uryWigginsHochbergTest function

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, ...)

Arguments

• 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 "x". Ignored with a warning if "x" is a list.
• p.adjust.method: method for adjusting p values (see p.adjust).
• formula: a formula of the form response ~ group where response gives the data values and group a vector or factor of the corresponding groups.
• data: an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
• 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 NAs. Defaults to getOption("na.action").

Returns

A list with class "PMCMR" containing the following components:

• method: a character string indicating what type of test was performed.
• data.name: a character string giving the name(s) of the data.
• statistic: lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
• p.value: lower-triangle matrix of the p-values for the pairwise tests.
• alternative: a character string describing the alternative hypothesis.
• p.adjust.method: a character string describing the method for p-value adjustment.
• model: a data frame of the input data.
• dist: a string that denotes the test distribution.

Details

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 ($1 \le j \le n_i$) in the $i$-th group ($1 \le i \le k$). Furthermore, the total sample size is $N = \sum_{i=1}^k n_i$. A total of $m = k(k-1)/2$

hypotheses can be tested: The null hypothesis is H$_{ij}: \mu_i = \mu_j ~~ (i \ne j)$ is tested against the alternative A$_{ij}: \mu_i \ne \mu_j$ (two-tailed). Ury-Wiggins and Hochberg all-pairs test statistics are given by

with $s^2_i$ the variance of the $i$-th group. The null hypothesis is rejected (two-tailed) if

with Welch's approximate equation for degree of freedom as

The p-values are computed from the TDist-distribution. The type of test depends on the selected p-value adjustment method (see also p.adjust):

• bonferroni: the Ury-Wiggins test is performed with Bonferroni adjusted p-values.
• hochberg: the Hochberg test is performed with Hochberg's adjusted p-values

Examples

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)

References

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.

See Also

dunnettT3Test tamhaneT2Test TDist