PMCMRplus (version 1.9.12)

uryWigginsHochbergTest: Ury, Wiggins, Hochberg Test

Description

Performs Ury-Wiggins and Hochberg's all-pairs comparison test for normally distributed data with unequal variances.

Usage

uryWigginsHochbergTest(x, ...)
# S3 method for default
uryWigginsHochbergTest(x, g, p.adjust.method = p.adjust.methods, ...)
# S3 method for formula
uryWigginsHochbergTest(
  formula,
  data,
  subset,
  na.action,
  p.adjust.method = p.adjust.methods,
  ...
)
# S3 method for aov
uryWigginsHochbergTest(x, p.adjust.method = p.adjust.methods, ...)

Value

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.

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").

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

$$t_{ij}\frac{{\overline{X}}_i-\overline{X_j}}{{(s_j^2/n_j+s_i^2/n_i)}^{1/2}},(i\ne j)$$

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

$$\mathrm{Pr}{\{|t_{ij}|\ge t_{v_{ij}{\alpha }^{\prime }/2}|\mathrm{H}\}}_{ij}=\alpha ,$$

with Welch's approximate equation for degree of freedom as

$$v_{ij}=\frac{{(s_i^2/n_i+s_j^2/n_j)}^2}{s_i^4/n_i^2(n_i-1)+s_j^4/n_j^2(n_j-1)}.$$

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

.

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

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)