunnTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Nemenyi's many-one comparison testans <- kwManyOneNdwTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Many one U testans <- manyOneUTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Chen Testans <- chenTest(weight ~ group, data = PlantGrowth,                    p.adjust.method = "holm")summary(ans)

Nemenyi-Damico-Wolfe Many-to-One Rank Comparison Test

Description

Performs Nemenyi-Damico-Wolfe non-parametric many-to-one comparisontest for Kruskal-type ranked data.

Usage

kwManyOneNdwTest(x, ...)## Default S3 method:kwManyOneNdwTest(  x,  g,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)## S3 method for class 'formula'kwManyOneNdwTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)

Arguments

Details

For many-to-one comparisons (pairwise comparisons with one control)in an one-factorial layout with non-normally distributedresiduals the Nemenyi-Damico-Wolfe non-parametric test can be performed.Let there bek groups including the control,then the number of treatment levels ism = k - 1.Thenm pairwise comparisons can be performed betweenthei-th treatment level and the control.H_i: \theta_0 = \theta_i is tested in the two-tailed case againstA_i: \theta_0 \ne \theta_i, ~~ (1 \le i \le m).

Ifp.adjust.method == "single-step" is selected,thep-values will be computedfrom the multivariate normal distribution. Otherwise,thep-values are computed from the standard normal distribution usingany of thep-adjustment methods as included inp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

This function is essentially the same askwManyOneDunnTest, butthere is no tie correction included. Therefore, the implementation ofDunn's test is superior, when ties are present.

References

Damico, J. A., Wolfe, D. A. (1989) Extended tables of the exact distribution ofa rank statistic for treatments versus control multiple comparisons in one-waylayout designs,Communications in Statistics - Theory and Methods18,3327–3353.

Nemenyi, P. (1963)Distribution-free Multiple Comparisons,Ph.D. thesis, Princeton University.

See Also

pmvt,TDist,kruskalTest,kwManyOneDunnTest,kwManyOneConoverTest

Examples

## Data set PlantGrowth## Global testkruskalTest(weight ~ group, data = PlantGrowth)## Conover's many-one comparison test## single-step means p-value from multivariate t distributionans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "single-step")summary(ans)## Conover's many-one comparison testans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Dunn's many-one comparison testans <- kwManyOneDunnTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Nemenyi's many-one comparison testans <- kwManyOneNdwTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Many one U testans <- manyOneUTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Chen Testans <- chenTest(weight ~ group, data = PlantGrowth,                    p.adjust.method = "holm")summary(ans)

Testing against Ordered Alternatives (Le's Test)

Description

Performs Le's test for testing against ordered alternatives.

Usage

leTest(x, ...)## Default S3 method:leTest(x, g, alternative = c("two.sided", "greater", "less"), ...)## S3 method for class 'formula'leTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  ...)

Arguments

Details

The null hypothesis, H_0: \theta_1 = \theta_2 = \ldots = \theta_kis tested against a simple order hypothesis,H_\mathrm{A}: \theta_1 \le \theta_2 \le \ldots \le\theta_k,~\theta_1 < \theta_k.

The p-values are estimated from the standard normal distribution.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

Le, C. T. (1988) A new rank test against ordered alternativesin k-sample problems,Biometrical Journal30, 87–92.

See Also

kruskalTest andshirleyWilliamsTestof the packagePMCMRplus,kruskal.test of the librarystats.

Examples

## Example from Sachs (1997, p. 402)x <- c(106, 114, 116, 127, 145,       110, 125, 143, 148, 151,       136, 139, 149, 160, 174)g <- gl(3,5)levels(g) <- c("A", "B", "C")## Chacko's testchackoTest(x, g)## Cuzick's testcuzickTest(x, g)## Johnson-Mehrotra testjohnsonTest(x, g)## Jonckheere-Terpstra testjonckheereTest(x, g)## Le's testleTest(x, g)## Spearman type testspearmanTest(x, g)## Murakami's BWS trend testbwsTrendTest(x, g)## Fligner-Wolfe testflignerWolfeTest(x, g)## Shan-Young-Kang testshanTest(x, g)

Least Significant Difference Test

Description

Performs the least significant difference all-pairs comparisonstest for normally distributed data with equal group variances.

Usage

lsdTest(x, ...)## Default S3 method:lsdTest(x, g, ...)## S3 method for class 'formula'lsdTest(formula, data, subset, na.action, ...)## S3 method for class 'aov'lsdTest(x, ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith normally distributed residuals and equal variancesthe least signifiant difference test can be performedafter a significant ANOVA F-test.LetX_{ij} denote a continuous random variablewith thej-the realization (1 \le j \le n_i)in thei-th group (1 \le i \le k). Furthermore, the totalsample size isN = \sum_{i=1}^k n_i. A total ofm = k(k-1)/2hypotheses can be tested: The null hypothesis isH_{ij}: \mu_i = \mu_j ~~ (i \ne j) is tested against the alternativeA_{ij}: \mu_i \ne \mu_j (two-tailed). Fisher's LSD all-pairs teststatistics are given by

t_{ij} \frac{\bar{X}_i - \bar{X_j}} {s_{\mathrm{in}} \left(1/n_j + 1/n_i\right)^{1/2}}, ~~ (i \ne j)

withs^2_{\mathrm{in}} the within-group ANOVA variance.The null hypothesis is rejected if|t_{ij}| > t_{v\alpha/2},withv = N - k degree of freedom. The p-values (two-tailed)are computed from theTDist distribution.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

As there is no p-value adjustment included, this function is equivalentto Fisher's protected LSD test, provided that the LSD test isonly applied after a significant one-way ANOVA F-test.If one is interested in other types of LSD test (i.e.with p-value adustment) see functionpairwise.t.test.

References

Sachs, L. (1997)Angewandte Statistik, New York: Springer.

See Also

TDist,pairwise.t.test

Examples

fit <- aov(weight ~ feed, chickwts)shapiro.test(residuals(fit))bartlett.test(weight ~ feed, chickwts)anova(fit)## also works with fitted objects of class aovres <- lsdTest(fit)summary(res)summaryGroup(res)

Mack-Wolfe Test for Umbrella Alternatives

Description

Performs Mack-Wolfe non-parametric test for umbrella alternatives.

Usage

mackWolfeTest(x, ...)## Default S3 method:mackWolfeTest(x, g, p = NULL, nperm = 1000, ...)## S3 method for class 'formula'mackWolfeTest(formula, data, subset, na.action, p = NULL, nperm = 1000, ...)

Arguments

Details

In dose-finding studies one may assume an increasing treatmenteffect with increasing dose level. However, the testsubject may actually succumb to toxic effects at high doses,which leads to decresing treatment effects.

The scope of the Mack-Wolfe Test is to test for umbrella alternativesfor either a known or unknown pointp (i.e. dose-level),where the peak (umbrella point) is present.

H_i: \theta_0 = \theta_i = \ldots = \theta_k is testedagainst the alternative A_i: \theta_1 \le \ldots \theta_p \ge\theta_k for somep, with at least one strict inequality.

Ifp = NULL (peak unknown), the upper-tailp-value is computedvia an asymptotic bootstrap permutation test.

If an integer value forp is given (peak known), theupper-tailp-value is computed from the standard normaldistribution (pnorm).

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

One may increase the number of permutations to e.g.nperm = 10000in order to get more precise p-values. However, this will be onthe expense of computational time.

References

Chen, I. Y. (1991) Notes on the Mack-Wolfe and Chen-WolfeTests for Umbrella Alternatives,Biom. J.33, 281–290.

Mack, G. A., Wolfe, D. A. (1981) K-sample rank tests forumbrella alternatives,J. Amer. Statist. Assoc.76, 175–181.

See Also

pnorm,sample.

Examples

## Example from Table 6.10 of Hollander and Wolfe (1999).## Plates with Salmonella bacteria of strain TA98 were exposed to## various doses of Acid Red 114 (in mu g / ml).## The data are the numbers of visible revertant colonies on 12 plates.## Assume a peak at D333 (i.e. p = 3).x <- c(22, 23, 35, 60, 59, 54, 98, 78, 50, 60, 82, 59, 22, 44,  33, 23, 21, 25)g <- as.ordered(rep(c(0, 100, 333, 1000, 3333, 10000), each=3))plot(x ~ g)mackWolfeTest(x=x, g=g, p=3)

Mandel's h Test According to E 691 ASTM

Description

The function calculates theconsistency statistics h and correspondingp-values for each group (lab) according toPractice E 691 ASTM.

Usage

mandelhTest(x, ...)## Default S3 method:mandelhTest(x, g, ...)## S3 method for class 'formula'mandelhTest(formula, data, subset, na.action, ...)

Arguments

Value

A list with class"mandel" containing the following components:

method

a character stringindicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

p.value

the p-value for the test.

statistic

the estimated quantiles of Mandel's statistic.

alternative

a character string describingthe alternative hypothesis.

grouplev

a character vector describing thelevels of the groups.

nrofrepl

the number of replicates for each group.

References

Practice E 691 (2005)Standard Practice forConducting an Interlaboratory Study to Determine thePrecision of a Test Method, ASTM International.

See Also

qmandelhpmandelh

Examples

data(Pentosan)mandelhTest(value ~ lab, data=Pentosan, subset=(material == "A"))

Mandel's k Test According to E 691 ASTM

Description

The function calculates theconsistency statistics k and correspondingp-values for each group (lab) according to Practice E 691 ASTM.

Usage

mandelkTest(x, ...)## Default S3 method:mandelkTest(x, g, ...)## S3 method for class 'formula'mandelkTest(formula, data, subset, na.action, ...)

Arguments

Value

A list with class"mandel" containing the following components:

method

a character stringindicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

p.value

the p-value for the test.

statistic

the estimated quantiles of Mandel's statistic.

alternative

a character string describingthe alternative hypothesis.

grouplev

a character vector describing thelevels of the groups.

nrofrepl

the number of replicates for each group.

References

Practice E 691 (2005)Standard Practice forConducting an Interlaboratory Study to Determine thePrecision of a Test Method, ASTM International.

See Also

qmandelkpmandelk

Examples

data(Pentosan)mandelkTest(value ~ lab, data=Pentosan, subset=(material == "A"))

Multiple Comparisons with One Control (U-test)

Description

Performs pairwise comparisons of multiple group levels withone control.

Usage

manyOneUTest(x, ...)## Default S3 method:manyOneUTest(  x,  g,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)## S3 method for class 'formula'manyOneUTest(  formula,  data,  subset,  na.action,  alternative = c("two.sided", "greater", "less"),  p.adjust.method = c("single-step", p.adjust.methods),  ...)

Arguments

Details

This functions performs Wilcoxon, Mann and Whitney's U-testfor a one factorial design where each factor level is tested againstone control (m = k -1 tests). As the data are re-rankedfor each comparison, this test is only suitable forbalanced (or almost balanced) experimental designs.

For the two-tailed test andp.adjust.method = "single-step"the multivariate normal distribution is used for controllingType 1 error and to calculate p-values. Otherwise,the p-values are calculated from the standard normal distributionwith any latter p-adjustment as available byp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels forg must be assigned in such a way,that they can be increasingly ordered from zero-dosecontrol to the highest dose level, e.g. integers{0, 1, 2, ..., k} or letters {a, b, c, ...}.Otherwise the function may not select the correct valuesfor intended zero-dose control.

It is safer, to i) label the factor levels as given above,and to ii) sort the data according to increasing dose-levelsprior to call the function (seeorder,factor).

References

OECD (ed. 2006)Current approaches in the statistical analysisof ecotoxicity data: A guidance to application, OECD Serieson testing and assessment, No. 54.

See Also

wilcox.test,pmvnorm,Normal

Examples

## Data set PlantGrowth## Global testkruskalTest(weight ~ group, data = PlantGrowth)## Conover's many-one comparison test## single-step means p-value from multivariate t distributionans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "single-step")summary(ans)## Conover's many-one comparison testans <- kwManyOneConoverTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Dunn's many-one comparison testans <- kwManyOneDunnTest(weight ~ group, data = PlantGrowth,                             p.adjust.method = "holm")summary(ans)## Nemenyi's many-one comparison testans <- kwManyOneNdwTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Many one U testans <- manyOneUTest(weight ~ group, data = PlantGrowth,                        p.adjust.method = "holm")summary(ans)## Chen Testans <- chenTest(weight ~ group, data = PlantGrowth,                    p.adjust.method = "holm")summary(ans)

Brown-Mood All Pairs Median Test

Description

Performs Brown-Mood All Pairs Median Test.

Usage

medianAllPairsTest(x, ...)## Default S3 method:medianAllPairsTest(x, g, p.adjust.method = p.adjust.methods, ...)## S3 method for class 'formula'medianAllPairsTest(  formula,  data,  subset,  na.action,  p.adjust.method = p.adjust.methods,  ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals Brown-Moodnon-parametric Median testcan be performed. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: \mu_i(x) = \mu_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: \mu_i(x) \ne \mu_j(x), ~~ i \ne j.

In this procedure the joined median is used for classification,but pairwise Pearson Chisquare-Tests are conducted. Any methodas given byp.adjust.methods can be usedto account for multiplicity.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Brown, G.W., Mood, A.M., 1951,On Median Tests for Linear Hypotheses,in:Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability.University of California Press, pp. 159–167.

See Also

chisq.test.

Examples

## Data set InsectSprays## Global testkruskalTest(count ~ spray, data = InsectSprays)## Conover's all-pairs comparison test## single-step means Tukey's p-adjustmentans <- kwAllPairsConoverTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "single-step")summary(ans)## Dunn's all-pairs comparison testans <- kwAllPairsDunnTest(count ~ spray, data = InsectSprays,                             p.adjust.method = "bonferroni")summary(ans)## Nemenyi's all-pairs comparison testans <- kwAllPairsNemenyiTest(count ~ spray, data = InsectSprays)summary(ans)## Brown-Mood all-pairs median testans <- medianAllPairsTest(count ~ spray, data = InsectSprays)summary(ans)

Brown-Mood Median Test

Description

Performs Brown-Mood Median Test.

Usage

medianTest(x, ...)## Default S3 method:medianTest(x, g, simulate.p.value = FALSE, B = 2000, ...)## S3 method for class 'formula'medianTest(  formula,  data,  subset,  na.action,  simulate.p.value = FALSE,  B = 2000,  ...)

Arguments

Details

The null hypothesis, H_0: \theta_1 = \theta_2 =\ldots = \theta_kis tested against the alternative,H_\mathrm{A}: \theta_i \ne \theta_j ~~(i \ne j), with at leastone unequality beeing strict.

Value

A list with class ‘htest’. For details seechisq.test.

References

Brown, G.W., Mood, A.M., 1951,On Median Tests for Linear Hypotheses,in:Proceedings of the Second Berkeley Symposium onMathematical Statistics and Probability.University of California Press, pp. 159–167.

See Also

chisq.test.

Examples

## Hollander & Wolfe (1973), 116.## Mucociliary efficiency from the rate of removal of dust in normal## subjects, subjects with obstructive airway disease, and subjects## with asbestosis.x <- c(2.9, 3.0, 2.5, 2.6, 3.2) # normal subjectsy <- c(3.8, 2.7, 4.0, 2.4)      # with obstructive airway diseasez <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosisg <- factor(x = c(rep(1, length(x)),                   rep(2, length(y)),                   rep(3, length(z))),             labels = c("ns", "oad", "a"))dat <- data.frame(   g = g,   x = c(x, y, z))## AD-TestadKSampleTest(x ~ g, data = dat)## BWS-TestbwsKSampleTest(x ~ g, data = dat)## Kruskal-Test## Using incomplete beta approximationkruskalTest(x ~ g, dat, dist="KruskalWallis")## Using chisquare distributionkruskalTest(x ~ g, dat, dist="Chisquare")## Not run: ## Check with kruskal.test from R statskruskal.test(x ~ g, dat)## End(Not run)## Using Conover's FkruskalTest(x ~ g, dat, dist="FDist")## Not run: ## Check with aov on ranksanova(aov(rank(x) ~ g, dat))## Check with oneway.testoneway.test(rank(x) ~ g, dat, var.equal = TRUE)## End(Not run)## Median Test asymptoticmedianTest(x ~ g, dat)## Median Test with simulated p-valuesset.seed(112)medianTest(x ~ g, dat, simulate.p.value = TRUE)

Madhava Rao-Raghunath Test for Testing Treatment vs. Control

Description

The function has implemented the nonparametric test ofMadhava Rao and Raghunath (2016) for testing paired two-samplesfor symmetry. The null hypothesisH: F(x,y) = F(y,x)is tested against the alternativeA: F(x,y) \ne F(y,x).

Usage

mrrTest(x, ...)## Default S3 method:mrrTest(x, y = NULL, m = NULL, ...)## S3 method for class 'formula'mrrTest(formula, data, subset, na.action, ...)

Arguments

Details

LetX_i andY_i, ~ i \le n denotecontinuous variables that were observedon the sameith test item (e.g. patient)withi = 1, \ldots n. Let

U_i = X_i + Y_i \qquad V_i = X_i - Y_i

LetU_{(i)} be theith order statistic,U_{(1)} \le U_{(2)} \le \ldots U_{(n)} andk thenumber of clusters, with the condition:

n = k ~ m.

Further, let the divider denoted_0 = -\infty,d_k = \infty, and else

d_j = \frac{ U_{(jm)} + U_{(jm+1)} }{2}, ~ 1 \le j \le k -1

The two counts are

n_j^{+} = \left\{ \begin{array}{lr} 1 & \mathrm{if}~ d_{j-1} < u_i < d_j, v_i > 0 \\ 0 & \end{array} \right.

and

n_j^{-} = \left\{ \begin{array}{lr} 1 & \mathrm{if}~ d_{j-1} < u_i < d_j, v_i \le 0 \\ 0 & \end{array} \right.

The test statistic is

M = \sum_{j = 1}^k \frac{\left(n_j^{+} - n_j^{-}\right)^2} {m}

The exact p-values for5 \le n \le 30 are taken from aninternal look-up table. The exact p-values were takenfrom Table 7, Appendix B of Madhava Rao and Raghunath (2016).

Ifm = NULL the function usesn = m forall prime numbers, otherwise it tries to find an value form in such a way, that fork = n / m all variablesare integer.

Value

A list with class"htest" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated quantile of the test statistic.

p.value

the p-value for the test.

parameter

the parameters of the test statistic, if any.

alternative

a character string describing the alternative hypothesis.

estimates

the estimates, if any.

null.value

the estimate under the null hypothesis, if any.

Note

The function returns an error code if a value formis provided that does not lead to an integer of the ratiok = n /m.

The function also returns an error code, if a tabulatedvalue for givenn,m and calculatedMcan not be found in the look-up table.

References

Madhava Rao, K.S., Ragunath, M. (2016) A Simple Nonparametric Testfor Testing Treatment Versus Control.J Stat Adv Theory Appl16,133–162.doi:10.18642/jsata_7100121717

Examples

## Madhava Rao and Raghunath (2016), p. 151## Inulin clearance of living donors## and recipients of their kidneysx <- c(61.4, 63.3, 63.7, 80.0, 77.3, 84.0, 105.0)y <- c(70.8, 89.2, 65.8, 67.1, 87.3, 85.1, 88.1)mrrTest(x, y)## formula method## Student's Sleep DatamrrTest(extra ~ group, data = sleep)

Lu-Smith All-Pairs Comparison Normal Scores Test

Description

Performs Lu-Smith all-pairs comparisonnormal scores test.

Usage

normalScoresAllPairsTest(x, ...)## Default S3 method:normalScoresAllPairsTest(  x,  g,  p.adjust.method = c("single-step", p.adjust.methods),  ...)## S3 method for class 'formula'normalScoresAllPairsTest(  formula,  data,  subset,  na.action,  p.adjust.method = c("single-step", p.adjust.methods),  ...)

Arguments

Details

For all-pairs comparisons in an one-factorial layoutwith non-normally distributed residuals Lu and Smith'snormal scores transformation can be used prior toan all-pairs comparison test. A total ofm = k(k-1)/2hypotheses can be tested. The null hypothesisH_{ij}: F_i(x) = F_j(x) is tested in the two-tailed testagainst the alternativeA_{ij}: F_i(x) \ne F_j(x), ~~ i \ne j.Forp.adjust.method = "single-step" theTukey's studentized range distribution is used to calculatep-values (seeTukey). Otherwise, thet-distribution is used for the calculation of p-valueswith a latter p-value adjustment asperformed byp.adjust.

Value

A list with class"PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimatedquantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-valueadjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

Lu, H., Smith, P. (1979) Distribution of normal scores statisticfor nonparametric one-way analysis of variance.Journal of the American Statistical Association74, 715–722.

See Also

normalScoresTest,normalScoresManyOneTest,normOrder.