nag_deviates_studentized_range (g01fmc) (PDF version)
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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) $\left|{q}_{0}-{q}_{*}\right|\le 0.000005×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,\left|{q}_{*}\right|\right)$.

8  Further Comments

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)

nag_deviates_studentized_range (g01fmc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual