PMCMRplus (version 1.9.3)

chackoTest: Testing against Ordered Alternatives (Chacko's Test)

Description

Performs Chacko's test for testing against ordered alternatives.

Usage

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

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 \(\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i\), then the test statistic is calculated as $$ H = \frac{1}{\sigma_R^2} \sum_{i=1}^k n_i \left(\bar{R^*}_i - \bar{R}\right), $$

where \(\bar{R^*}_i\) is the isotonic mean of the \(i\)-th group and \(\sigma_R^2 = N \left(N + 1\right) / 12\) the expected variance (without ties). H\(_0\) is rejected, if \(H > \chi^2_{v,\alpha}\) with \(v = k -1\) degree of freedom. The p-values are estimated from the chi-square distribution.

References

Chacko, V. J. (1963) Testing homogeneity against ordered alternatives, Ann Math Statist 34, 945--956.

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)

# }