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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_studentized_range (g01fm)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

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, 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. 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,  
where
C=vv/2Γ v/22v/2- 1 ,   ϕ y=12πe-y2/2   and   Φ y=-yϕ tdt.  
For a given probability p0, the deviate q0 is found as the solution to the equation
Pq0;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.
Constraint: 0.0<p<1.0.
2:     v – double scalar
v, the number of degrees of freedom.
Constraint: v1.0.
3:     ir int64int32nag_int scalar
r, the number of groups.
Constraint: ir2.

Optional Input Parameters

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail int64int32nag_int scalar
ifail=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 ifail=1, then nag_stat_inv_cdf_studentized_range (g01fm) returns 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
On entry,p0.0,
orp1.0,
orv<1.0,
orir<2.
   ifail=2
The function was unable to find an upper bound for the value of q0. This will be caused by p0 being too close to 1.0.
W  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.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

The returned solution, q*, to equation (1) is determined so that at least one of the following criteria apply.
(a) Pq*;v,r-p00.000005
(b) q0-q*0.000005×max1.0,q*.

Further Comments

To obtain the factors for Duncan's multiple-range test, equation (1) has to be solved for p1, where p1=p0r-1, so on input p should be set to p0r-1.

Example

Three values of p, ν and r are read in and the Studentized range deviates or quantiles are computed and printed.
function g01fm_example


fprintf('g01fm example results\n\n');

p  = [    0.95     0.30    0.9];
v  = [   10       60       5  ];
ir = [int64(5)  12       4  ];
quantile  = p;

fprintf('     p       v    ir   quantile\n');
for j = 1:numel(p)
   [quantile(j), ifail] = g01fm( ...
				 p(j), v(j), ir(j));
end

fprintf('%8.3f%8.3f%4d %8.3f\n', [p; v; double(ir); quantile]);


g01fm example results

     p       v    ir   quantile
   0.950  10.000   5    4.654
   0.300  60.000  12    2.810
   0.900   5.000   4    4.264

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Chapter Introduction
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