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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_stat_prob_studentized_range (g01em)

## Purpose

nag_stat_prob_studentized_range (g01em) returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

## Syntax

[result, ifail] = g01em(q, v, ir)
[result, ifail] = nag_stat_prob_studentized_range(q, v, ir)

## Description

The externally Studentized range, $q$, for a sample, ${x}_{1},{x}_{2},\dots ,{x}_{r}$, is defined as:
 $q = max⁡xi - min⁡xi σ^e ,$
where ${\stackrel{^}{\sigma }}_{e}$ is an independent estimate of the standard error of the ${x}_{i}$'s. 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, ${\stackrel{-}{T}}_{\mathrm{largest}}$ and ${\stackrel{-}{T}}_{\mathrm{smallest}}$, divided by the square root of the mean-square experimental error, $M{S}_{\mathrm{error}}$, 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
 $C=vv/2Γ v/22v/2- 1 , ϕ y=12π e-y2/2 and Φ y=∫-∞yϕ t dt.$
The above two-dimensional integral is evaluated using nag_quad_2d_fin (d01da) with the upper and lower limits computed to give stated accuracy (see Accuracy).
If the degrees of freedom $v$ are greater than $2000$ the probability integral can be approximated by its asymptotic form:
 $Pq;r=r∫-∞∞ϕyΦy-Φy-q r-1dy.$
This integral is evaluated using nag_quad_1d_inf (d01am).

## References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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:     $\mathrm{q}$ – double scalar
$q$, the Studentized range statistic.
Constraint: ${\mathbf{q}}>0.0$.
2:     $\mathrm{v}$ – double scalar
$v$, the number of degrees of freedom for the experimental error.
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).
If on exit ${\mathbf{ifail}}={\mathbf{1}}$, then nag_stat_prob_studentized_range (g01em) returns to $0.0$.

## Error Indicators and Warnings

Note: nag_stat_prob_studentized_range (g01em) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

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, ${\mathbf{q}}\le 0.0$, or ${\mathbf{v}}<1.0$, or ${\mathbf{ir}}<2$.
W  ${\mathbf{ifail}}=2$
There is some doubt as to whether full accuracy has been achieved.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The returned value will have absolute accuracy to at least four decimal places (usually five), unless ${\mathbf{ifail}}={\mathbf{2}}$. When ${\mathbf{ifail}}={\mathbf{2}}$ it is usual that the returned value will be a good estimate of the true value.

None.

## Example

The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of $q$, $\nu$ and $r$.
```function g01em_example

fprintf('g01em example results\n\n');

% Probability for Studentized range statistic distribution
q  = [ 4.6543;  2.8099; 4.2636];
v  = [10;      60.0;    5.0];
ir = [int64(5); 12;   4];

fprintf('   q       v     ir    probability\n');
for j = 1:numel(q);

[p, ifail] = g01em( ...
q(j) , v(j),  ir(j));

fprintf('%8.4f%6.1f%4d%12.4f\n', q(j), v(j), ir(j), p);
end

```
```g01em example results

q       v     ir    probability
4.6543  10.0   5      0.9500
2.8099  60.0  12      0.3000
4.2636   5.0   4      0.9000
```