Performs Tukey's all-pairs comparisons test for normally distributed data with equal group variances.
tukeyTest(x, ...)# S3 method for default
tukeyTest(x, g, ...)
# S3 method for formula
tukeyTest(formula, data, subset, na.action, ...)
# S3 method for aov
tukeyTest(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.
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").
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.
For all-pairs comparisons in an one-factorial layout with normally distributed residuals and equal variances Tukey's 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). Tukey's all-pairs test statistics are given by
$$ t_{ij} \frac{\bar{X}_i - \bar{X_j}} {s_{\mathrm{in}} \left(1/n_j + 1/n_i\right)^{1/2}}, ~~ (i \ne j) $$
with \(s^2_{\mathrm{in}}\) the within-group ANOVA variance.
The null hypothesis is rejected if \(|t_{ij}| > q_{vm\alpha} / \sqrt{2}\),
with \(v = N - k\) degree of freedom. The p-values are computed
from the Tukey distribution.
Sachs, L. (1997) Angewandte Statistik, New York: Springer.
Tukey, J. (1949) Comparing Individual Means in the Analysis of Variance, Biometrics 5, 99--114.
# NOT RUN {
fit <- aov(weight ~ feed, chickwts)
shapiro.test(residuals(fit))
bartlett.test(weight ~ feed, chickwts)
anova(fit)
## also works with fitted objects of class aov
res <- tukeyTest(fit)
summary(res)
summaryGroup(res)
# }
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