Ansari-Bradley Test

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

Usage

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

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.

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.)

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 \(s\) is the ratio of scales and hence \(s^2\) 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 \(s\) in the Ansari-Bradley test but for \(s^2\) in the F test.

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.

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)
#> Warning: cannot compute exact p-value with ties
#> 
#> 	Ansari-Bradley test
#> 
#> data:  ramsay and jung.parekh
#> AB = 185.5, p-value = 0.1815
#> alternative hypothesis: true ratio of scales is not equal to 1
#> 
ansari.exact(ramsay, jung.parekh)
#> 
#> 	Ansari-Bradley test
#> 
#> data:  ramsay and jung.parekh
#> AB = 185.5, p-value = 0.1881
#> alternative hypothesis: true ratio of scales is not equal to 1
#> 

ansari.exact(rnorm(20), rnorm(20, 0, 2), conf.int = TRUE)
#> 
#> 	Ansari-Bradley test
#> 
#> data:  rnorm(20) and rnorm(20, 0, 2)
#> AB = 219, p-value = 0.6488
#> alternative hypothesis: true ratio of scales is not equal to 1
#> 95 percent confidence interval:
#>  0.3263856 2.9455026
#> sample estimates:
#> ratio of scales 
#>       0.8317342 
#>