MTest() R function from [PMCMRplus]

Extended One-Sided Studentised Range Test

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


MTest(x, ...)

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

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

## S3 method for class '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").

Returns

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, c(" $\\theta_m' - \\mu_m \\le \\mu_j - \\mu_i \\le \\mu_l' - \\mu_l\n$ ", " $\\qquad (l \\le i \\le m~\\mathrm{and}~ m' \\le j \\le l')$ "). 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} = \max_{i \le m < m' \le j} \frac{\left(\bar{x}_{m'} - \bar{x}_m \right)}{s_{\mathrm{in}} / \sqrt{n}},%SEE PDF

with $n = n_i; ~ N = \sum_i^k n_i ~~ (1 \le i \le k)$ , $\bar{x}_i$ 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} = \max_{i \le m < m' \le j} \frac{\left(\bar{x}_{m'} - \bar{x}_m \right)}{s_{\mathrm{in}} \left(1/n_m + 1/n_{m'}\right)^{1/2}},%SEE PDF

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 ( $v$ ).

Note

The function will give a warning for the unbalanced case and returns the critical value $h_{k,\alpha,\infty} / \sqrt{2}$ .

Examples


##
md <- aov(weight ~ group, PlantGrowth)
anova(md)
osrtTest(md)
MTest(md)

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.

MTest function