# Symmetry Tests Testing the symmetry of a numeric repeated measurements variable in a complete block design. ```r ## S3 method for class 'formula' sign_test(formula, data, subset = NULL, ...) ## S3 method for class 'SymmetryProblem' sign_test(object, ...) ## S3 method for class 'formula' wilcoxsign_test(formula, data, subset = NULL, ...) ## S3 method for class 'SymmetryProblem' wilcoxsign_test(object, zero.method = c("Pratt", "Wilcoxon"), ...) ## S3 method for class 'formula' friedman_test(formula, data, subset = NULL, ...) ## S3 method for class 'SymmetryProblem' friedman_test(object, ...) ## S3 method for class 'formula' quade_test(formula, data, subset = NULL, ...) ## S3 method for class 'SymmetryProblem' quade_test(object, ...) ``` ## Arguments - `formula`: a formula of the form `y ~ x | block` where `y` is a numeric variable, `x` is a factor with two (`sign_test` and `wilcoxsign_test`) or more levels and `block` is an optional factor (which is generated automatically if omitted). - `data`: an optional data frame containing the variables in the model formula. - `subset`: an optional vector specifying a subset of observations to be used. Defaults to `NULL`. - `object`: an object inheriting from class `"SymmetryProblem"`. - `zero.method`: a character, the method used to handle zeros: either `"Pratt"` (default) or `"Wilcoxon"`; see Details . - `...`: further arguments to be passed to `symmetry_test()`. ## Details `sign_test()`, `wilcoxsign_test()`, `friedman_test()` and `quade_test()` provide the sign test, the Wilcoxon signed-rank test, the Friedman test, the Page test and the Quade test. A general description of these methods is given by Hollander and Wolfe (1999). The null hypothesis of symmetry is tested. The response variable and the measurement conditions are given by `y` and `x`, respectively, and `block` is a factor where each level corresponds to exactly one subject with repeated measurements. For `sign_test` and `wilcoxsign_test`, formulae of the form `y ~ x | block` and `y ~ x` are allowed. The latter form is interpreted as `y` is the first and `x` the second measurement on the same subject. If `x` is an ordered factor, the default scores, `1:nlevels(x)`, can be altered using the `scores` argument (see `symmetry_test()`); this argument can also be used to coerce nominal factors to class `"ordered"`. In this case, a linear-by-linear association test is computed and the direction of the alternative hypothesis can be specified using the `alternative` argument. For the Friedman test, this extension was given by Page (1963) and is known as the Page test. For `wilcoxsign_test()`, the default method of handling zeros (`zero.method = "Pratt"`), due to Pratt (1959), first rank-transforms the absolute differences (including zeros) and then discards the ranks corresponding to the zero-differences. The proposal by Wilcoxon (1949, p. 6) first discards the zero-differences and then rank-transforms the remaining absolute differences (`zero.method = "Wilcoxon"`). The conditional null distribution of the test statistic is used to obtain $p$-values and an asymptotic approximation of the exact distribution is used by default (`distribution = "asymptotic"`). Alternatively, the distribution can be approximated via Monte Carlo resampling or computed exactly for univariate two-sample problems by setting `distribution` to `"approximate"` or `"exact"`, respectively. See `asymptotic()`, `approximate()` and `exact()` for details. ## Returns An object inheriting from class `"IndependenceTest"`. ## Note Starting with `coin` version 1.0-16, the `zero.method` argument replaced the (now removed) `ties.method` argument. The current default is `zero.method = "Pratt"` whereas earlier versions had `ties.method = "HollanderWolfe"`, which is equivalent to `zero.method = "Wilcoxon"`. ## References Hollander, M. and Wolfe, D. A. (1999). **Nonparametric Statistical Methods**, Second Edition. New York: John Wiley & Sons. Page, E. B. (1963). Ordered hypotheses for multiple treatments: a significance test for linear ranks. **Journal of the American Statistical Association** 58 (301), 216--230. tools:::Rd_expr_doi("10.1080/01621459.1963.10500843") Pratt, J. W. (1959). Remarks on zeros and ties in the Wilcoxon signed rank procedures. **Journal of the American Statistical Association** 54 (287), 655--667. tools:::Rd_expr_doi("10.1080/01621459.1959.10501526") Quade, D. (1979). Using weighted rankings in the analysis of complete blocks with additive block effects. **Journal of the American Statistical Association** 74 (367), 680--683. tools:::Rd_expr_doi("10.1080/01621459.1979.10481670") Wilcoxon, F. (1949). **Some Rapid Approximate Statistical Procedures**. New York: American Cyanamid Company. ## Examples ```r ## Example data from ?wilcox.test y1 <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30) y2 <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29) ## One-sided exact sign test (st <- sign_test(y1 ~ y2, distribution = "exact", alternative = "greater")) midpvalue(st) # mid-p-value ## One-sided exact Wilcoxon signed-rank test (wt <- wilcoxsign_test(y1 ~ y2, distribution = "exact", alternative = "greater")) statistic(wt, type = "linear") midpvalue(wt) # mid-p-value ## Comparison with R's wilcox.test() function wilcox.test(y1, y2, paired = TRUE, alternative = "greater") ## Data with explicit group and block information dta <- data.frame(y = c(y1, y2), x = gl(2, length(y1)), block = factor(rep(seq_along(y1), 2))) ## For two samples, the sign test is equivalent to the Friedman test... sign_test(y ~ x | block, data = dta, distribution = "exact") friedman_test(y ~ x | block, data = dta, distribution = "exact") ## ...and the signed-rank test is equivalent to the Quade test wilcoxsign_test(y ~ x | block, data = dta, distribution = "exact") quade_test(y ~ x | block, data = dta, distribution = "exact") ## Comparison of three methods ("round out", "narrow angle", and "wide angle") ## for rounding first base. ## Hollander and Wolfe (1999, p. 274, Tab. 7.1) rounding <- 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(1:3, 22), labels = c("Round Out", "Narrow Angle", "Wide Angle")), block = gl(22, 3) ) ## Asymptotic Friedman test friedman_test(times ~ methods | block, data = rounding) ## Parallel coordinates plot with(rounding, { matplot(t(matrix(times, ncol = 3, byrow = TRUE)), type = "l", lty = 1, col = 1, ylab = "Time", xlim = c(0.5, 3.5), axes = FALSE) axis(1, at = 1:3, labels = levels(methods)) axis(2) }) ## Where do the differences come from? ## Wilcoxon-Nemenyi-McDonald-Thompson test (Hollander and Wolfe, 1999, p. 295) ## Note: all pairwise comparisons (st <- symmetry_test(times ~ methods | block, data = rounding, ytrafo = function(data) trafo(data, numeric_trafo = rank_trafo, block = rounding$block), xtrafo = mcp_trafo(methods = "Tukey"))) ## Simultaneous test of all pairwise comparisons ## Wide Angle vs. Round Out differ (Hollander and Wolfe, 1999, p. 296) pvalue(st, method = "single-step") # subset pivotality is violated ## Strength Index of Cotton ## Hollander and Wolfe (1999, p. 286, Tab. 7.5) cotton <- data.frame( 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), potash = ordered(rep(c(144, 108, 72, 54, 36), 3), levels = c(144, 108, 72, 54, 36)), block = gl(3, 5) ) ## One-sided asymptotic Page test friedman_test(strength ~ potash | block, data = cotton, alternative = "greater") ## One-sided approximative (Monte Carlo) Page test friedman_test(strength ~ potash | block, data = cotton, alternative = "greater", distribution = approximate(nresample = 10000)) ## Data from Quade (1979, p. 683) dta <- data.frame( y = c(52, 45, 38, 63, 79, 50, 45, 57, 39, 53, 51, 43, 47, 50, 56, 62, 72, 49, 49, 52, 40), x = factor(rep(LETTERS[1:3], 7)), b = factor(rep(1:7, each = 3)) ) ## Approximative (Monte Carlo) Friedman test ## Quade (1979, p. 683) friedman_test(y ~ x | b, data = dta, distribution = approximate(nresample = 10000)) # chi^2 = 6.000 ## Approximative (Monte Carlo) Quade test ## Quade (1979, p. 683) (qt <- quade_test(y ~ x | b, data = dta, distribution = approximate(nresample = 10000))) # W = 8.157 ## Comparison with R's quade.test() function quade.test(y ~ x | b, data = dta) ## quade.test() uses an F-statistic b <- nlevels(qt@statistic@block) A <- sum(qt@statistic@ytrans^2) B <- sum(statistic(qt, type = "linear")^2) / b (b - 1) * B / (A - B) # F = 8.3765 ```

SymmetryTests function