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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_inv_cdf_studentized_range (g01fm)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_stat_inv_cdf_studentized_range (g01fm) returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.


[result, ifail] = g01fm(p, v, ir)
[result, ifail] = nag_stat_inv_cdf_studentized_range(p, v, ir)


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,  
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.


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


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


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.

On entry,p0.0,
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.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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.


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));

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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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