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

Syntax

C#
public static double g01fm(
	double p,
	double v,
	int ir,
	out int ifail
)
Visual Basic
Public Shared Function g01fm ( _
	p As Double, _
	v As Double, _
	ir As Integer, _
	<OutAttribute> ByRef ifail As Integer _
) As Double
Visual C++
public:
static double g01fm(
	double p, 
	double v, 
	int ir, 
	[OutAttribute] int% ifail
)
F#
static member g01fm : 
        p : float * 
        v : float * 
        ir : int * 
        ifail : int byref -> float 

Parameters

p
Type: System..::..Double
On entry: the lower tail probability for the Studentized range statistic, p0.
Constraint: 0.0<p<1.0.
v
Type: System..::..Double
On entry: v, the number of degrees of freedom.
Constraint: v1.0.
ir
Type: System..::..Int32
On entry: r, the number of groups.
Constraint: ir2.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Return Value

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

Description

The externally Studentized range, q, for a sample, x1,x2,,xr, is defined as
q=maxxi-minxiσ^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-smallestMSerror/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-vx2/2r-ϕyΦy-Φy-qxr-1dydx,
where
C=vv/2Γv/22v/2-1,  ϕy=12πe-y2/2  and  Φy=-yϕtdt.
For a given probability p0, the deviate q0 is found as the solution to the equation
Pq0;v,r=p0, (1)
using 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

Error Indicators and Warnings

Note: g01fm may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
If on exit ifail=1, then g01fm returns 0.0.
ifail=1
On entry,p0.0,
orp1.0,
orv<1.0,
orir<2.
ifail=2
The method was unable to find an upper bound for the value of q0. This will be caused by p0 being too close to 1.0.
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.
ifail=-9000
An error occured, see message report.

Accuracy

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*.

Parallelism and Performance

None.

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.

Example

Three values of p, ν and r are read in and the Studentized range deviates or quantiles are computed and printed.

Example program (C#): g01fme.cs

Example program data: g01fme.d

Example program results: g01fme.r

See Also