PMCMRplus (version 1.9.6)

dunnettTest: Dunnett's Many-to-One Comparisons Test

Description

Performs Dunnett's multiple comparisons test with one control.

Usage

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"), ...)

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.

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.

alternative

the alternative hypothesis. Defaults to two.sided.

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").

Details

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

$$t_i\frac{{\overline{X}}_i-\overline{X_0}}{s_{\mathrm{in}}{(1/n_0+1/n_i)}^{1/2}},(1\le i\le k)$$

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

$${\rho }_{ij}=\sqrt{\frac{n_i{n}_j}{(n_i+n_0)(n_j+n_0)}}(i\ne j).$$

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.

References

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.

See Also

pmvt pDunnett

Examples

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"))