gamesHowellTest {PMCMRplus}

Games-Howell Test

Description

Performs Games-Howell all-pairs comparison test for normally distributed data with unequal group variances.

Usage

gamesHowellTest(x, ...)

## Default S3 method:
gamesHowellTest(x, g, ...)

## S3 method for class 'formula'
gamesHowellTest(formula, data, subset, na.action, ...)

## S3 method for class 'aov'
gamesHowellTest(x, ...)

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.

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 between-groups variances the Games-Howell Test 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). Games-Howell Test all-pairs test statistics are given by

t_{ij} \frac{\bar{X}_i - \bar{X_j}} {\left( s^2_j / n_j + s^2_i / n_i \right)^{1/2}}, ~~ (i \ne j)

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

\mathrm{Pr} \left\{ |t_{ij}| \sqrt{2} \ge q_{m v_{ij} \alpha} | \mathrm{H} \right\}_{ij} = \alpha,

with Welch's approximate solution for calculating the degree of freedom.

v_{ij} = \frac{\left( s^2_i / n_i + s^2_j / n_j \right)^2} {s^4_i / n^2_i \left(n_i - 1\right) + s^4_j / n^2_j \left(n_j - 1\right)}.

The p-values are computed from the Tukey distribution.

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.

See Also

Tukey

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 <- gamesHowellTest(fit)
summary(res)
summaryGroup(res)