ansari.test
Ansari-Bradley Test
Performs the Ansari-Bradley two-sample test for a difference in scale parameters.
- Keywords
- htest
Usage
ansari.test(x, ...)
"ansari.test"(x, y, alternative = c("two.sided", "less", "greater"), exact = NULL, conf.int = FALSE, conf.level = 0.95, ...)
"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 ~ rhswherelhsis a numeric variable giving the data values andrhsa 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 formulaformula. By default the variables are taken fromenvironment(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 togetOption("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
- 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.)
"htest" containing the following components:
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 \href{https://CRAN.R-project.org/package=#1}{\pkg{#1}}coincoin
for exact and approximate conditional p-values for the
Ansari-Bradley test, as well as different methods for handling ties.
Examples
library(stats)
## 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)