ansari.test {stats} R Documentation

### Description

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

### Usage

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, ...)


### 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 matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(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 and there are no ties. 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.

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 "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

David F. Bauer (1972). Constructing confidence sets using rank statistics. Journal of the American Statistical Association, 67, 687–690. doi:10.1080/01621459.1972.10481279.

Jaroslav Hajek, Zbynek Sidak and Pranab K. Sen (1999). Theory of Rank Tests. San Diego, London: Academic Press.

Myles Hollander and Douglas A. Wolfe (1973). Nonparametric Statistical Methods. New York: John Wiley & Sons. Pages 83–92.

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.

ansari_test in package coin for exact and approximate 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)


[Package stats version 4.3.0 Index]