Performs Dunnett's multiple comparisons test with one control.
dunnettTest(x, ...)# S3 method for default
dunnettTest(x, g, alternative = c("two.sided", "greater", "less"), ...)
# S3 method for formula
dunnettTest(
formula,
data,
subset,
na.action,
alternative = c("two.sided", "greater", "less"),
...
)
# S3 method for aov
dunnettTest(x, alternative = c("two.sided", "greater", "less"), ...)
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.
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.
a vector or factor object giving the group for the
corresponding elements of "x".
Ignored with a warning if "x" is a list.
the alternative hypothesis. Defaults to two.sided.
a formula of the form response ~ group where
response gives the data values and group a vector or
factor of the corresponding groups.
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).
an optional vector specifying a subset of observations to be used.
a function which indicates what should happen when
the data contain NAs. Defaults to getOption("na.action").
For many-to-one comparisons in an one-factorial layout with normally distributed residuals Dunnett's test can be used. Let \(X_{0j}\) denote a continuous random variable with the \(j\)-the realization of the control group (\(1 \le j \le n_0\)) and \(X_{ij}\) the \(j\)-the realization in the \(i\)-th treatment group (\(1 \le i \le k\)). Furthermore, the total sample size is \(N = n_0 + \sum_{i=1}^k n_i\). A total of \(m = k\) hypotheses can be tested: The null hypothesis is H\(_{i}: \mu_i = \mu_0\) is tested against the alternative A\(_{i}: \mu_i \ne \mu_0\) (two-tailed). Dunnett's test statistics are given by
$$ t_{i} \frac{\bar{X}_i - \bar{X_0}} {s_{\mathrm{in}} \left(1/n_0 + 1/n_i\right)^{1/2}}, ~~ (1 \le i \le k) $$
with \(s^2_{\mathrm{in}}\) the within-group ANOVA variance. The null hypothesis is rejected if \(|t_{ij}| > |T_{kv\rho\alpha}|\) (two-tailed), with \(v = N - k\) degree of freedom and \(rho\) the correlation:
$$ \rho_{ij} = \sqrt{\frac{n_i n_j} {\left(n_i + n_0\right) \left(n_j+ n_0\right)}} ~~ (i \ne j) .$$
The p-values are computed with the function pDunnett
that is a wrapper to the the multivariate-t distribution as implemented in the function
pmvt.
Dunnett, C. W. (1955) A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association 50, 1096–1121.
OECD (ed. 2006) Current approaches in the statistical analysis of ecotoxicity data: A guidance to application - Annexes. OECD Series on testing and assessment, No. 54.
pmvt pDunnett
fit <- aov(Y ~ DOSE, data = trout)
shapiro.test(residuals(fit))
bartlett.test(Y ~ DOSE, data = trout)
## works with fitted object of class aov
summary(dunnettTest(fit, alternative = "less"))
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