LocationTests

0th

Percentile

Independent Two- and K-Sample Location Tests

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

Keywords
htest
Usage
## S3 method for class 'formula':
oneway_test(formula, data, subset = NULL, weights = NULL, \dots)
## S3 method for class 'IndependenceProblem':
oneway_test(object, ...)

## S3 method for class 'formula': wilcox_test(formula, data, subset = NULL, weights = NULL, \dots) ## S3 method for class 'IndependenceProblem': wilcox_test(object, conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'formula': normal_test(formula, data, subset = NULL, weights = NULL, \dots) ## S3 method for class 'IndependenceProblem': normal_test(object, ties.method = c("mid-ranks", "average-scores"), conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'formula': median_test(formula, data, subset = NULL, weights = NULL, \dots) ## S3 method for class 'IndependenceProblem': median_test(object, conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'formula': kruskal_test(formula, data, subset = NULL, weights = NULL, \dots) ## S3 method for class 'IndependenceProblem': kruskal_test(object, distribution = c("asymptotic", "approximate"), ...)

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 shift alternatives. For a general description of the test procedures documented here we refer to Hollander & Wolfe (1999).

The test procedures apply a rank transformation to the response values y, except of oneway_test which computes a test statistic using the untransformed response values.

The asymptotic null distribution is computed by default for all procedures. Exact p-values may be computed for the two-sample problems and can be approximated via Monte-Carlo resampling for all procedures. 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 linear rank tests for two samples (wilcox_test, normal_test and median_test) can be used to test the two-sided hypothesis $H_0: Y_1 - Y_2 = 0$, where $Y_i$ is the median of the responses in the ith group. Confidence intervals for the difference in location are available for the rank-based procedures and are computed according to Bauer (1972). In case alternative = "less", the null hypothesis $H_0: Y_1 - Y_2 \ge 0$ is tested and alternative = "greater" corresponds to a null hypothesis $H_0: Y_1 - Y_2 \le 0$.

In case x is an ordered factor, kruskal_test computes the linear-by-linear association test for ordered alternatives.

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

Value

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.

Aliases
Examples
### Tritiated Water Diffusion Across Human Chorioamnion
  ### Hollander & Wolfe (1999), Table 4.1, page 110
  water_transfer <- data.frame(
      pd = c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46,
             1.15, 0.88, 0.90, 0.74, 1.21),
      age = factor(c(rep("At term", 10), rep("12-26 Weeks", 5))))

  ### Wilcoxon-Mann-Whitney test, cf. Hollander & Wolfe (1999), page 111
  ### exact p-value and confidence interval for the difference in location
  ### (At term - 12-26 Weeks)
  wt <- wilcox_test(pd ~ age, data = water_transfer, 
                    distribution = "exact", conf.int = TRUE)
  print(wt)

  ### extract observed Wilcoxon statistic, i.e, the sum of the
  ### ranks for age = "12-26 Weeks"
  statistic(wt, "linear")

  ### its expectation
  expectation(wt)

  ### and variance
  covariance(wt)

  ### and, finally, the exact two-sided p-value
  pvalue(wt)

  ### Confidence interval for difference (12-26 Weeks - At term)
  wilcox_test(pd ~ age, data = water_transfer, 
              xtrafo = function(data) 
                  trafo(data, factor_trafo = function(x) 
                      as.numeric(x == levels(x)[2])),
              distribution = "exact", conf.int = TRUE)

  ### Permutation test, asymptotic p-value
  oneway_test(pd ~ age, data = water_transfer)

  ### approximate p-value (with 99\% confidence interval)
  pvalue(oneway_test(pd ~ age, data = water_transfer, 
                     distribution = approximate(B = 9999)))
  ### exact p-value
  pt <- oneway_test(pd ~ age, data = water_transfer, distribution = "exact")
  pvalue(pt)

  ### plot density and distribution of the standardized 
  ### test statistic
  layout(matrix(1:2, nrow = 2))
  s <- support(pt)
  d <- sapply(s, function(x) dperm(pt, x))
  p <- sapply(s, function(x) pperm(pt, x))
  plot(s, d, type = "S", xlab = "Teststatistic", ylab = "Density")
  plot(s, p, type = "S", xlab = "Teststatistic", ylab = "Cumm. Probability")


  ### Length of YOY Gizzard Shad from Kokosing Lake, Ohio,
  ### sampled in Summer 1984, Hollander & Wolfe (1999), Table 6.3, page 200
  YOY <- data.frame(length = c(46, 28, 46, 37, 32, 41, 42, 45, 38, 44, 
                               42, 60, 32, 42, 45, 58, 27, 51, 42, 52, 
                               38, 33, 26, 25, 28, 28, 26, 27, 27, 27, 
                               31, 30, 27, 29, 30, 25, 25, 24, 27, 30),
                    site = factor(c(rep("I", 10), rep("II", 10),
                                    rep("III", 10), rep("IV", 10))))

  ### Kruskal-Wallis test, approximate exact p-value
  kw <- kruskal_test(length ~ site, data = YOY, 
                     distribution = approximate(B = 9999))
  kw
  pvalue(kw)

  ### Nemenyi-Damico-Wolfe-Dunn test (joint ranking)
  ### Hollander & Wolfe (1999), page 244 
  ### (where Steel-Dwass results are given)
  if (require("multcomp")) {

    NDWD <- oneway_test(length ~ site, data = YOY,
        ytrafo = function(data) trafo(data, numeric_trafo = rank),
        xtrafo = function(data) trafo(data, factor_trafo = function(x)
            model.matrix(~x - 1) %*% t(contrMat(table(x), "Tukey"))),
        teststat = "max", distribution = approximate(B = 90000))

    ### global p-value
    print(pvalue(NDWD))

    ### sites (I = II) != (III = IV) at alpha = 0.01 (page 244)
    print(pvalue(NDWD, method = "single-step"))
  }
Documentation reproduced from package coin, version 1.0-7, License: GPL-2

Community examples

Looks like there are no examples yet.