| % File src/library/stats/man/ansari.test.Rd |
| % Part of the R package, https://www.R-project.org |
| % Copyright 1995-2018 R Core Team |
| % Distributed under GPL 2 or later |
| |
| \name{ansari.test} |
| \alias{ansari.test} |
| \alias{ansari.test.default} |
| \alias{ansari.test.formula} |
| \title{Ansari-Bradley Test} |
| \description{ |
| Performs the Ansari-Bradley two-sample test for a difference in scale |
| parameters. |
| } |
| \usage{ |
| ansari.test(x, \dots) |
| |
| \method{ansari.test}{default}(x, y, |
| alternative = c("two.sided", "less", "greater"), |
| exact = NULL, conf.int = FALSE, conf.level = 0.95, |
| \dots) |
| |
| \method{ansari.test}{formula}(formula, data, subset, na.action, \dots) |
| } |
| \arguments{ |
| \item{x}{numeric vector of data values.} |
| \item{y}{numeric vector of data values.} |
| \item{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{exact}{a logical indicating whether an exact p-value |
| should be computed.} |
| \item{conf.int}{a logical,indicating whether a confidence interval |
| should be computed.} |
| \item{conf.level}{confidence level of the interval.} |
| \item{formula}{a formula of the form \code{lhs ~ 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{data}{an optional matrix or data frame (or similar: see |
| \code{\link{model.frame}}) containing the variables in the |
| formula \code{formula}. By default the variables are taken from |
| \code{environment(formula)}.} |
| \item{subset}{an optional vector specifying a subset of observations |
| to be used.} |
| \item{na.action}{a function which indicates what should happen when |
| the data contain \code{NA}s. Defaults to |
| \code{getOption("na.action")}.} |
| \item{\dots}{further arguments to be passed to or from methods.} |
| } |
| \details{ |
| 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 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 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. |
| } |
| \value{ |
| A list with class \code{"htest"} containing the following components: |
| \item{statistic}{the value of the Ansari-Bradley test statistic.} |
| \item{p.value}{the p-value of the test.} |
| \item{null.value}{the ratio of scales \eqn{s} under the null, 1.} |
| \item{alternative}{a character string describing the alternative |
| hypothesis.} |
| \item{method}{the string \code{"Ansari-Bradley test"}.} |
| \item{data.name}{a character string giving the names of the data.} |
| \item{conf.int}{a confidence interval for the scale parameter. |
| (Only present if argument \code{conf.int = TRUE}.)} |
| \item{estimate}{an estimate of the ratio of scales. |
| (Only present if argument \code{conf.int = TRUE}.)} |
| } |
| \note{ |
| 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. |
| } |
| \references{ |
| David F. Bauer (1972). |
| Constructing confidence sets using rank statistics. |
| \emph{Journal of the American Statistical Association}, |
| \bold{67}, 687--690. |
| \doi{10.1080/01621459.1972.10481279}. |
| |
| Jaroslav Hajek, Zbynek Sidak and Pranab K. Sen (1999). |
| \emph{Theory of Rank Tests}. |
| San Diego, London: Academic Press. |
| |
| Myles Hollander and Douglas A. Wolfe (1973). |
| \emph{Nonparametric Statistical Methods}. |
| New York: John Wiley & Sons. |
| Pages 83--92. |
| } |
| \seealso{ |
| \code{\link{fligner.test}} for a rank-based (nonparametric) |
| \eqn{k}-sample test for homogeneity of variances; |
| \code{\link{mood.test}} for another rank-based two-sample test for a |
| difference in scale parameters; |
| \code{\link{var.test}} and \code{\link{bartlett.test}} for parametric |
| tests for the homogeneity in variance. |
| |
| \code{\link[coin:ScaleTests]{ansari_test}} in package \CRANpkg{coin} |
| for exact and approximate \emph{conditional} p-values for the |
| Ansari-Bradley test, as well as different methods for handling ties. |
| } |
| \examples{ |
| ## 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) |
| } |
| \keyword{htest} |