Ury, Wiggins, Hochberg Test
Performs Ury-Wiggins and Hochberg's all-pairs comparison test for normally distributed data with unequal variances.
uryWigginsHochbergTest(x, ...) ## Default S3 method: uryWigginsHochbergTest(x, g, p.adjust.method = p.adjust.methods, ...) ## S3 method for class 'formula' uryWigginsHochbergTest( formula, data, subset, na.action, p.adjust.method = p.adjust.methods, ... ) ## S3 method for class 'aov' uryWigginsHochbergTest(x, p.adjust.method = p.adjust.methods, ...)
x |
a numeric vector of data values, a list of numeric data vectors or a fitted model object, usually an aov fit. |
... |
further arguments to be passed to or from methods. |
g |
a vector or factor object giving the group for the
corresponding elements of |
p.adjust.method |
method for adjusting p values
(see |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
subset |
an optional vector specifying a subset of observations to be used. |
na.action |
a function which indicates what should happen when
the data contain |
For all-pairs comparisons in an one-factorial layout with normally distributed residuals but unequal groups variances the tests of Ury-Wiggins and Hochberg can be performed. Let X_{ij} denote a continuous random variable with the j-the realization (1 ≤ j ≤ n_i) in the i-th group (1 ≤ i ≤ k). Furthermore, the total sample size is N = ∑_{i=1}^k n_i. A total of m = k(k-1)/2 hypotheses can be tested: The null hypothesis is H_{ij}: μ_i = μ_j ~~ (i \ne j) is tested against the alternative A_{ij}: μ_i \ne μ_j (two-tailed). Ury-Wiggins and Hochberg all-pairs test statistics are given by
SEE PDF
with s^2_i the variance of the i-th group. The null hypothesis is rejected (two-tailed) if
SEE PDF
with Welch's approximate equation for degree of freedom as
SEE PDF
the Ury-Wiggins test is performed with Bonferroni adjusted p-values.
the Hochberg test is performed with Hochberg's adjusted p-values
.
A list with class "PMCMR" containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
lower-triangle matrix of the p-values for the pairwise tests.
a character string describing the alternative hypothesis.
a character string describing the method for p-value adjustment.
a data frame of the input data.
a string that denotes the test distribution.
Hochberg, Y. (1976) A Modification of the T-Method of Multiple Comparisons for a One-Way Layout With Unequal Variances, Journal of the American Statistical Association 71, 200–203.
Ury, H. and Wiggins, A. D. (1971) Large Sample and Other Multiple Comparisons Among Means, British Journal of Mathematical and Statistical Psychology 24, 174–194.
fit <- aov(weight ~ feed, chickwts) shapiro.test(residuals(fit)) bartlett.test(weight ~ feed, chickwts) # var1 = varN anova(fit) ## also works with fitted objects of class aov res <- uryWigginsHochbergTest(fit) summary(res) summaryGroup(res)
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