R: One-Sided Studentized Range Test

osrtTest {PMCMRplus} | R Documentation

## One-Sided Studentized Range Test

### Description

Performs Hayter's one-sided studentized range test against an ordered alternative for normal data with equal variances.

### Usage

    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"), ...)

### 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 `NA`s. Defaults to `getOption("na.action")`.

### Details

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`.

### Value

A list with class `"osrt"` that contains the following components:

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 α = 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.

### Note

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}.

### 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.

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.

### See Also

`link{hayterStoneTest}` `MTest`

### Examples

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

[Package PMCMRplus version 1.7.0 Index]