PMCMRplus (version 1.9.3)

hayterStoneTest: Hayter-Stone Test

Description

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

Usage

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

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 htest 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.

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: \(\theta_i < \theta_j\), with at least one 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 statistic \(h\) is distributed as \(h_{k,\alpha,v}\). Thus, the null hypothesis is rejected, if \(h > h_{k,\alpha,v}\), 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 asymptotic permutation test is conducted and a \(p\)-value is 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, J Amer Stat Assoc 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

osrtTest, hsAllPairsTest, sample, pHayStonLSA

Examples

# 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)
# }