\HeaderA{ansari.test}{Ansari-Bradley Test}{ansari.test}
\methaliasA{ansari.test.default}{ansari.test}{ansari.test.default}
\methaliasA{ansari.test.formula}{ansari.test}{ansari.test.formula}
\keyword{htest}{ansari.test}
\begin{Description}\relax
Performs the Ansari-Bradley two-sample test for a difference in scale
parameters.
\end{Description}
\begin{Usage}
\begin{verbatim}
ansari.test(x, ...)

## Default S3 method:
ansari.test(x, y, alternative = c("two.sided", "less", "greater"),
            exact = NULL, conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'formula':
ansari.test(formula, data, subset, na.action, ...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] numeric vector of data values.
\item[\code{y}] numeric vector of data values.
\item[\code{alternative}] indicates the alternative hypothesis and must be
one of \code{"two.sided"}, \code{"greater"} or \code{"less"}.  You
can specify just the initial letter.
\item[\code{exact}] a logical indicating whether an exact \eqn{p}{}-value
should be computed.
\item[\code{conf.int}] a logical,indicating whether a confidence interval
should be computed.
\item[\code{conf.level}] confidence level of the interval.
\item[\code{formula}] a formula of the form \code{lhs \textasciitilde{} rhs} where \code{lhs}
is a numeric variable giving the data values and \code{rhs} a factor
with two levels giving the corresponding groups.
\item[\code{data}] an optional matrix or data frame (or similar: see
\code{\LinkA{model.frame}{model.frame}}) containing the variables in the
formula \code{formula}.  By default the variables are taken from
\code{environment(formula)}.
\item[\code{subset}] an optional vector specifying a subset of observations
to be used.
\item[\code{na.action}] a function which indicates what should happen when
the data contain \code{NA}s.  Defaults to
\code{getOption("na.action")}.
\item[\code{...}] further arguments to be passed to or from methods.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
Suppose that \code{x} and \code{y} are independent samples from
distributions with densities \eqn{f((t-m)/s)/s}{} and \eqn{f(t-m)}{},
respectively, where \eqn{m}{} is an unknown nuisance parameter and
\eqn{s}{}, the ratio of scales, is the parameter of interest.  The
Ansari-Bradley test is used for testing the null that \eqn{s}{} equals
1, the two-sided alternative being that \eqn{s \ne 1}{s != 1} (the
distributions differ only in variance), and the one-sided alternatives
being \eqn{s > 1}{} (the distribution underlying \code{x} has a larger
variance, \code{"greater"}) or \eqn{s < 1}{} (\code{"less"}).

By default (if \code{exact} is not specified), an exact \eqn{p}{}-value
is computed if both samples contain less than 50 finite values and
there are no ties.  Otherwise, a normal approximation is used.

Optionally, a nonparametric confidence interval and an estimator for
\eqn{s}{} are computed.  If exact \eqn{p}{}-values are available, an exact
confidence interval is obtained by the algorithm described in Bauer
(1972), and the Hodges-Lehmann estimator is employed.  Otherwise, the
returned confidence interval and point estimate are based on normal
approximations.

Note that mid-ranks are used in the case of ties rather than average
scores as employed in Hollander \& Wolfe (1973).  See, e.g., Hajek,
Sidak and Sen (1999), pages 131ff, for more information.
\end{Details}
\begin{Value}
A list with class \code{"htest"} containing the following components:
\begin{ldescription}
\item[\code{statistic}] the value of the Ansari-Bradley test statistic.
\item[\code{p.value}] the \eqn{p}{}-value of the test.
\item[\code{null.value}] the ratio of scales \eqn{s}{} under the null, 1.
\item[\code{alternative}] a character string describing the alternative
hypothesis.
\item[\code{method}] the string \code{"Ansari-Bradley test"}.
\item[\code{data.name}] a character string giving the names of the data.
\item[\code{conf.int}] a confidence interval for the scale parameter.
(Only present if argument \code{conf.int = TRUE}.)
\item[\code{estimate}] an estimate of the ratio of scales.
(Only present if argument \code{conf.int = TRUE}.)
\end{ldescription}
\end{Value}
\begin{Note}\relax
To compare results of the Ansari-Bradley test to those of the F test
to compare two variances (under the assumption of normality), observe
that \eqn{s}{} is the ratio of scales and hence \eqn{s^2}{} is the ratio
of variances (provided they exist), whereas for the F test the ratio
of variances itself is the parameter of interest.  In particular,
confidence intervals are for \eqn{s}{} in the Ansari-Bradley test but
for \eqn{s^2}{} in the F test.
\end{Note}
\begin{References}\relax
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.

Myles Hollander \& Douglas A. Wolfe (1973),
\emph{Nonparametric statistical inference}.
New York: John Wiley \& Sons.
Pages 83--92.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{fligner.test}{fligner.test}} for a rank-based (nonparametric)
\eqn{k}{}-sample test for homogeneity of variances;
\code{\LinkA{mood.test}{mood.test}} for another rank-based two-sample test for a
difference in scale parameters;
\code{\LinkA{var.test}{var.test}} and \code{\LinkA{bartlett.test}{bartlett.test}} for parametric
tests for the homogeneity in variance.

\code{\LinkA{ansari\_test}{ansari.Rul.test}} in package \pkg{coin} for exact and
approximate \emph{conditional} \eqn{p}{}-values for the Ansari-Bradley
test, as well as different methods for handling ties.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## Hollander & Wolfe (1973, p. 86f):
## Serum iron determination using Hyland control sera
ramsay <- c(111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
            101, 96, 97, 102, 107, 113, 116, 113, 110, 98)
jung.parekh <- c(107, 108, 106, 98, 105, 103, 110, 105, 104,
            100, 96, 108, 103, 104, 114, 114, 113, 108, 106, 99)
ansari.test(ramsay, jung.parekh)

ansari.test(rnorm(10), rnorm(10, 0, 2), conf.int = TRUE)

## try more points - failed in 2.4.1
ansari.test(rnorm(100), rnorm(100, 0, 2), conf.int = TRUE)
\end{ExampleCode}
\end{Examples}

