PMCMRplus (version 1.9.3)

shanTest: Testing against Ordered Alternatives (Shan-Young-Kang Test)

Description

Performs the Shan-Young-Kang test for testing against ordered alternatives.

Usage

shanTest(x, ...)
# S3 method for default
shanTest(x, g, alternative = c("greater", "less"), ...)
# S3 method for formula
shanTest(
  formula,
  data,
  subset,
  na.action,
  alternative = c("greater", "less"),
  ...
)

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".

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

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.

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 \(\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i\), the the test statistic is

$$ S = \sum_{i = 1}^{k-1} \sum_{j = i + 1}^k D_{ij}, $$

with $$ D_{ij} = \sum_{l = 1}^{n_i} \sum_{m=1}^{n_j} \left(R_{jm} - R_{il} \right)~ \mathrm{I}\left(X_{jm} > X_{il} \right), $$

where

$$ \mathrm{I}(u) = \left\{ \begin{array}{c} 1, \qquad \forall~ u > 0 \\ 0, \qquad \forall~ u \le 0 \end{array} \right. .$$

The test statistic is asymptotically normal distributed: $$ z = \frac{S - \mu_{\mathrm{S}}}{\sqrt{s^2_{\mathrm{S}}}} $$

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

References

Shan, G., Young, D., Kang, L. (2014) A New Powerful Nonparametric Rank Test for Ordered Alternative Problem. PLOS ONE 9, e112924. https://doi.org/10.1371/journal.pone.0112924

See Also

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

Examples

# NOT RUN {
## 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)

# }