swh:1:snp:813359ba77493c9d5dd1abad9a1f53490a8abf57
Tip revision: 60fafa14b471f505570c2c85e69cb2cf2e495536 authored by Torsten Hothorn on 15 June 2005, 00:00:00 UTC
version 0.2-13
version 0.2-13
Tip revision: 60fafa1
LocationTests.Rd
\name{LocationTests}
\alias{oneway_test}
\alias{oneway_test.formula}
\alias{oneway_test.IndependenceProblem}
\alias{wilcox_test.formula}
\alias{wilcox_test.IndependenceProblem}
\alias{wilcox_test}
\alias{normal_test.formula}
\alias{normal_test.IndependenceProblem}
\alias{normal_test}
\alias{median_test.formula}
\alias{median_test.IndependenceProblem}
\alias{median_test}
\alias{kruskal_test.formula}
\alias{kruskal_test.IndependenceProblem}
\alias{kruskal_test}
\title{ Independent Two- and K-Sample Location Tests }
\description{
Testing the equality of the distributions of a numeric response in
two or more independent groups against shift alternatives.
}
\usage{
\method{oneway_test}{formula}(formula, data, subset = NULL, weights = NULL, \dots)
\method{oneway_test}{IndependenceProblem}(object,
alternative = c("two.sided", "less", "greater"),
distribution = c("asymptotic", "approximate", "exact"), ...)
\method{wilcox_test}{formula}(formula, data, subset = NULL, weights = NULL, \dots)
\method{wilcox_test}{IndependenceProblem}(object,
alternative = c("two.sided", "less", "greater"),
distribution = c("asymptotic", "approximate", "exact"),
conf.int = FALSE, conf.level = 0.95, ...)
\method{normal_test}{formula}(formula, data, subset = NULL, weights = NULL, \dots)
\method{normal_test}{IndependenceProblem}(object,
alternative = c("two.sided", "less", "greater"),
distribution = c("asymptotic", "approximate", "exact"),
ties.method = c("mid-ranks", "average-scores"),
conf.int = FALSE, conf.level = 0.95, ...)
\method{median_test}{formula}(formula, data, subset = NULL, weights = NULL, \dots)
\method{median_test}{IndependenceProblem}(object,
alternative = c("two.sided", "less", "greater"),
distribution = c("asymptotic", "approximate", "exact"),
conf.int = FALSE, conf.level = 0.95, ...)
\method{kruskal_test}{formula}(formula, data, subset = NULL, weights = NULL, \dots)
\method{kruskal_test}{IndependenceProblem}(object,
distribution = c("asymptotic", "approximate"), ...)
}
\arguments{
\item{formula}{a formula of the form \code{y ~ x | block} where \code{y}
is a numeric variable giving the data values and \code{x} a factor
with two or more levels giving the corresponding groups. \code{block} is an
optional factor for stratification.}
\item{data}{an optional data frame containing the variables in the
model formula.}
\item{subset}{an optional vector specifying a subset of observations
to be used.}
\item{weights}{an optional formula of the form \code{~ w} defining
integer valued weights for the observations.}
\item{object}{an object of class \code{IndependenceProblem}.}
\item{alternative}{a character, the alternative hypothesis must be
one of \code{"two.sided"} (default), \code{"greater"} or
\code{"less"}. You can specify just the initial letter.}
\item{distribution}{a character, the null distribution of the test statistic
can be computed \code{exact}ly or can be approximated by its
asymptotic distribution (\code{asymptotic})
or via Monte-Carlo resampling (\code{approximate}).
Alternatively, the functions
\code{\link{exact}}, \code{\link{approximate}} or \code{\link{asymptotic}} can be
used to specify how the exact conditional distribution of the test statistic
should be calculated or approximated.}
\item{ties.method}{a character, two methods are available to adjust scores for ties,
either the score generating function is applied to \code{mid-ranks}
or the scores computed based on random ranks are averaged for all tied
values (\code{average-scores}).}
\item{conf.int}{a logical indicating whether a confidence interval
for the difference in location should be computed.}
\item{conf.level}{confidence level of the interval.}
\item{\dots}{further arguments to be passed to or from methods.}
}
\details{
The null hypothesis of the equality of the distribution of \code{y} in
the groups given by \code{x} is tested. In particular, the methods
documented here are designed to detect shift alternatives. For a general
description of the test procedures documented here we refer to Hollander &
Wolfe (1999).
The test procedures apply a rank transformation to the response values
\code{y}, except of \code{oneway_test} which computes a test statistic
using the untransformed response values.
The asymptotic null distribution is computed by default for all
procedures. Exact p-values may be computed for the two-sample problems and
can be approximated via Monte-Carlo resampling
for all procedures. Exact p-values
are computed either by the shift algorithm (Streitberg & Roehmel, 1986,
1987) or by the split-up algorithm (van de Wiel, 2001).
The linear rank tests for two samples (\code{wilcox_test},
\code{normal_test} and \code{median_test}) can be used to test the
two-sided hypothesis \eqn{H_0: Y_1 - Y_2 = 0}, where \eqn{Y_i} is the median
of the responses in the ith group. Confidence intervals for the difference
in location are available for the rank-based procedures and are computed
according to Bauer (1972). In case \code{alternative = "less"}, the
null hypothesis \eqn{H_0: Y_1 - Y_2 \ge 0} is tested and
\code{alternative = "greater"} corresponds to a null hypothesis
\eqn{H_0: Y_1 - Y_2 \le 0}.
In case \code{x} is an ordered factor, \code{kruskal_test} computes the
linear-by-linear association test for ordered alternatives.
For the adjustment of scores for tied values see Hajek, Sidak and Sen
(1999), page 131ff.
}
\value{
An object inheriting from class \code{\link{IndependenceTest-class}} with
methods \code{\link{show}}, \code{\link{statistic}}, \code{\link{expectation}},
\code{\link{covariance}} and \code{\link{pvalue}}. The null distribution
can be inspected by \code{\link{pperm}}, \code{\link{dperm}},
\code{\link{qperm}} and \code{\link{support}} methods. Confidence
intervals can be extracted by \code{confint}.
}
\references{
Myles Hollander & Douglas A. Wolfe (1999),
\emph{Nonparametric Statistical Methods, 2nd Edition}.
New York: John Wiley & Sons.
Bernd Streitberg & Joachim R\"ohmel (1986),
Exact distributions for permutations and rank tests:
An introduction to some recently published algorithms.
\emph{Statistical Software Newsletter} \bold{12}(1), 10--17.
Bernd Streitberg & Joachim R\"ohmel (1987),
Exakte Verteilungen f\"ur Rang- und Randomisierungstests
im allgemeinen $c$-Stichprobenfall.
\emph{EDV in Medizin und Biologie} \bold{18}(1), 12--19.
Mark A. van de Wiel (2001), The split-up algorithm: a fast
symbolic method for computing p-values of rank statistics.
\emph{Computational Statistics} \bold{16}, 519--538.
David F. Bauer (1972), Constructing confidence sets using
rank statistics. \emph{Journal of the American Statistical Association}
\bold{67}, 687--690.
Jaroslav Hajek, Zbynek Sidak & Pranab K. Sen (1999),
\emph{Theory of Rank Tests}. San Diego, London: Academic Press.
}
\examples{
### Tritiated Water Diffusion Across Human Chorioamnion
### Hollander & Wolfe (1999), Table 4.1, page 110
water_transfer <- data.frame(
pd = c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46,
1.15, 0.88, 0.90, 0.74, 1.21),
age = factor(c(rep("At term", 10), rep("12-26 Weeks", 5))))
### Wilcoxon-Mann-Whitney test, cf. Hollander & Wolfe (1999), page 111
### exact p-value and confidence interval for the difference in location
### (At term - 12-26 Weeks)
wt <- wilcox_test(pd ~ age, data = water_transfer,
distribution = "exact", conf.int = TRUE)
print(wt)
### extract observed Wilcoxon statistic, i.e, the sum of the
### ranks for age = "12-26 Weeks"
statistic(wt, "linear")
### its expectation
expectation(wt)
### and variance
covariance(wt)
### and, finally, the exact two-sided p-value
pvalue(wt)
### Confidence interval for difference (12-26 Weeks - At term)
wilcox_test(pd ~ age, data = water_transfer,
xtrafo = function(data)
trafo(data, factor_trafo = function(x) as.numeric(x == levels(x)[2])),
distribution = "exact", conf.int = TRUE)
### Permutation test, asymptotic p-value
oneway_test(pd ~ age, data = water_transfer)
### approximate p-value (with 99\% confidence interval)
pvalue(oneway_test(pd ~ age, data = water_transfer,
distribution = approximate(B = 9999)))
### exact p-value
pt <- oneway_test(pd ~ age, data = water_transfer, distribution = "exact")
pvalue(pt)
### plot density and distribution of the standardised
### test statistic
layout(matrix(1:2, nrow = 2))
s <- support(pt)
d <- sapply(s, function(x) dperm(pt, x))
p <- sapply(s, function(x) pperm(pt, x))
plot(s, d, type = "S", xlab = "Teststatistic", ylab = "Density")
plot(s, p, type = "S", xlab = "Teststatistic", ylab = "Cumm. Probability")
### Length of YOY Gizzard Shad from Kokosing Lake, Ohio,
### sampled in Summer 1984, Hollander & Wolfe (1999), Table 6.3, page 200
YOY <- data.frame(length = c(46, 28, 46, 37, 32, 41, 42, 45, 38, 44,
42, 60, 32, 42, 45, 58, 27, 51, 42, 52,
38, 33, 26, 25, 28, 28, 26, 27, 27, 27,
31, 30, 27, 29, 30, 25, 25, 24, 27, 30),
site = factor(c(rep("I", 10), rep("II", 10),
rep("III", 10), rep("IV", 10))))
### Kruskal-Wallis test, approximate exact p-value
kw <- kruskal_test(length ~ site, data = YOY,
distribution = approximate(B = 9999))
kw
pvalue(kw)
### Nemenyi-Damico-Wolfe-Dunn test (joint ranking)
### Hollander & Wolfe (1999), page 244
### (where Steel-Dwass results are given)
if (require(multcomp)) {
NDWD <- oneway_test(length ~ site, data = YOY,
ytrafo = function(data) trafo(data, numeric_trafo = rank),
xtrafo = function(data) trafo(data, factor_trafo = function(x)
model.matrix(~x - 1) \%*\% t(contrMat(table(x), "Tukey"))),
teststat = "maxtype", distribution = approximate(B = 90000))
### global p-value
print(pvalue(NDWD))
### sites (I = II) != (III = IV) at alpha = 0.01 (page 244)
print(pvalue(NDWD, adjusted = TRUE))
}
}
\keyword{htest}