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NAG Toolbox

NAG Toolbox: nag_stat_prob_studentized_range (g01em)

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, qq, for a sample, x1,x2,,xrx1,x2,,xr, is defined as:
q = ( max xi min xi )/(σ̂e) ,
q = maxxi - minxi σ^e ,
where σ̂eσ^e is an independent estimate of the standard error of the xixi'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, 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)=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 Section [Accuracy]).
If the degrees of freedom vv are greater than 20002000 the probability integral can be approximated by its asymptotic form:
P(q ; r) = rφ(y)[Φ(y)Φ(yq)]r1dy.
P(q;r)=r-ϕ(y)[Φ(y)-Φ(y-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
qq, the Studentized range statistic.
Constraint: q > 0.0q>0.0.
2:     v – double scalar
vv, the number of degrees of freedom for the experimental error.
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]).
If on exit ifail = 1ifail=1, then nag_stat_prob_studentized_range (g01em) returns to 0.00.0.

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 = 1ifail=1
On entry,q0.0q0.0,
orv < 1.0v<1.0,
orir < 2ir<2.
W ifail = 2ifail=2
There is some doubt as to whether full accuracy has been achieved.

Accuracy

The returned value will have absolute accuracy to at least four decimal places (usually five), unless ifail = 2ifail=2. When ifail = 2ifail=2 it is usual that the returned value will be a good estimate of the true value.

Further Comments

None.

Example

function nag_stat_prob_studentized_range_example
q = 4.6543;
v = 10;
ir = int64(5);
[result, ifail] = nag_stat_prob_studentized_range(q, v, ir)
 

result =

    0.9500


ifail =

                    0


function g01em_example
q = 4.6543;
v = 10;
ir = int64(5);
[result, ifail] = g01em(q, v, ir)
 

result =

    0.9500


ifail =

                    0



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Chapter Introduction
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