ansari.test              package:stats              R Documentation

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
          '"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 'NA's.  Defaults to 'getOption("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 '"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'.)

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 & Pranab K. Sen (1999), _Theory of
     Rank Tests_. San Diego, London: Academic Press.

     Myles Hollander & 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)

