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PMCMRplus (version 1.9.3)

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

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 $$ \hat{h}_{ij} = \max_{i \le m < m' \le j} \frac{\left(\bar{x}_{m'} - \bar{x}_m \right)} {s_{\mathrm{in}} \left(1/n_m + 1/n_{m'}\right)^{1/2}}, $$

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

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

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