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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$q$, for a sample, x1,x2,,xr${x}_{1},{x}_{2},\dots ,{x}_{r}$, is defined as:
 q = ( max xi − min xi )/(σ̂e) , $q = max⁡xi - min⁡xi σ^e ,$
where σ̂e${\stackrel{^}{\sigma }}_{e}$ is an independent estimate of the standard error of the xi${x}_{i}$'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, T1,T2,,Tr${\stackrel{-}{T}}_{1},{\stackrel{-}{T}}_{2},\dots ,{\stackrel{-}{T}}_{r}$, the Studentized range statistic is defined to be the difference between the largest and smallest means, Tlargest${\stackrel{-}{T}}_{\mathrm{largest}}$ and Tsmallest${\stackrel{-}{T}}_{\mathrm{smallest}}$, divided by the square root of the mean-square experimental error, MSerror$M{S}_{\mathrm{error}}$, over the number of observations in each group, n$n$, i.e.,
 q = (Tlargest − Tsmallest)/(sqrt(MSerror / n)). $q=T-largest-T-smallest MSerror/n .$
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(q ; v,r)$P\left(q;v,r\right)$, for v$v$ degrees of freedom and r$r$ groups can be written as:
P(q ; v,r) = Cxv1evx2 / 2
 ( ∞ ) r ∫ φ(y)[Φ(y) − Φ(y − qx)]r − 1dy − ∞
dx,
0
$P(q;v,r)=C∫0∞xv-1e-vx2/2 {r∫-∞∞ϕ(y)[Φ(y)-Φ(y-qx)] r-1dy}dx,$
where
 y C = (vv / 2)/(Γ (v / 2)2v / 2 − 1),  φ(y) = 1/(sqrt(2π ))e − y2 / 2  and  Φ(y) = ∫ φ(t)dt. − ∞
$C=vv/2Γ (v/2)2v/2- 1 , ϕ (y)=12π e-y2/2 and Φ (y)=∫-∞yϕ (t) dt.$
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 Section [Accuracy]).
If the degrees of freedom v$v$ are greater than 2000$2000$ the probability integral can be approximated by its asymptotic form:
 ∞ P(q ; r) = r ∫ φ(y)[Φ(y) − Φ(y − q)]r − 1dy. − ∞
$P(q;r)=r∫-∞∞ϕ(y)[Φ(y)-Φ(y-q)] r-1dy.$
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:     q – double scalar
q$q$, the Studentized range statistic.
Constraint: q > 0.0${\mathbf{q}}>0.0$.
2:     v – double scalar
v$v$, the number of degrees of freedom for the experimental error.
Constraint: v1.0${\mathbf{v}}\ge 1.0$.
3:     ir – int64int32nag_int scalar
r$r$, the number of groups.
Constraint: ir2${\mathbf{ir}}\ge 2$.

None.

None.

### Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ 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$0.0$.

## 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.

ifail = 1${\mathbf{ifail}}=1$
 On entry, q ≤ 0.0${\mathbf{q}}\le 0.0$, or v < 1.0${\mathbf{v}}<1.0$, or ir < 2${\mathbf{ir}}<2$.
W ifail = 2${\mathbf{ifail}}=2$
There is some doubt as to whether full accuracy has been achieved.

## 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

```function nag_stat_prob_studentized_range_example
q = 4.6543;
v = 10;
ir = int64(5);
[result, ifail] = nag_stat_prob_studentized_range(q, v, ir)
```
```

result =

0.9500

ifail =

0

```
```function g01em_example
q = 4.6543;
v = 10;
ir = int64(5);
[result, ifail] = g01em(q, v, ir)
```
```

result =

0.9500

ifail =

0

```