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NAG Toolbox: nag_stat_inv_cdf_studentized_range (g01fm)

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, qq, for a sample, x1,x2,,xrx1,x2,,xr, is defined as
q = ( max (xi) min (xi) )/(σ̂e) ,
q = max(xi) - min(xi) σ^e ,
where σ̂eσ^e is an independent estimate of the standard error of the xixi. 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, T1,T2,,TrT-1,T-2,,T-r, the Studentized range statistic is defined to be the difference between the largest and smallest means, TlargestT-largest and TsmallestT-smallest, divided by the square root of the mean-square experimental error, MSerrorMSerror, over the number of observations in each group, nn, i.e.,
q = (TlargestTsmallest)/(sqrt(MSerror / n)).
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, P(q ; v,r)P(q;v,r), for vv degrees of freedom and rr groups, can be written as:
P(q ; v,r) = Cxv1evx2 / 2
( )
rφ(y)(Φ(y)Φ(yqx))r1dy
dx,
0
P(q;v,r)=C0xv-1e-vx2/2 (r-ϕ(y)(Φ(y)-Φ(y-qx)) r-1dy)dx,
where
y
C = (vv / 2)/(Γ (v / 2)2v / 2 1),  φ(y) = 1/(sqrt(2π))ey2 / 2  and  Φ(y) = φ(t)dt.
C=vv/2Γ (v/2)2v/2- 1 ,   ϕ (y)=12πe-y2/2   and   Φ (y)=-yϕ (t)dt.
For a given probability p0p0, the deviate q0q0 is found as the solution to the equation
P(q0;v,r) = p0,
P(q0;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, p0p0.
Constraint: 0.0 < p < 1.00.0<p<1.0.
2:     v – double scalar
vv, the number of degrees of freedom.
Constraint: v1.0v1.0.
3:     ir – int64int32nag_int scalar
rr, the number of groups.
Constraint: ir2ir2.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     result – double scalar
The result of the function.
2:     ifail – int64int32nag_int scalar
ifail = 0ifail=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 = 1ifail=1, then nag_stat_inv_cdf_studentized_range (g01fm) returns 0.00.0.

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

  ifail = 1ifail=1
On entry,p0.0p0.0,
orp1.0p1.0,
orv < 1.0v<1.0,
orir < 2ir<2.
  ifail = 2ifail=2
The function was unable to find an upper bound for the value of q0q0. This will be caused by p0p0 being too close to 1.01.0.
W ifail = 3ifail=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.

Accuracy

The returned solution, q*q*, to equation (1) is determined so that at least one of the following criteria apply.
(a) |P(q*;v,r)p0|0.000005|P(q*;v,r)-p0|0.000005
(b) |q0q*|0.000005 × max (1.0,|q*|)|q0-q*|0.000005×max(1.0,|q*|).

Further Comments

To obtain the factors for Duncan's multiple-range test, equation (1) has to be solved for p1p1, where p1 = p0r1p1=p0r-1, so on input p should be set to p0r1p0r-1.

Example

function nag_stat_inv_cdf_studentized_range_example
p = 0.95;
v = 10;
ir = int64(5);
[result, ifail] = nag_stat_inv_cdf_studentized_range(p, v, ir)
 

result =

    4.6543


ifail =

                    0


function g01fm_example
p = 0.95;
v = 10;
ir = int64(5);
[result, ifail] = g01fm(p, v, ir)
 

result =

    4.6543


ifail =

                    0



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

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