PMCMRplus (version 1.7.1)

tukeyTest: Tukey's Multiple Comparison Test

Description

Performs Tukey's all-pairs comparisons test for normally distributed data with equal group variances.

Usage

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, ...)

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").

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.

Details

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

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

with $s_{\mathrm{in}}^2$ 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.

References

Sachs, L. (1997) Angewandte Statistik, New York: Springer.

Tukey, J. (1949) Comparing Individual Means in the Analysis of Variance, Biometrics 5, 99--114.

See Also

Tukey, TukeyHSD

Examples

# 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)
# }