The externally Studentized range,
$q$, for a sample,
${x}_{1},{x}_{2},\dots ,{x}_{r}$, is defined as
where
${\hat{\sigma}}_{e}$ is an independent estimate of the standard error of the
${x}_{i}$. The most common use of this statistic is in the testing of means from a balanced design. In this case for a set of group means,
${\stackrel{-}{T}}_{1},{\stackrel{-}{T}}_{2},\dots ,{\stackrel{-}{T}}_{r}$, the Studentized range statistic is defined to be the difference between the largest and smallest means,
${\stackrel{-}{T}}_{\text{largest}}$ and
${\stackrel{-}{T}}_{\text{smallest}}$, divided by the square root of the mean-square experimental error,
$M{S}_{\text{error}}$, over the number of observations in each group,
$n$, i.e.,
The Studentized range statistic can be used as part of a multiple comparisons procedure such as the Newman–Keuls procedure or Duncan's multiple range test (see
Montgomery (1984) and
Winer (1970)).
For a Studentized range statistic the probability integral,
$P\left(q;v,r\right)$, for
$v$ degrees of freedom and
$r$ groups, can be written as:
where
For a given probability
${p}_{0}$, the deviate
${q}_{0}$ is found as the solution to the equation
using
a root-finding procedure.
Initial estimates are found using the approximation given in
Lund and Lund (1983) and a simple search procedure.
Lund R E and Lund J R (1983) Algorithm AS 190: probabilities and upper quartiles for the studentized range Appl. Statist. 32(2) 204–210
The returned solution,
${q}_{*}$, to equation
(1) is determined so that at least one of the following criteria apply.
(a) |
$\left|P\left({q}_{*}\text{;}v,r\right)-{p}_{0}\right|\le 0.000005$ |
(b) |
$\left|{q}_{0}-{q}_{*}\right|\le 0.000005\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,\left|{q}_{*}\right|\right)$. |
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
To obtain the factors for Duncan's multiple-range test, equation
(1) has to be solved for
${p}_{1}$, where
${p}_{1}={p}_{0}^{r-1}$, so on input
p should be set to
${p}_{0}^{r-1}$.