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PMCMRplus (version 1.9.3)

hsAllPairsTest: Hayter-Stone All-Pairs Comparison Test

Description

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

Usage

hsAllPairsTest(x, ...)

# S3 method for default hsAllPairsTest( x, g, alternative = c("greater", "less"), method = c("look-up", "boot", "asympt"), nperm = 10000, ... )

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

Arguments

x

a numeric vector of data values, or a list of numeric data vectors.

further arguments to be passed to or from methods.

g

a vector or factor object giving the group for the corresponding elements of "x". Ignored with a warning if "x" is a list.

alternative

the alternative hypothesis. Defaults to greater.

method

a character string specifying the test statistic to use. Defaults to "look-up" that uses published Table values of Williams (1972).

nperm

number of permutations for the asymptotic permutation test. Defaults to 1000. Ignored, if method = "look-up".

formula

a formula of the form response ~ group where response gives the data values and group a vector or factor of the corresponding groups.

data

an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).

subset

an optional vector specifying a subset of observations to be used.

na.action

a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Value

Either a list of class "PMCMR" or a list with class "osrt" that contains 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 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.

Details

Let \(X\) be an identically and idepentendly distributed variable that was \(n\) times observed at \(k\) increasing treatment levels. Hayter and Stone (1991) proposed a non-parametric procedure to 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 statistic \(S_{ij}\) is distributed as \(h_{k,\alpha,v}\). Thus, the null hypothesis is rejected, if \(S_{ij} > h_{k,\alpha,v}\), with \(v = \infty\) degree of freedom.

If method = "look-up" the function will 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 (\(k\)) and the degree of freedoms (\(v = \infty\)).

If method = "boot" an asymetric permutation test is conducted and \(p\)-values are returned.

If method = "asympt" is selected the asymptotic \(p\)-value is estimated as implemented in the function pHayStonLSA of the package NSM3.

References

Hayter, A. J.(1990) A One-Sided Studentised Range Test for Testing Against a Simple Ordered Alternative, Journal of the American Statistical Association 85, 778--785.

Hayter, A.J., Stone, G. (1991) Distribution free multiple comparisons for monotonically ordered treatment effects. Austral J Statist 33, 335--346.

See Also

hayterStoneTest sample

Examples

Run this code
# NOT RUN {
## 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)
# }

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