ansari.test
Ansari-Bradley Test
Description
Performs the Ansari-Bradley two-sample test for a difference in scale parameters.
Usage
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, ...)
Arguments
x | numeric vector of data values. |
y | numeric vector of data values. |
alternative | indicates the alternative hypothesis and must be one of |
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 |
data | an optional matrix or data frame (or similar: see |
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 |
... | 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 "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 |
data.name | a character string giving the names of the data. |
conf.int | a confidence interval for the scale parameter. (Only present if argument |
estimate | an estimate of the ratio of scales. (Only present if argument |
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. doi: 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
## 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)
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Licensed under the GNU General Public License.