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 & Pranab K. Sen (1999), Theory of Rank Tests. San Diego, London: Academic Press.

Myles Hollander & 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 2.8.1 Index]