exactRankTests (version 0.8-35)

ansari.exact: Ansari-Bradley Test

Description

Performs the Ansari-Bradley two-sample test for a difference in scale parameters for possibly tied observations.

Usage

# S3 method for default
ansari.exact(x, y, alternative = c("two.sided", "less", "greater"),
            exact = NULL, conf.int = FALSE, conf.level = 0.95, …)
# S3 method for formula
ansari.exact(formula, data, subset, na.action, …)

Arguments

x

numeric vector of data values.

y

numeric vector of data values.

alternative

indicates the alternative hypothesis and must be one of "two.sided", "greater" or "less". You can specify just the initial letter.

exact

a logical indicating whether an exact p-value should be computed.

conf.int

a logical,indicating whether a confidence interval should be computed.

conf.level

confidence level of the interval.

formula

a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs a factor with two levels giving the corresponding groups.

data

an optional data frame containing the variables in the model 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").

…

further arguments to be passed to or from methods.

Value

A list with class "htest" containing the following components:

statistic

the value of the Ansari-Bradley test statistic.

p.value

the p-value of the test.

null.value

the ratio of scales \(s\) under the null, 1.

alternative

a character string describing the alternative hypothesis.

method

the string "Ansari-Bradley test".

data.name

a character string giving the names of the data.

conf.int

a confidence interval for the scale parameter. (Only present if argument conf.int = TRUE.)

estimate

an estimate of the ratio of scales. (Only present if argument conf.int = TRUE.)

Details

Suppose that x and y are independent samples from distributions with densities \(f((t-m)/s)/s\) and \(f(t-m)\), respectively, where \(m\) is an unknown nuisance parameter and \(s\), the ratio of scales, is the parameter of interest. The Ansari-Bradley test is used for testing the null that \(s\) equals 1, the two-sided alternative being that \(s \ne 1\) (the distributions differ only in variance), and the one-sided alternatives being \(s > 1\) (the distribution underlying x has a larger variance, "greater") or \(s < 1\) ("less").

By default (if exact is not specified), an exact p-value is computed if both samples contain less than 50 finite values. Otherwise, a normal approximation is used.

Optionally, a nonparametric confidence interval and an estimator for \(s\) 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.

References

Myles Hollander & Douglas A. Wolfe (1973), Nonparametric statistical inference. New York: John Wiley & Sons. Pages 83--92.

David F. Bauer (1972), Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67, 687--690.

See Also

fligner.test for a rank-based (nonparametric) \(k\)-sample test for homogeneity of variances; mood.test for another rank-based two-sample test for a difference in scale parameters; var.test and bartlett.test for parametric tests for the homogeneity in variance.

Examples

# NOT RUN {
## 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.exact(ramsay, jung.parekh)

ansari.exact(rnorm(20), rnorm(20, 0, 2), conf.int = TRUE)
# }