Performs Games-Howell all-pairs comparison test for normally distributed data with unequal group variances.
gamesHowellTest(x, ...)# S3 method for default
gamesHowellTest(x, g, ...)
# S3 method for formula
gamesHowellTest(formula, data, subset, na.action, ...)
# S3 method for aov
gamesHowellTest(x, ...)
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.
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.
a vector or factor object giving the group for the
corresponding elements of "x".
Ignored with a warning if "x" is a list.
a formula of the form response ~ group where
response gives the data values and group a vector or
factor of the corresponding groups.
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).
an optional vector specifying a subset of observations to be used.
a function which indicates what should happen when
the data contain NAs. Defaults to getOption("na.action").
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.
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)
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