nag_prob_studentized_range (g01emc) (PDF version)
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g01 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_prob_studentized_range (g01emc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_prob_studentized_range (g01emc) returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

2  Specification

#include <nag.h>
#include <nagg01.h>
double  nag_prob_studentized_range (double q, double v, Integer ir, NagError *fail)

3  Description

The externally Studentized range, q, for a sample, x1,x2,,xr, is defined as:
q = maxxi - minxi σ^e ,
where σ^e is an independent estimate of the standard error of the xi'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, T-1,T-2,,T-r, the Studentized range statistic is defined to be the difference between the largest and smallest means, T-largest and T-smallest, divided by the square root of the mean-square experimental error, MSerror, 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, Pq;v,r, for v degrees of freedom and r groups can be written as:
Pq;v,r=C0xv-1e-vx2/2 r-ϕyΦy-Φy-qx r-1dydx,
C=vv/2Γ v/22v/2- 1 ,   ϕ y=12π e-y2/2   and   Φ y=-yϕ t dt.
The above two-dimensional integral is evaluated using numerical quadrature with the upper and lower limits computed to give stated accuracy (see Section 7).
If the degrees of freedom v are greater than 2000 the probability integral can be approximated by its asymptotic form:
Pq;r=r-ϕyΦy-Φy-q r-1dy.
This integral is evaluated using nag_1d_quad_inf_1 (d01smc).

4  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

5  Arguments

1:     qdoubleInput
On entry: q, the Studentized range statistic.
Constraint: q>0.0.
2:     vdoubleInput
On entry: v, the number of degrees of freedom for the experimental error.
Constraint: v1.0.
3:     irIntegerInput
On entry: r, the number of groups.
Constraint: ir2.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
If on exit fail.code= NE_INT or NE_REAL, then nag_prob_studentized_range (g01emc) returns to 0.0.

6  Error Indicators and Warnings

Warning – There is some doubt as to whether full accuracy has been achieved.
On entry, ir=value.
Constraint: ir2.
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.
On entry, q=value.
Constraint: q>0.0.
On entry, v=value.
Constraint: v1.0.

7  Accuracy

The returned value will have absolute accuracy to at least four decimal places (usually five), unless fail.code= NE_ACCURACY. When fail.code= NE_ACCURACY it is usual that the returned value will be a good estimate of the true value.

8  Further Comments


9  Example

The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of q, ν and r.

9.1  Program Text

Program Text (g01emce.c)

9.2  Program Data

Program Data (g01emce.d)

9.3  Program Results

Program Results (g01emce.r)

nag_prob_studentized_range (g01emc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012