duncanTest | R Documentation |
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, ...)
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
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.
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.
Duncan, D. B. (1955) Multiple range and multiple F tests, Biometrics 11, 1–42.
Tukey
, TukeyHSD
tukeyTest
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)
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