PMCMRplus (version 1.3.0)

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, ...)

Arguments

x

a numeric vector of data values, or a list of numeric data vectors.

…

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, \(df = N - 2\). 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").

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.

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. A total of \(m = k(k-1)/2\) hypotheses can be tested. The null hypothesis H\(_{ij}: \mu_i(x) = \mu_j(x)\) is tested in the two-tailed test against the alternative A\(_{ij}: \mu_i(x) \ne \mu_j(x), ~~ i \ne j\).

T2 test uses Welch's approximate solution for calculating the degree of freedom. T2' test uses the usual \(df = N - 2\) approximation. A warning message appears in the modified T2' test, if none of in Tamhane (1979) given conditions for nearly balanced sample sizes and nearly balanced standard errors is true.

The p-values are computed from the t-distribution and adjusted according to Dunn-Sidak.

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.

See Also

dunnettT3Test

Examples

# NOT RUN {
set.seed(245)
mn <- rep(c(1, 2^(1:4)), each=5)
sd <- rep(1:5, each=5)
x <- mn + rnorm(25, sd = sd)
g <- factor(rep(1:5, each=5))

fit <- aov(x ~ g)
shapiro.test(residuals(fit))
bartlett.test(x ~ g) # var1 != varN
anova(fit)
summary(T2 <- tamhaneT2Test(x, g))
T2
## compare with pairwise.t.test
WT <- pairwise.t.test(x, g, 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$p.value)
m <- length(p.raw[!is.na(p.raw)])
PADJ <- matrix(ans <- p.adj.sidak(p.raw, m),
               nrow = 4, ncol = 4)
colnames(PADJ) <- colnames(WT$p.value)
rownames(PADJ) <- rownames(WT$p.value)
PADJ

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


# }