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
wherelhs
is a numeric variable giving the data values andrhs
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 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
NA
s. 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 \(s \ne 1\) (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)
# }