Package 'PMCMRplus'

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 layout with non-normally distributed residuals Anderson-Darling's all-pairs comparison test can be used. A total of $m = k(k-1)/2$ hypotheses can be tested. The null hypothesis H$_{ij}: F_i(x) = F_j(x)$ is tested in the two-tailed test against the alternative A$_{ij}: F_i(x) \ne F_j(x), ~~ i \ne j$.

This function is a wrapper function that sequentially calls adKSampleTest for each pair. The calculated p-values for Pr(>|T2N|) can be adjusted to account for Type I error multiplicity using any method as implemented in p.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 estimated quantiles 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-value adjustment.

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 Association 82, 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)

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$_0: F_1 = F_2 = \ldots = F_k$ is tested against the alternative, H$_\mathrm{A}: F_i \ne F_j ~~(i \ne j)$, with at least one unequality beeing strict.

This function only evaluates version 1 of the k-sample Anderson-Darling test (i.e. Eq. 6) of Scholz and Stephens (1987). The p-values are estimated with the extended empirical function as implemented in ad.pval of the package kSamples.

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 Association 82, 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 subjects
y <- c(3.8, 2.7, 4.0, 2.4)      # with obstructive airway disease
z <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosis
g <- 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-Test
adKSampleTest(x ~ g, data = dat)

## BWS-Test
bwsKSampleTest(x ~ g, data = dat)

## Kruskal-Test
## Using incomplete beta approximation
kruskalTest(x ~ g, dat, dist="KruskalWallis")
## Using chisquare distribution
kruskalTest(x ~ g, dat, dist="Chisquare")

## Not run: 
## Check with kruskal.test from R stats
kruskal.test(x ~ g, dat)

## End(Not run)
## Using Conover's F
kruskalTest(x ~ g, dat, dist="FDist")

## Not run: 
## Check with aov on ranks
anova(aov(rank(x) ~ g, dat))
## Check with oneway.test
oneway.test(rank(x) ~ g, dat, var.equal = TRUE)

## End(Not run)

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 distributed residuals Anderson-Darling's non-parametric test can be performed. Let there be $k$ groups including the control, then the number of treatment levels is $m = k - 1$. Then $m$ pairwise comparisons can be performed between the $i$-th treatment level and the control. H$_i: F_0 = F_i$ is tested in the two-tailed case against A$_i: F_0 \ne F_i, ~~ (1 \le i \le m)$.

This function is a wrapper function that sequentially calls adKSampleTest for each pair. The calculated p-values for Pr(>|T2N|) can be adjusted to account for Type I error inflation using any method as implemented in p.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 estimated quantiles 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-value adjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels for g must be assigned in such a way, that they can be increasingly ordered from zero-dose control 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 values for 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-levels prior to call the function (see order, factor).

References

Scholz, F.W., Stephens, M.A. (1987) K-Sample Anderson-Darling Tests. Journal of the American Statistical Association 82, 918–924.

See Also

adKSampleTest, adAllPairsTest, ad.pval.

Examples

## Data set PlantGrowth
## Global test
adKSampleTest(weight ~ group, data = PlantGrowth)

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

Description

A dose-response experiment was conducted using Atrazine at 9 different dose-levels including the zero-dose control and the biomass of algae (Selenastrum capricornutum) as the response variable. Three replicates were measured at day 0, 1 and 2. The fluorescence method (Mayer et al. 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 analysis of ecotoxicity data: A guidance to application - Annexes, OECD Series on testing and assessment, No. 54, (ENV/JM/MONO(2006)18/ANN).

See Also

demo(algae)

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 arithmetic mean. The extend of the whiskers are $\pm z_{(1-\alpha/2)} \times s_{\mathrm{E},i}$, where the latter denotes the standard error of the $i$th group. Symbolic letters are depicted on top of the bars, whereas different letters indicate significant differences between groups for the selected level of alpha.

Note

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

Examples

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

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 layout with non-normally distributed residuals Baumgartner-Weiß-Schindler all-pairs comparison test can be used. A total of $m = k(k-1)/2$ hypotheses can be tested. The null hypothesis H$_{ij}: F_i(x) = F_j(x)$ is tested in the two-tailed test against the alternative A$_{ij}: F_i(x) \ne F_j(x), ~~ i \ne j$.

This function is a wrapper function that sequentially calls bws_test for each pair. The default test method ("BWS") is the original Baumgartner-Weiß-Schindler test statistic B. For method == "Murakami" it is the modified BWS statistic denoted B*. The calculated p-values for Pr(>|B|) or Pr(>|B*|) can be adjusted to account for Type I error inflation using any method as implemented in p.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 estimated quantiles 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-value adjustment.

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 the general two-sample problem, Biometrics 54, 1129–1135.

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

See Also

bws_test.

Examples

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

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

Let $X_{ij} ~ (1 \le i \le k,~ 1 \le 1 \le n_i)$ denote an identically and independently distributed variable that is obtained from an unknown continuous distribution $F_i(x)$. Let $R_{ij}$ be the rank of $X_{ij}$, where $X_{ij}$ is jointly ranked from $1$ to $N, ~ N = \sum_{i=1}^k n_i$. In the $k$-sample test the null hypothesis, H: $F_i = F_j$ is tested against the alternative, A: $F_i \ne F_j ~~(i \ne j)$ with at least one inequality beeing strict. Murakami (2006) has generalized the two-sample Baumgartner-Weiß-Schindler test (Baumgartner et al. 1998) and proposed a modified statistic $B_k^*$ 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

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

The $p$-values are estimated via an assymptotic boot-strap method. It should be noted that the $B_k^*$ detects both differences in the unknown location parameters and / or differences in the unknown scale parameters of the $k$-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 = 10000 in order to get more precise p-values. However, this will be on the expense of computational time.

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for the general two-sample problem, Biometrics 54, 1129–1135.

Murakami, H. (2006) K-sample rank test based on modified Baumgartner statistic and its power comparison, 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 subjects
y <- c(3.8, 2.7, 4.0, 2.4)      # with obstructive airway disease
z <- c(2.8, 3.4, 3.7, 2.2, 2.0) # with asbestosis
g <- 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-Test
adKSampleTest(x ~ g, data = dat)

## BWS-Test
bwsKSampleTest(x ~ g, data = dat)

## Kruskal-Test
## Using incomplete beta approximation
kruskalTest(x ~ g, dat, dist="KruskalWallis")
## Using chisquare distribution
kruskalTest(x ~ g, dat, dist="Chisquare")

## Not run: 
## Check with kruskal.test from R stats
kruskal.test(x ~ g, dat)

## End(Not run)
## Using Conover's F
kruskalTest(x ~ g, dat, dist="FDist")

## Not run: 
## Check with aov on ranks
anova(aov(rank(x) ~ g, dat))
## Check with oneway.test
oneway.test(rank(x) ~ g, dat, var.equal = TRUE)

## End(Not run)

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 distributed residuals Baumgartner-Weiß-Schindler's non-parametric test can be performed. Let there be $k$ groups including the control, then the number of treatment levels is $m = k - 1$. Then $m$ pairwise comparisons can be performed between the $i$-th treatment level and the control. H$_i: F_0 = F_i$ is tested in the two-tailed case against A$_i: F_0 \ne F_i, ~~ (1 \le i \le m)$.

This function is a wrapper function that sequentially calls bws_stat and bws_cdf for each pair. For the default test method ("BWS") the original Baumgartner-Weiß-Schindler test statistic B and its corresponding Pr(>|B|) is calculated. For method == "BWS" only a two-sided test is possible.

For method == "Murakami" the modified BWS statistic denoted B* and its corresponding Pr(>|B*|) is computed by sequentially calling murakami_stat and murakami_cdf. For method == "Murakami" only a two-sided test is possible.

If alternative == "greater" then the alternative, if one population 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 its corresponding Pr(>B*) or Pr(<B*) is computed by sequentally calling murakami_stat and murakami_cdf with flavor = 2.

The p-values can be adjusted to account for Type I error inflation using any method as implemented in p.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 estimated quantiles 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-value adjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels for g must be assigned in such a way, that they can be increasingly ordered from zero-dose control 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 values for 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-levels prior to call the function (see order, factor).

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for the general two-sample problem, Biometrics 54, 1129–1135.

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

Neuhäuser, M. (2001) One-Side Two-Sample and Trend Tests Based on a Modified Baumgartner-Weiss-Schindler Statistic. J Nonparametric Stat 13, 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 test
set.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 BWStest
BWStest::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)

Description

Performs Murakami's modified Baumgartner-Weiß-Schindler test 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$_0: F_1(u) = F_2(u) = \ldots = F_k(u) ~~ u \in R$ is tested against a simple order hypothesis, H$_\mathrm{A}: F_1(u) \le F_2(u) \le \ldots \le F_k(u),~F_1(u) < F_k(u), ~~ u \in R$.

The p-values are estimated through an assymptotic boot-strap method using the function sample.

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 = 10000 in order to get more precise p-values. However, this will be on the expense of computational time.

Factor labels for g must be assigned in such a way, that they can be increasingly ordered from zero-dose control 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 values for 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-levels prior to call the function (see order, factor).

References

Baumgartner, W., Weiss, P., Schindler, H. (1998) A nonparametric test for the general two-sample problem, Biometrics 54, 1129–1135.

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

Neuhäuser, M. (2001) One-Side Two-Sample and Trend Tests Based on a Modified Baumgartner-Weiss-Schindler Statistic. J Nonparametric Stat 13, 729–739.

See Also

sample, bwsAllPairsTest, bwsManyOneTest.

kruskalTest and shirleyWilliamsTest of the package PMCMRplus, kruskal.test of the library stats.

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 test
chackoTest(x, g)

## Cuzick's test
cuzickTest(x, g)

## Johnson-Mehrotra test
johnsonTest(x, g)

## Jonckheere-Terpstra test
jonckheereTest(x, g)

## Le's test
leTest(x, g)

## Spearman type test
spearmanTest(x, g)

## Murakami's BWS trend test
bwsTrendTest(x, g)

## Fligner-Wolfe test
flignerWolfeTest(x, g)

## Shan-Young-Kang test
shanTest(x, g)

Description

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

According to the authors, the procedure shall only be applied after Chacko's test (see chackoTest) indicates global 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 against the alternative A$_{ij}: \theta_i < \theta_j$ for any $1 \le i < j \le k$.

Let $R_{ij}$ be the rank of $X_{ij}$, where $X_{ij}$ is jointly ranked from $\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i$, then the test statistics for all-pairs comparisons and a balanced design is calculated as

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

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

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

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-values as given in the tables of Hayter (1990) for $\alpha = 0.05$ (one-sided) are looked up according to the number of groups ($k-1$) and the degree of freedoms ($v = \infty$).

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

Value

Either a list of class "osrt" if dist = "h" or a list of class "PMCMR" if dist = "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 estimated quantiles 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-value adjustment.

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 the critical value $h_{k-1,\alpha,\infty} / \sqrt{2}$ if applicable.

References

Hayter, A. J.(1990) A One-Sided Studentised Range Test for Testing Against a Simple Ordered Alternative, J Amer Stat Assoc 85, 778–785.

Nashimoto, K., Wright, F.T. (2007) Nonparametric Multiple-Comparison Methods for Simply Ordered Medians. Comput Stat Data Anal 51, 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 table
shirleyWilliamsTest(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-test
NPMTest(x, g)

## Hayter-Stone test
hayterStoneTest(x, g)

## all-pairs comparisons
hsAllPairsTest(x, g)

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$_0: \theta_1 = \theta_2 = \ldots = \theta_k$ is tested against a simple order hypothesis, H$_\mathrm{A}: \theta_1 \le \theta_2 \le \ldots \le \theta_k,~\theta_1 < \theta_k$.

Let $R_{ij}$ be the rank of $X_{ij}$, where $X_{ij}$ is jointly ranked from $\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i$, 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 the $i$-th group and $\sigma_R^2 = N \left(N + 1\right) / 12$ the expected variance (without ties). H$_0$ is rejected, if $H > \chi^2_{v,\alpha}$ with $v = k -1$ degree of freedom. The p-values are estimated 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.

Source

The source code for the application of the pool adjacent violators theorem to calculate the isotonic means was taken from the file "pava.f", which is included in the package Iso:

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: Isotonic Regression in Two Independent Variables, Appl Statist 34, 352–357.

The Algorith AS 206 is available from StatLib http://lib.stat.cmu.edu/apstat/. The Royal Statistical Society holds the copyright to these routines, but has given its permission for their distribution provided that no fee is charged.

Note

Factor labels for g must be assigned in such a way, that they can be increasingly ordered from zero-dose control 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 values for 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-levels prior to call the function (see order, factor).

The function does neither check nor correct for ties.

References

Chacko, V. J. (1963) Testing homogeneity against ordered alternatives, Ann Math Statist 34, 945–956.

See Also

kruskalTest and shirleyWilliamsTest of the package PMCMRplus, kruskal.test of the library stats.

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 test
chackoTest(x, g)

## Cuzick's test
cuzickTest(x, g)

## Johnson-Mehrotra test
johnsonTest(x, g)

## Jonckheere-Terpstra test
jonckheereTest(x, g)

## Le's test
leTest(x, g)

## Spearman type test
spearmanTest(x, g)

## Murakami's BWS trend test
bwsTrendTest(x, g)

## Fligner-Wolfe test
flignerWolfeTest(x, g)

## Shan-Young-Kang test
shanTest(x, g)

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 for testing several treatment levels with a zero control. Let there be $k$ groups including the control and let the zero dose level be indicated with $i = 0$ and the highest dose level with $i = m$, then the following m = k - 1 hypotheses are tested:

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

Let $Y_{ij1}, Y_{ij2}, \ldots, Y_{ijn_{ij}}$ $(i = 1, 2, \dots, b, j = 0, 1, \ldots, k ~ \mathrm{and} ~ n_{ij} \geq 1)$ be a i.i.d. random variable of at least ordinal scale. Further,the zero dose control is indicated with $j = 0$.

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 returns $I(a) = 1, ~ \mathrm{if}~ a > 0, 0.5 ~ \mathrm{if} a = 0$ otherwise $0$.

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 of $T_j$ are

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

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

with $g_i$ the number of ties in the $i$th block and $t_u$ the size of the tied group $u$.

The test statistic $T_j^*$ is asymptotically multivariate normal distributed.

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

If p.adjust.method = "single-step" than the p-values are calculated with the probability function of the multivariate normal distribution with $\Sigma = I_k$. Otherwise the standard normal distribution is used to calculate p-values and any method as available by p.adjust or by the step-down procedure as proposed by Chen (1999), if p.adjust.method = "SD1" can be used to 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 estimated quantiles 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-value adjustment.

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 of the Minimum Effective Dose for Randomized Block Designs. Commun Stat-Simul Comput 31, 301–312.

See Also

Normal pmvnorm

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"))

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 for testing several treatment levels with a zero control. Let $X_{0j}$ denote a variable with the $j$-th realization of the control group ($1 \le j \le n_0$) and $X_{ij}$ the $j$-the realization in the $i$-th treatment group ($1 \le i \le k$). The variables are i.i.d. of a least ordinal scale with $F(x) = F(x_0) = F(x_i), ~ (1 \le i \le k)$. A total of $m = k$ 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 statistics $T_i$ 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 returns $I(a) = 1, ~ \mathrm{if}~ a > 0, 0.5 ~ \mathrm{if} a = 0$ otherwise $0$.

The expected $i$th mean is

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

with $N_j = \sum_{j =0}^i n_j$ and the $i$th 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 statistic $T_i^*$ 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 available by p.adjust or by the step-down procedure as proposed by Chen (1999), if p.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 estimated quantiles 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-value adjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

Note

Factor labels for g must be assigned in such a way, that they can be increasingly ordered from zero-dose control 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 values for 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-levels prior to call the function (see order, factor).

References

Chen, Y.-I., 1999, Nonparametric Identification of the Minimum Effective Dose. Biometrics 55, 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 level
df <- 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) <- 0:5
ans <- chenTest(x ~ g, data = df, alternative = "greater",
                p.adjust.method = "SD1")
summary(ans)

Description

Distribution function and quantile function for 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 and qcochran gives the quantile function.

References

Cochran, W.G. (1941) The distribution of the largest of a set of estimated variances 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)

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$_0: \sigma_1^2 = \sigma_2^2 = \ldots = \sigma_k^2$ is tested against the alternative (greater) H$_{\mathrm{A}}: \sigma_p > \sigma_i ~~ (i \le k, i \ne p)$ with at least one inequality being strict.

The p-value is computed with the function pcochran.

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 estimated variances 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"))

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$_0: \theta_1 = \theta_2 = \ldots = \theta_k$ is 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 for g must be assigned in such a way, that they can be increasingly ordered from zero-dose control 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 values for 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-levels prior to call the function (see order, factor).

References

Cuzick, J. (1995) A Wilcoxon-type test for trend, Statistics in Medicine 4, 87–90.

See Also

kruskalTest and shirleyWilliamsTest of the package PMCMRplus, kruskal.test of the library stats.

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 test
chackoTest(x, g)

## Cuzick's test
cuzickTest(x, g)

## Johnson-Mehrotra test
johnsonTest(x, g)

## Jonckheere-Terpstra test
jonckheereTest(x, g)

## Le's test
leTest(x, g)

## Spearman type test
spearmanTest(x, g)

## Murakami's BWS trend test
bwsTrendTest(x, g)

## Fligner-Wolfe test
flignerWolfeTest(x, g)

## Shan-Young-Kang test
shanTest(x, g)

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)

Description

Performs Grubbs double outlier test.

Usage

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

Arguments

Details

Let $X$ denote an identically and independently distributed continuous variate with realizations $x_i ~~ (1 \le i \le k)$. Further, let the increasingly ordered realizations denote $x_{(1)} \le x_{(2)} \le \ldots \le x_{(n)}$. Then the 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 proposed as

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

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

The p-value is computed with the function pdgrubbs.

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")

Description

Performs the all-pairs comparison test for different factor levels 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 layout with non-normally distributed residuals the DSCF all-pairs comparison test can be used. A total of $m = k(k-1)/2$ hypotheses can be tested. The null hypothesis H$_{ij}: F_i(x) = F_j(x)$ is tested in the two-tailed test against the alternative A$_{ij}: F_i(x) \ne F_j(x), ~~ i \ne j$. As opposed to the all-pairs comparison procedures that depend on Kruskal ranks, the DSCF test is basically an extension of the 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 estimated quantiles 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-value adjustment.

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 multiple comparisons in the one-way analysis of variance, Communications in Statistics - Theory and Methods 20, 127–139.

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

Steel, R. G. D. (1960) A rank sum test for comparing all pairs of treatments, Technometrics 2, 197–207

See Also

Tukey, pairwise.wilcox.test

Description

Performs Duncan's all-pairs comparisons test for normally distributed data 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 layout with normally distributed residuals and equal variances Duncan's multiple range test can be performed. Let $X_{ij}$ denote a continuous random variable with the $j$-the realization ($1 \le j \le n_i$) in the $i$-th group ($1 \le i \le k$). Furthermore, the total sample size is $N = \sum_{i=1}^k n_i$. A total of $m = k(k-1)/2$ hypotheses can be tested: The null hypothesis is H$_{ij}: \mu_i = \mu_j ~~ (i \ne j)$ is tested against the alternative A$_{ij}: \mu_i \ne \mu_j$ (two-tailed). Duncan's all-pairs test statistics are given by

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

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

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

with $v = N - k$ degree of freedom, the range $m' = 1 + |i - j|$ and $\alpha'$ the Bonferroni adjusted alpha-error. The p-values are computed from the Tukey 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 estimated quantiles 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-value adjustment.

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, Biometrics 11, 1–42.

See Also

Tukey, TukeyHSD tukeyTest

Examples

fit <- aov(weight ~ feed, chickwts)
shapiro.test(residuals(fit))
bartlett.test(weight ~ feed, chickwts)
anova(fit)

## also works with fitted objects of class aov
res <- duncanTest(fit)
summary(res)
summaryGroup(res)

Description

Performs Dunnett's all-pairs comparison test for normally distributed data 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, ...)