R: Ansari-Bradley Test

ansari.test {stats} | R Documentation

## Ansari-Bradley Test

### 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 `NA`s. 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 != 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.

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.

### 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.

`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 2.15.1 Index]