duncanTest() R function from [PMCMRplus]

Duncan's Multiple Range Test

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


duncanTest(x, ...)

## Default S3 method:
duncanTest(x, g, ...)

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

## S3 method for class '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").

Returns

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)%SEE PDF

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\},%SEE PDF

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.

Examples


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)

References

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

duncanTest function