coin (version 1.0-24)

SymmetryTests: Symmetry Tests

Description

Testing the symmetry of a response for repeated measurements in a complete block design.

Usage

## S3 method for class 'formula':
friedman_test(formula, data, subset = NULL, \dots)
## S3 method for class 'SymmetryProblem':
friedman_test(object, distribution = c("asymptotic", "approximate"), 
              ...) 
## S3 method for class 'formula':
wilcoxsign_test(formula, data, subset = NULL, \dots)
## S3 method for class 'IndependenceProblem':
wilcoxsign_test(object, zero.method = c("Pratt", "Wilcoxon"), 
              ties.method = NULL, ...)

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 (wilcoxsign_test) or more levels giving the corresponding groups. bloc

data

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

subset

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

object

an object inheriting from class SymmetryProblem.

distribution

a character, the null distribution of the test statistic can be approximated by its asymptotic distribution (asympt) or via Monte-Carlo resampling (approx). Alternatively, the functions

zero.method

a character specifying the way zeros, for example induced by tied repeated measurements, are handled, see below.

ties.method

now defunct, see below.

...

further arguments to be passed to or from methods.

Value

• An object inheriting from class IndependenceTest with methods show, pvalue and statistic.

Details

The null hypothesis of symmetry of y across x is tested. In case x is an ordered factor friedman_test performs the Page test, scores can be altered by the scores argument (see symmetry_test).

For wilcoxsign_test, formulae of the form y ~ x | block and y ~ x are allowed. The latter is interpreted in the sense that y is the first and x the second measurement on the same observation. By default, following Pratt (1959), we rank the absolute differences (including zeros) first, then discard the ranks corresponding to zero differences, but keep the other ranks as they are. Alternatively, zero differences are discarded first and then rank the remaining absolute differences (the original proposal by Wilcoxon, as explained in Hollander & Wolfe, 1999).

The default zero.method = "Pratt" was introduced in coin 1.0-16. Earlier versions had ties.method = "HollanderWolfe" which is now equivalent to zero.method = "Wilcoxon".

References

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

John W. Pratt (1959). Remarks on zeros and ties in the Wilcoxon signed rank procedures. Journal of the American Statistical Association, 54(287), 655--667.

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.

Examples

### Hollander & Wolfe (1999), Table 7.1, page 274
  ### Comparison of three methods ("round out", "narrow angle", and
  ###  "wide angle") for rounding first base. 
  RoundingTimes <- data.frame(
      times = c(5.40, 5.50, 5.55,
                5.85, 5.70, 5.75,
                5.20, 5.60, 5.50,
                5.55, 5.50, 5.40,
                5.90, 5.85, 5.70,
                5.45, 5.55, 5.60,
                5.40, 5.40, 5.35,
                5.45, 5.50, 5.35,
                5.25, 5.15, 5.00,
                5.85, 5.80, 5.70,
                5.25, 5.20, 5.10,
                5.65, 5.55, 5.45,
                5.60, 5.35, 5.45,
                5.05, 5.00, 4.95,
                5.50, 5.50, 5.40,
                5.45, 5.55, 5.50,
                5.55, 5.55, 5.35,
                5.45, 5.50, 5.55,
                5.50, 5.45, 5.25,
                5.65, 5.60, 5.40,
                5.70, 5.65, 5.55,
                6.30, 6.30, 6.25),
      methods = factor(rep(c("Round Out", "Narrow Angle", "Wide Angle"), 22)),
      block = factor(rep(1:22, rep(3, 22))))

  ### classical global test
  friedman_test(times ~ methods | block, data = RoundingTimes)

  ### parallel coordinates plot
  matplot(t(matrix(RoundingTimes$times, ncol = 3, byrow = TRUE)), 
          type = "l", col = 1, lty = 1, axes = FALSE, ylab = "Time", 
          xlim = c(0.5, 3.5))
  axis(1, at = 1:3, labels = levels(RoundingTimes$methods))
  axis(2)

  ### where do the differences come from?
  ### Wilcoxon-Nemenyi-McDonald-Thompson test
  ### Hollander & Wolfe (1999), page 295
  if (require("multcomp")) {

      ### all pairwise comparisons
      rtt <- symmetry_test(times ~ methods | block, data = RoundingTimes,
           teststat = "max",
           xtrafo = function(data)
               trafo(data, factor_trafo = function(x)
                   model.matrix(~ x - 1) %*% t(contrMat(table(x), "Tukey"))
               ),
           ytrafo = function(data)
               trafo(data, numeric_trafo = rank, block = RoundingTimes$block)
      )

      ### a global test, again
      print(pvalue(rtt))

      ### simultaneous P-values for all pair comparisons
      ### Wide Angle vs. Round Out differ (Hollander and Wolfe, 1999, page 296)
      print(pvalue(rtt, method = "single-step"))
}

  ### Strength Index of Cotton, Hollander & Wolfe (1999), Table 7.5, page 286
  sc <- data.frame(block = factor(c(rep(1, 5), rep(2, 5), rep(3, 5))),
                   potash = ordered(rep(c(144, 108, 72, 54, 36), 3)),
                   strength = c(7.46, 7.17, 7.76, 8.14, 7.63,
                                7.68, 7.57, 7.73, 8.15, 8.00,
                                7.21, 7.80, 7.74, 7.87, 7.93))

  ### Page test for ordered alternatives
  ft <- friedman_test(strength ~ potash | block, data = sc)
  ft

  ### one-sided p-value
  1 - pnorm(sqrt(statistic(ft)))

  ### approximate null distribution via Monte-Carlo
  pvalue(friedman_test(strength ~ potash | block, data = sc, 
                       distribution = approximate(B = 9999)))

  ### example from ?wilcox.test
  x <- c(1.83,  0.50,  1.62,  2.48, 1.68, 1.88, 1.55, 3.06, 1.30)
  y <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)
  wilcoxsign_test(x ~ y, alternative = "greater", distribution = exact())
  wilcox.test(x, y, paired = TRUE, alternative = "greater")

  ### with explicit group and block information
  xydat <- data.frame(y = c(y, x), x = gl(2, length(x)), 
                      block = factor(rep(1:length(x), 2)))
  wilcoxsign_test(y ~ x | block, data = xydat, 
                  alternative = "greater", distribution = exact())