PMCMRplus (version 1.7.1)

MTest: Extended One-Sided Studentised Range Test

Description

Performs Nashimoto-Wright's extended one-sided studentised range test against an ordered alternative for normal data with equal variances.

Usage

MTest(x, ...)
# S3 method for default
MTest(x, g, alternative = c("greater", "less"), ...)
# S3 method for formula
MTest(
  formula,
  data,
  subset,
  na.action,
  alternative = c("greater", "less"),
  ...
)
# S3 method for aov
MTest(x, alternative = c("greater", "less"), ...)

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.

alternative

the alternative hypothesis. Defaults to greater.

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

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

the estimated statistic(s)

crit.value

critical values for $\alpha =0.05$.

alternative

a character string describing the alternative hypothesis.

parameter

the parameter(s) of the test distribution.

dist

a string that denotes the test distribution.

There are print and summary methods available.

Details

The procedure uses the property of a simple order, ${\theta }_m^{\prime }-{\mu }_m\le {\mu }_j-{\mu }_i\le {\mu }_l^{\prime }-{\mu }_l(l\le i\le m\mathrm{and}{m}^{\prime }\le j\le l^{\prime })$. The null hypothesis H${}_{ij}:{\mu }_i={\mu }_j$ is tested against the alternative A${}_{ij}:{\mu }_i<{\mu }_j$ for any $1\le i<j\le k$.

The all-pairs comparisons test statistics for a balanced design are $${\hat{h}}_{ij}=\underset{i\le m<m^{\prime }\le j}{max}\frac{({\overline{x}}_{m^{\prime }}-{\overline{x}}_m)}{s_{\mathrm{in}}/\sqrt{n}},$$

with $n=n_i;N=\sum _i^k{n}_i(1\le i\le k)$, ${\overline{x}}_i$ the arithmetic mean of the $i$th group, and $s_{\mathrm{in}}^2$ the within ANOVA variance. The null hypothesis is rejected, if $\hat{h}>h_{k,\alpha ,v}$, with $v=N-k$ degree of freedom.

For the unbalanced case with moderate imbalance the test statistic is $${\hat{h}}_{ij}=\underset{i\le m<m^{\prime }\le j}{max}\frac{({\overline{x}}_{m^{\prime }}-{\overline{x}}_m)}{s_{\mathrm{in}}{(1/n_m+1/n_{m^{\prime }})}^{1/2}},$$

The null hypothesis is rejected, if ${\hat{h}}_{ij}>h_{k,\alpha ,v}/\sqrt{2}$.

The function does not return p-values. Instead the critical h-values as given in the tables of Hayter (1990) for $\alpha =0.05$ (one-sided) are looked up according to the number of groups ($k$) and the degree of freedoms ($v$).

References

Hayter, A. J.(1990) A One-Sided Studentised Range Test for Testing Against a Simple Ordered Alternative, Journal of the American Statistical Association 85, 778--785.

Nashimoto, K., Wright, F.T., (2005) Multiple comparison procedures for detecting differences in simply ordered means. Comput. Statist. Data Anal. 48, 291--306.

See Also

osrtTest, NPMTest

Examples

# NOT RUN {
##
md <- aov(weight ~ group, PlantGrowth)
anova(md)
osrtTest(md)
MTest(md)
# }