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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_inv_cdf_studentized_range (g01fm)

## Purpose

nag_stat_inv_cdf_studentized_range (g01fm) returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.

## Syntax

[result, ifail] = g01fm(p, v, ir)
[result, ifail] = nag_stat_inv_cdf_studentized_range(p, v, ir)

## Description

The externally Studentized range, $q$, for a sample, ${x}_{1},{x}_{2},\dots ,{x}_{r}$, is defined as
 $q = maxxi - minxi σ^e ,$
where ${\stackrel{^}{\sigma }}_{e}$ 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.,
 $q=T-largest-T-smallest MSerror/n .$
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:
 $Pq;v,r=C∫0∞xv-1e-vx2/2 r∫-∞∞ϕyΦy-Φy-qx r-1dydx,$
where
 $C=vv/2Γ v/22v/2- 1 , ϕ y=12πe-y2/2 and Φ y=∫-∞yϕ tdt.$
For a given probability ${p}_{0}$, the deviate ${q}_{0}$ is found as the solution to the equation
 $Pq0;v,r=p0,$ (1)
using nag_roots_contfn_brent_rcomm (c05az) . Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.

## References

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
Montgomery D C (1984) Design and Analysis of Experiments Wiley
Winer B J (1970) Statistical Principles in Experimental Design McGraw–Hill

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{p}$ – double scalar
The lower tail probability for the Studentized range statistic, ${p}_{0}$.
Constraint: $0.0<{\mathbf{p}}<1.0$.
2:     $\mathrm{v}$ – double scalar
$v$, the number of degrees of freedom.
Constraint: ${\mathbf{v}}\ge 1.0$.
3:     $\mathrm{ir}$int64int32nag_int scalar
$r$, the number of groups.
Constraint: ${\mathbf{ir}}\ge 2$.

None.

### Output Parameters

1:     $\mathrm{result}$ – double scalar
The result of the function.
2:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_stat_inv_cdf_studentized_range (g01fm) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
If on exit ${\mathbf{ifail}}={\mathbf{1}}$, then nag_stat_inv_cdf_studentized_range (g01fm) returns $0.0$.

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{p}}\le 0.0$, or ${\mathbf{p}}\ge 1.0$, or ${\mathbf{v}}<1.0$, or ${\mathbf{ir}}<2$.
${\mathbf{ifail}}=2$
The function was unable to find an upper bound for the value of ${q}_{0}$. This will be caused by ${p}_{0}$ being too close to $1.0$.
W  ${\mathbf{ifail}}=3$
There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

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) $\left|{q}_{0}-{q}_{*}\right|\le 0.000005×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,\left|{q}_{*}\right|\right)$.

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 ${p}_{0}^{r-1}$.

## Example

Three values of $p$, $\nu$ and $r$ are read in and the Studentized range deviates or quantiles are computed and printed.
```function g01fm_example

fprintf('g01fm example results\n\n');

p  = [    0.95     0.30    0.9];
v  = [   10       60       5  ];
ir = [int64(5)  12       4  ];
quantile  = p;

fprintf('     p       v    ir   quantile\n');
for j = 1:numel(p)
[quantile(j), ifail] = g01fm( ...
p(j), v(j), ir(j));
end

fprintf('%8.3f%8.3f%4d %8.3f\n', [p; v; double(ir); quantile]);

```
```g01fm example results

p       v    ir   quantile
0.950  10.000   5    4.654
0.300  60.000  12    2.810
0.900   5.000   4    4.264
```