coin (version 1.0-7)

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, 
              ties.method = c("HollanderWolfe", "Pratt"), ...)

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, zero differences are discarded first and then rank the remaining absolute differences (as explained in Hollander & Wolfe, 1999). Alternatively, 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.

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())