Learn R Programming

PMCMRplus (version 1.9.3)

duncanTest: Duncan's Multiple Range Test

Description

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

Usage

duncanTest(x, ...)

# S3 method for default duncanTest(x, g, ...)

# S3 method for formula duncanTest(formula, data, subset, na.action, ...)

# S3 method for aov duncanTest(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 Duncan's multiple range 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). Duncan's all-pairs test statistics are given by

$$ t_{(i)(j)} \frac{\bar{X}_{(i)} - \bar{X}_{(j)}} {s_{\mathrm{in}} \left(r\right)^{1/2}}, ~~ (i < j) $$

with \(s^2_{\mathrm{in}}\) the within-group ANOVA variance, \(r = k / \sum_{i=1}^k n_i\) and \(\bar{X}_{(i)}\) the increasingly ordered means \(1 \le i \le k\). The null hypothesis is rejected if

$$ \mathrm{Pr} \left\{ |t_{(i)(j)}| \ge q_{vm'\alpha'} | \mathrm{H} \right\}_{(i)(j)} = \alpha' = \min \left\{1,~ 1 - (1 - \alpha)^{(1 / (m' - 1))} \right\}, $$

with \(v = N - k\) degree of freedom, the range \(m' = 1 + |i - j|\) and \(\alpha'\) the Bonferroni adjusted alpha-error. The p-values are computed from the Tukey distribution.

References

Duncan, D. B. (1955) Multiple range and multiple F tests, Biometrics 11, 1--42.

See Also

Tukey, TukeyHSD tukeyTest

Examples

Run this code
# 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 <- duncanTest(fit)
summary(res)
summaryGroup(res)
# }

Run the code above in your browser using DataLab