Performs Nashimoto-Wright's extended one-sided studentised range test against an ordered alternative for normal data with equal variances.
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"), ...)
a numeric vector of data values, or a list of numeric data vectors.
further arguments to be passed to or from methods.
a vector or factor object giving the group for the
corresponding elements of "x".
Ignored with a warning if "x" is a list.
the alternative hypothesis. Defaults to greater.
a formula of the form response ~ group where
response gives the data values and group a vector or
factor of the corresponding groups.
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).
an optional vector specifying a subset of observations to be used.
a function which indicates what should happen when
the data contain NAs. Defaults to getOption("na.action").
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
the estimated statistic(s)
critical values for \(\alpha = 0.05\).
a character string describing the alternative hypothesis.
the parameter(s) of the test distribution.
a string that denotes the test distribution.
There are print and summary methods available.
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 (\(v\)).
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.
# NOT RUN {
##
md <- aov(weight ~ group, PlantGrowth)
anova(md)
osrtTest(md)
MTest(md)
# }
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