shanTest: Testing against Ordered Alternatives (Shan-Young-Kang Test)
In PMCMRplus: Calculate Pairwise Multiple Comparisons of Mean Rank Sums Extended

shanTest

R Documentation

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

Description

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

Usage

shanTest(x, ...)

## Default S3 method:
shanTest(x, g, alternative = c("greater", "less"), ...)

## S3 method for class '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 `NA`s. Defaults to `getOption("na.action")`.

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.

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

The variance estimation (see Theorem 2.1, Shan et al. 2014) can become negative for certain combinations of N,~n_i,~k \qquad (1 \le i \le k). In these cases the function will return a warning and the returned p-value will be NaN.

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

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

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)

PMCMRplus documentation built on Nov. 27, 2023, 1:08 a.m.