coin (version 1.0-24)

ScaleTests: Independent Two- and K-Sample Scale Tests

Description

Testing the equality of the distributions of a numeric response in two or more independent groups against scale alternatives.

Usage

## S3 method for class 'formula':
ansari_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
ansari_test(object, 
    alternative = c("two.sided", "less", "greater"),
    ties.method = c("mid-ranks", "average-scores"),
    conf.int = FALSE, conf.level = 0.95, ...)
## S3 method for class 'formula':
fligner_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
fligner_test(object, 
    ties.method = c("mid-ranks", "average-scores"),
    distribution = c("asymptotic", "approximate"), 
    ...)

Arguments

formula

a formula of the form y ~ x | block where y is a numeric variable giving the data values and x a factor with two or more levels giving the corresponding groups. block is an optional factor fo

data

an optional data frame containing the variables in the model formula.

subset

an optional vector specifying a subset of observations to be used.

weights

an optional formula of the form ~ w defining integer valued weights for the observations.

object

an object of class IndependenceProblem.

alternative

a character, the alternative hypothesis must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.

distribution

a character, the null distribution of the test statistic can be computed exactly or can be approximated by its asymptotic distribution (asymptotic) or via Monte-Carlo resampling (approximate). Alte

ties.method

a character, two methods are available to adjust scores for ties, either the score generating function is applied to mid-ranks or the scores computed based on random ranks are averaged for all tied values (average-scor

conf.int

a logical indicating whether a confidence interval for the difference in location should be computed.

conf.level

confidence level of the interval.

...

further arguments to be passed to or from methods.

Value

• An object inheriting from class IndependenceTest-class with methods show, statistic, expectation, covariance and pvalue. The null distribution can be inspected by pperm, dperm, qperm and support methods. Confidence intervals can be extracted by confint.

Details

The null hypothesis of the equality of the distribution of y in the groups given by x is tested. In particular, the methods documented here are designed to detect scale alternatives. For a general description of the test procedures documented here we refer to Hollander & Wolfe (1999).

The asymptotic null distribution is computed by default for both procedures. Exact p-values may be computed for the Ansari-Bradley test can be approximated via Monte-Carlo for the Fligner-Killeen procedure. Exact p-values are computed either by the shift algorithm (Streitberg & R"ohmel, 1986, 1987) or by the split-up algorithm (van de Wiel, 2001).

The Ansari-Bradley test can be used to test the two-sided hypothesis $var(Y_1) / var(Y_2) = 1$, where $var(Y_i)$ is the variance of the responses in the ith group. Confidence intervals for the ratio of scales are available for the Ansari-Bradley test and are computed according to Bauer (1972). In case alternative = "less", the null hypothesis $var(Y_1) / var(Y_2) \ge 1$ is tested and alternative = "greater" corresponds to $var(Y_1) / var(Y_2) \le 1$.

For the adjustment of scores for tied values see Hajek, Sidak and Sen (1999), page 131ff.

References

Myles Hollander & Douglas A. Wolfe (1999). Nonparametric Statistical Methods, 2nd Edition. New York: John Wiley & Sons.

Bernd Streitberg & Joachim R"ohmel (1986). Exact distributions for permutations and rank tests: An introduction to some recently published algorithms. Statistical Software Newsletter 12(1), 10--17.

Bernd Streitberg & Joachim R"ohmel (1987). Exakte Verteilungen f"ur Rang- und Randomisierungstests im allgemeinen $c$-Stichprobenfall. EDV in Medizin und Biologie 18(1), 12--19.

Mark A. van de Wiel (2001). The split-up algorithm: a fast symbolic method for computing p-values of rank statistics. Computational Statistics 16, 519--538.

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.

Examples

### Serum Iron Determination Using Hyland Control Sera
  ### Hollander & Wolfe (1999), page 147
  sid <- data.frame(
      serum = c(111, 107, 100, 99, 102, 106, 109, 108, 104, 99,
                101, 96, 97, 102, 107, 113, 116, 113, 110, 98,
                107, 108, 106, 98, 105, 103, 110, 105, 104,
                100, 96, 108, 103, 104, 114, 114, 113, 108, 106, 99),
      method = factor(gl(2, 20), labels = c("Ramsay", "Jung-Parekh")))

  ### Ansari-Bradley test, asymptotical p-value
  ansari_test(serum ~ method, data = sid)

  ### exact p-value
  ansari_test(serum ~ method, data = sid, distribution = "exact")


  ### Platelet Counts of Newborn Infants
  ### Hollander & Wolfe (1999), Table 5.4, page 171
  platalet_counts <- data.frame(
      counts = c(120, 124, 215, 90, 67, 95, 190, 180, 135, 399, 
                 12, 20, 112, 32, 60, 40),
      treatment = factor(c(rep("Prednisone", 10), rep("Control", 6))))

  ### Lepage test, Hollander & Wolfe (1999), page 172 
  lt <- independence_test(counts ~ treatment, data = platalet_counts,
      ytrafo = function(data) trafo(data, numeric_trafo = function(x)       
          cbind(rank(x), ansari_trafo(x))),
      teststat = "quad", distribution = approximate(B = 9999))

  lt

  ### where did the rejection come from? Use maximum statistic
  ### instead of a quadratic form
  ltmax <- independence_test(counts ~ treatment, data = platalet_counts,
      ytrafo = function(data) trafo(data, numeric_trafo = function(x) 
          matrix(c(rank(x), ansari_trafo(x)), ncol = 2,
                 dimnames = list(1:length(x), c("Location", "Scale")))),
      teststat = "max")

  ### points to a difference in location
  pvalue(ltmax, method = "single-step")

  ### Funny: We could have used a simple Bonferroni procedure
  ### since the correlation between the Wilcoxon and Ansari-Bradley 
  ### test statistics is zero
  covariance(ltmax)