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
c05azf
.
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
If on entry
${\mathbf{ifail}}=0$ or
$-1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
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) |
$\left|{q}_{0}-{q}_{*}\right|\le 0.000005\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,\left|{q}_{*}\right|\right)$. |
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
${p}_{0}^{r-1}$.