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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_prob_studentized_range (g01em)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


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


[result, ifail] = g01em(q, v, ir)
[result, ifail] = nag_stat_prob_studentized_range(q, 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'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 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).


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


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


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.

On entry,q0.0,
W  ifail=2
There is some doubt as to whether full accuracy has been achieved.
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 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



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);

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

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

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