# NAG Library Routine Document

## 1Purpose

g01fmf returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.

## 2Specification

Fortran Interface
 Function g01fmf ( p, v, ir,
 Real (Kind=nag_wp) :: g01fmf Integer, Intent (In) :: ir Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: p, v
#include <nagmk26.h>
 double g01fmf_ (const double *p, const double *v, const Integer *ir, Integer *ifail)

## 3Description

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 c05azf . Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.

## 4References

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

## 5Arguments

1:     $\mathbf{p}$ – Real (Kind=nag_wp)Input
On entry: the lower tail probability for the Studentized range statistic, ${p}_{0}$.
Constraint: $0.0<{\mathbf{p}}<1.0$.
2:     $\mathbf{v}$ – Real (Kind=nag_wp)Input
On entry: $v$, the number of degrees of freedom.
Constraint: ${\mathbf{v}}\ge 1.0$.
3:     $\mathbf{ir}$ – IntegerInput
On entry: $r$, the number of groups.
Constraint: ${\mathbf{ir}}\ge 2$.
4:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g01fmf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
If on exit ${\mathbf{ifail}}={\mathbf{1}}$, then g01fmf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ir}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ir}}\ge 2$.
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{p}}<1.0$.
On entry, ${\mathbf{v}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{v}}\ge 1.0$.
${\mathbf{ifail}}=2$
The routine was unable to find an upper bound for the value of ${q}_{0}$. This will be caused by ${p}_{0}$ being too close to $1.0$.
${\mathbf{ifail}}=3$
There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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

## 8Parallelism and Performance

g01fmf is not threaded in any implementation.

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

## 10Example

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

### 10.1Program Text

Program Text (g01fmfe.f90)

### 10.2Program Data

Program Data (g01fmfe.d)

### 10.3Program Results

Program Results (g01fmfe.r)