g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_deviates_studentized_range (g01fmc)

## 1  Purpose

nag_deviates_studentized_range (g01fmc) returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.

## 2  Specification

 #include #include
 double nag_deviates_studentized_range (double p, double v, Integer ir, NagError *fail)

## 3  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 a root-finding procedure. Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.

## 4  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

## 5  Arguments

1:     pdoubleInput
On entry: the lower tail probability for the Studentized range statistic, ${p}_{0}$.
Constraint: $0.0<{\mathbf{p}}<1.0$.
2:     vdoubleInput
On entry: $v$, the number of degrees of freedom.
Constraint: ${\mathbf{v}}\ge 1.0$.
3:     irIntegerInput
On entry: $r$, the number of groups.
Constraint: ${\mathbf{ir}}\ge 2$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ACCURACY
Warning – There is some doubt as to whether full accuracy has been achieved.
NE_INIT_ESTIMATE
Unable to find initial estimate.
NE_INT
On entry, ${\mathbf{ir}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ir}}\ge 2$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL
On entry, ${\mathbf{p}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{p}}<1.0$.
On entry, ${\mathbf{v}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{v}}\ge 1.0$.

## 7  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 ${p}_{0}^{r-1}$.

## 9  Example

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

### 9.1  Program Text

Program Text (g01fmce.c)

### 9.2  Program Data

Program Data (g01fmce.d)

### 9.3  Program Results

Program Results (g01fmce.r)