tamhaneDunnettTest function

Tamhane-Dunnett Many-to-One Comparison Test

Performs Tamhane-Dunnett's multiple comparisons test with one control. For many-to-one comparisons in an one-factorial layout with normally distributed residuals and unequal variances Tamhane-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). Tamhane-Dunnett's test statistics are given by

[REMOVE_ME]$t_{i} \frac{\bar{X}_i - \bar{X_0}}{\left( s^2_0 / n_0 + s^2_i / n_i \right)^{1/2} } ~~(1 \le i \le k)$

The null hypothesis is rejected if $|t_{i}| > T_{kv_{i}\rho_{ij}\alpha}$ (two-tailed), with

[REMOVE_ME]$v_i = n_0 + n_i - 2$

degree of freedom and the correlation

[REMOVE_ME]$\rho_{ii} = 1, ~ \rho_{ij} = 0 ~ (i \ne j).$

The p-values are computed from the multivariate-t distribution as implemented in the function pmvt distribution.

tamhaneDunnettTest(x, ...)

## Default S3 method:
tamhaneDunnettTest(x, g, alternative = c("two.sided", "greater", "less"), ...)

## S3 method for class 'formula'
tamhaneDunnettTest(
  formula,
  data,
  subset,
  na.action,
  alternative = c("two.sided", "greater", "less"),
  ...
)

## S3 method for class 'aov'
tamhaneDunnettTest(x, alternative = c("two.sided", "greater", "less"), ...)

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").

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.

Description

Performs Tamhane-Dunnett's multiple comparisons test with one control. For many-to-one comparisons in an one-factorial layout with normally distributed residuals and unequal variances Tamhane-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). Tamhane-Dunnett's test statistics are given by

The null hypothesis is rejected if $|t_{i}| > T_{kv_{i}\rho_{ij}\alpha}$ (two-tailed), with

degree of freedom and the correlation

The p-values are computed from the multivariate-t distribution as implemented in the function pmvt distribution.

Examples

set.seed(245)
mn <- c(1, 2, 2^2, 2^3, 2^4)
x <- rep(mn, each=5) + rnorm(25)
g <- factor(rep(1:5, each=5))

fit <- aov(x ~ g - 1)
shapiro.test(residuals(fit))
bartlett.test(x ~ g - 1)
anova(fit)
## works with object of class aov
summary(tamhaneDunnettTest(fit, alternative = "greater"))

References

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, welchManyOneTTest