Tukey's Multiple Comparison Test
Performs Tukey's all-pairs comparisons test for normally distributed data with equal group variances.
tukeyTest(x, ...) ## Default S3 method: tukeyTest(x, g, ...) ## S3 method for class 'formula' tukeyTest(formula, data, subset, na.action, ...) ## S3 method for class 'aov' tukeyTest(x, ...)
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 |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
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 |
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 ≤ j ≤ n_i) in the i-th group (1 ≤ i ≤ k). Furthermore, the total sample size is N = ∑_{i=1}^k n_i. A total of m = k(k-1)/2 hypotheses can be tested: The null hypothesis is H_{ij}: μ_i = μ_j ~~ (i \ne j) is tested against the alternative A_{ij}: μ_i \ne μ_j (two-tailed). Tukey's all-pairs test statistics are given by
SEE PDF
with s^2_{\mathrm{in}} the within-group ANOVA variance.
The null hypothesis is rejected if |t_{ij}| > q_{vmα} / √{2},
with v = N - k degree of freedom. The p-values are computed
from the Tukey distribution.
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.
Sachs, L. (1997) Angewandte Statistik, New York: Springer.
Tukey, J. (1949) Comparing Individual Means in the Analysis of Variance, Biometrics 5, 99–114.
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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