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NAG Toolbox: nag_stat_prob_studentized_range (g01em)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_stat_prob_studentized_range (g01em) returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

Syntax

[result, ifail] = g01em(q, v, ir)
[result, ifail] = nag_stat_prob_studentized_range(q, 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'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,  
where
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 nag_quad_2d_fin (d01da) with the upper and lower limits computed to give stated accuracy (see Accuracy).
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_quad_1d_inf (d01am).

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

Parameters

Compulsory Input Parameters

1:     q – double scalar
q, the Studentized range statistic.
Constraint: q>0.0.
2:     v – double scalar
v, the number of degrees of freedom for the experimental error.
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).
If on exit ifail=1, then nag_stat_prob_studentized_range (g01em) returns to 0.0.

Error Indicators and Warnings

Note: nag_stat_prob_studentized_range (g01em) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

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,q0.0,
orv<1.0,
orir<2.
W  ifail=2
There is some doubt as to whether full accuracy has been achieved.
   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 value will have absolute accuracy to at least four decimal places (usually five), unless ifail=2. When ifail=2 it is usual that the returned value will be a good estimate of the true value.

Further Comments

None.

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.
function g01em_example


fprintf('g01em example results\n\n');

% Probability for Studentized range statistic distribution
q  = [ 4.6543;  2.8099; 4.2636];
v  = [10;      60.0;    5.0];
ir = [int64(5); 12;   4];

fprintf('   q       v     ir    probability\n');
for j = 1:numel(q);

  [p, ifail] = g01em( ...
                      q(j) , v(j),  ir(j));

  fprintf('%8.4f%6.1f%4d%12.4f\n', q(j), v(j), ir(j), p);
end


g01em example results

   q       v     ir    probability
  4.6543  10.0   5      0.9500
  2.8099  60.0  12      0.3000
  4.2636   5.0   4      0.9000

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Chapter Contents
Chapter Introduction
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