tamhaneT2Test: Tamhane's T2 Test

Description

Performs Tamhane's T2 (or T2') all-pairs comparison test for normally distributed data with unequal variances.

Usage

tamhaneT2Test(x, ...)
# S3 method for default
tamhaneT2Test(x, g, welch = TRUE, ...)
# S3 method for formula
tamhaneT2Test(formula, data, subset, na.action, welch = TRUE, ...)
# S3 method for aov
tamhaneT2Test(x, welch = TRUE, ...)

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.
welch: indicates, whether Welch's approximate solution for calculating the degree of freedom shall be used or, as usually, $d f = N - 2$ . Defaults to TRUE.
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 all-pairs comparisons in an one-factorial layout with normally distributed residuals but unequal groups variances the T2 test (or T2' test) of Tamhane can be performed. Let $X_{i j}$ denote a continuous random variable with the $j$ -the realization ( $1 \leq j \leq n_{i}$ ) in the $i$ -th group ( $1 \leq i \leq k$ ). 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 $_{i j} : μ_{i} = μ_{j} (i \neq j)$ is tested against the alternative A $_{i j} : μ_{i} \neq μ_{j}$ (two-tailed). Tamhane T2 all-pairs test statistics are given by

$t_{i j} \frac{{\bar{X}}_{i} - \bar{X_{j}}}{{(s_{j}^{2} / n_{j} + s_{i}^{2} / n_{i})}^{1 / 2}}, (i \neq j)$

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

$\Pr {| t_{i j} | \geq t_{v_{i j} α^{'} / 2} | H}_{i j} = α .$

T2 test uses Welch's approximate solution for calculating the degree of freedom.

$v_{i j} = \frac{{(s_{i}^{2} / n_{i} + s_{j}^{2} / n_{j})}^{2}}{s_{i}^{4} / n_{i}^{2} (n_{i} - 1) + s_{j}^{4} / n_{j}^{2} (n_{j} - 1)} .$

T2' test applies the following approximation for the degree of freedom $v_{i j} = n_{i} + n_{j} - 2$

The p-values are computed from the TDist-distribution and adjusted according to Dunn-Sidak. $p_{i j}^{'} = min {1, (1 - (1 - p_{i j})^{m})}$

References

Tamhane, A. C. (1979) A Comparison of Procedures for Multiple Comparisons of Means with Unequal Variances, Journal of the American Statistical Association 74, 471--480.

Examples

Run this code

fit <- aov(weight ~ feed, chickwts)
shapiro.test(residuals(fit))
bartlett.test(weight ~ feed, chickwts) # var1 = varN
anova(fit)

## also works with fitted objects of class aov
res <- tamhaneT2Test(fit)
summary(res)
summaryGroup(res)
res

## compare with pairwise.t.test
WT <- pairwise.t.test(chickwts$weight,
                      chickwts$feed,
                      pool.sd = FALSE,
                      p.adjust.method = "none")
p.adj.sidak <- function(p, m) sapply(p, function(p) min(1, 1 - (1 - p)^m))
p.raw <- as.vector(WT$p.value)
m <- length(p.raw[!is.na(p.raw)])
PADJ <- matrix(ans <- p.adj.sidak(p.raw, m),
               nrow = 5, ncol = 5)
colnames(PADJ) <- colnames(WT$p.value)
rownames(PADJ) <- rownames(WT$p.value)
PADJ

## same without Welch's approximate solution
summary(T2b <- tamhaneT2Test(fit, welch = FALSE))

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