coin (version 0.4-2)

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, 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 (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

...

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).

References

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

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 = "maxtype",
         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)))