| chackoTest {PMCMRplus} | R Documentation |
Testing against Ordered Alternatives (Chacko's Test)
Description
Performs Chacko's test for testing against ordered alternatives.
Usage
chackoTest(x, ...)
## Default S3 method:
chackoTest(x, g, alternative = c("greater", "less"), ...)
## S3 method for class '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 |
alternative |
the alternative hypothesis. Defaults to |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
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 |
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.
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.
Source
The source code for the application of the pool adjacent violators
theorem to calculate the isotonic means
was taken from the file "pava.f", which is included in the
package Iso:
Rolf Turner (2015). Iso: Functions to Perform Isotonic Regression. R package version 0.0-17. https://CRAN.R-project.org/package=Iso.
The file "pava.f" is a Ratfor modification of Algorithm AS 206.1:
Bril, G., Dykstra, R., Pillers, C., Robertson, T. (1984) Statistical Algorithms: Algorithm AS 206: Isotonic Regression in Two Independent Variables, Appl Statist 34, 352–357.
The Algorith AS 206 is available from StatLib http://lib.stat.cmu.edu/apstat/. The Royal Statistical Society holds the copyright to these routines, but has given its permission for their distribution provided that no fee is charged.
Note
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).
The function does neither check nor correct for ties.
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
## 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)