ansari.test
AnsariBradley Test
Performs the AnsariBradley twosample 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 pvalue 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 $f((tm)/s)/s$ and $f(tm)$,
respectively, where $m$ is an unknown nuisance parameter and
$s$, the ratio of scales, is the parameter of interest. The
AnsariBradley test is used for testing the null that $s$ equals
1, the twosided alternative being that $s != 1$ (the
distributions differ only in variance), and the onesided 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 pvalue
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 pvalues are available, an exact confidence interval is obtained by the algorithm described in Bauer (1972), and the HodgesLehmann estimator is employed. Otherwise, the returned confidence interval and point estimate are based on normal approximations.
Note that midranks 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 AnsariBradley test statistic.
 p.value
 the pvalue 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
"AnsariBradley 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 AnsariBradley 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 AnsariBradley 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, 687690.
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 8392.
See Also
fligner.test
for a rankbased (nonparametric)
$k$sample test for homogeneity of variances;
mood.test
for another rankbased twosample 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.Rproject.org/package=#1}{\pkg{#1}}coincoin
for exact and approximate conditional pvalues for the
AnsariBradley 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)