PMCMRplus (version 1.9.10)

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"),
  ...
)

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.

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").

Details

The null hypothesis, H${}_0:{\theta }_1={\theta }_2=\dots ={\theta }_k$ is tested against a simple order hypothesis, H${}_{\mathrm{A}}:{\theta }_1\le {\theta }_2\le \dots \le {\theta }_k,{\theta }_1<{\theta }_k$.

Let $R_{ij}$ be the rank of $X_{ij}$, where $X_{ij}$ is jointly ranked from $\{1,2,\dots ,N\},N=\sum _{i=1}^k{n}_i$, the the test statistic is

$$S=\sum _{i=1}^{k-1}\sum _{j=i+1}^k{D}_{ij},$$

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

where

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

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

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

## 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)