dunnettT3Test() R function from [PMCMRplus]

Dunnett's T3 Test

Performs Dunnett's all-pairs comparison test for normally distributed data with unequal variances.


dunnettT3Test(x, ...)

## Default S3 method:
dunnettT3Test(x, g, ...)

## S3 method for class 'formula'
dunnettT3Test(formula, data, subset, na.action, ...)

## S3 method for class 'aov'
dunnettT3Test(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 but unequal groups variances the T3 test of Dunnett 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). Dunnett T3 all-pairs test statistics are given by

t_{ij} \frac{\bar{X}_i - \bar{X_j}}{\left( s^2_j / n_j + s^2_i / n_i \right)^{1/2}}, ~~(i \ne j)%SEE PDF

with $s^2_i$ the variance of the $i$ -th group. The null hypothesis is rejected (two-tailed) if

\mathrm{Pr} \left\{ |t_{ij}| \ge T_{v_{ij}\rho_{ij}\alpha'/2} | \mathrm{H} \right\}_{ij} =\alpha,%SEE PDF

with Welch's approximate solution for calculating the degree of freedom.

v_{ij} = \frac{\left( s^2_i / n_i + s^2_j / n_j \right)^2}{s^4_i / n^2_i \left(n_i - 1\right) + s^4_j / n^2_j \left(n_j - 1\right)}.%SEE PDF

The $p$ -values are computed from the studentized maximum modulus distribution that is the equivalent of the multivariate t distribution with $\rho_{ii} = 1, ~ \rho_{ij} = 0 ~ (i \ne j)$ . The function pmvt is used to calculate the $p$ -values.

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 <- dunnettT3Test(fit)
summary(res)
summaryGroup(res)

References

C. W. Dunnett (1980) Pair wise multiple comparisons in the unequal variance case, Journal of the American Statistical Association 75 , 796--800.

dunnettT3Test function