# ansari.test

##### Ansari-Bradley Test

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

- Keywords
- htest

##### Usage

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

# S3 method for 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 \(s \ne 1\) (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:

the value of the Ansari-Bradley test statistic.

the p-value of the test.

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

a character string describing the alternative hypothesis.

the string `"Ansari-Bradley test"`

.

a character string giving the names of the data.

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

.)

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.
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.

##### 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

`library(stats)`

```
# 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.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)
# }
```

*Documentation reproduced from package stats, version 3.6.2, License: Part of R 3.6.2*