Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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$q$, 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${x}_{i}$. 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${\stackrel{-}{T}}_{\text{largest}}$ and Tsmallest${\stackrel{-}{T}}_{\text{smallest}}$, divided by the square root of the mean-square experimental error, MSerror$M{S}_{\text{error}}$, 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. − ∞
$C=vv/2Γ (v/2)2v/2- 1 , ϕ (y)=12πe-y2/2 and Φ (y)=∫-∞yϕ (t)dt.$
For a given probability p0${p}_{0}$, the deviate q0${q}_{0}$ is found as the solution to the equation
 P(q0;v,r) = p0, $P(q0;v,r)=p0,$ (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:     p – double scalar
The lower tail probability for the Studentized range statistic, p0${p}_{0}$.
Constraint: 0.0 < p < 1.0$0.0<{\mathbf{p}}<1.0$.
2:     v – double scalar
v$v$, the number of degrees of freedom.
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]).

## 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$0.0$.

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

ifail = 1${\mathbf{ifail}}=1$
 On entry, p ≤ 0.0${\mathbf{p}}\le 0.0$, or p ≥ 1.0${\mathbf{p}}\ge 1.0$, or v < 1.0${\mathbf{v}}<1.0$, or ir < 2${\mathbf{ir}}<2$.
ifail = 2${\mathbf{ifail}}=2$
The function was unable to find an upper bound for the value of q0${q}_{0}$. This will be caused by p0${p}_{0}$ being too close to 1.0$1.0$.
W ifail = 3${\mathbf{ifail}}=3$
There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.

## Accuracy

The returned solution, q*${q}_{*}$, to equation (1) is determined so that at least one of the following criteria apply.
 (a) |P(q*;v,r) − p0| ≤ 0.000005$|P\left({q}_{*}\text{;}v,r\right)-{p}_{0}|\le 0.000005$ (b) |q0 − q*| ≤ 0.000005 × max (1.0,|q*|)$|{q}_{0}-{q}_{*}|\le 0.000005×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,|{q}_{*}|\right)$.

To obtain the factors for Duncan's multiple-range test, equation (1) has to be solved for p1${p}_{1}$, where p1 = p0r1${p}_{1}={p}_{0}^{r-1}$, so on input p should be set to p0r1${p}_{0}^{r-1}$.

## Example

```function nag_stat_inv_cdf_studentized_range_example
p = 0.95;
v = 10;
ir = int64(5);
[result, ifail] = nag_stat_inv_cdf_studentized_range(p, v, ir)
```
```

result =

4.6543

ifail =

0

```
```function g01fm_example
p = 0.95;
v = 10;
ir = int64(5);
[result, ifail] = g01fm(p, v, ir)
```
```

result =

4.6543

ifail =

0

```