One-Sided Studentized Range Test
Performs Hayter's one-sided studentized range test against an ordered alternative for normal data with equal variances.
osrtTest(x, ...)
## Default S3 method:
osrtTest(x, g, alternative = c("greater", "less"), ...)
## S3 method for class 'formula'
osrtTest(
formula,
data,
subset,
na.action,
alternative = c("greater", "less"),
...
)
## S3 method for class 'aov'
osrtTest(x, alternative = c("greater", "less"), ...)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 |
alternative |
the alternative hypothesis. Defaults to |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
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 |
Hayter's one-sided studentized range test (OSRT) can be used for testing several treatment levels with a zero control in a balanced one-factorial design with normally distributed variables that have a common variance. The null hypothesis, H: μ_i = μ_j ~~ (i < j) is tested against a simple order alternative, A: μ_i < μ_j, with at least one inequality being strict.
The test statistic is calculated as,
SEE PDF.
with k the number of groups, n = n_1, n_2, …, n_k and s_{\mathrm{in}}^2 the within ANOVA variance. The null hypothesis is rejected, if \hat{h} > h_{k,α,v}, with v = N - k degree of freedom.
For the unbalanced case with moderate imbalance the test statistic is
SEE PDF.
The function does not return p-values. Instead the critical h-values
as given in the tables of Hayter (1990) for α = 0.05 (one-sided)
are looked up according to the number of groups (k) and
the degree of freedoms (v).
Non tabulated values are linearly interpolated with the function
approx.
A list with class "osrt" that contains the following components:
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 α = 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.
Hayter (1990) has tabulated critical h-values for balanced designs only. For some unbalanced designs some k = 3 critical h-values can be found in Hayter et al. 2001. ' The function will give a warning for the unbalanced case and returns the critical value h_{k,α,v} / √{2}.
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.
Hayter, A.J., Miwa, T., Liu, W. (2001) Efficient Directional Inference Methodologies for the Comparisons of Three Ordered Treatment Effects. J Japan Statist Soc 31, 153–174.
link{hayterStoneTest} MTest
## md <- aov(weight ~ group, PlantGrowth) anova(md) osrtTest(md) MTest(md)
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