| ansari.test {stats} | R Documentation |
Performs the Ansari-Bradley two-sample test for a difference in scale parameters.
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, ...)
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 NAs. Defaults to
getOption("na.action"). |
... |
further arguments to be passed to or from methods. |
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.
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.) |
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.
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 inference. New York: John Wiley & Sons. Pages 83–92.
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.
## 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)