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 (Declaration)
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: v≥1.0.

ir: Type: System..::.Int32

On entry: r, the number of groups.

Constraint: ir≥2.

ifail: Type: System..::.Int32 %

On exit: ifail=0 unless the method detects an error (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 - minxi σ^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-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=C∫0∞xv-1e-vx2/2 r∫-∞∞ϕyΦy-Φy-qx r-1dydx,

where

C=vv/2Γ v/22v/2- 1 , ϕ y=12πe-y2/2 and Φ y=∫-∞yϕ tdt.

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, p≤0.0,
or p≥1.0,
or v<1.0,
or ir<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.

Accuracy

The returned solution, q*, to equation (1) is determined so that at least one of the following criteria apply.

(a)	Pq*;v,r-p0≤0.000005
(b)	q0-q≤0.000005×max1.0,q.

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

On entry,	p≤0.0,
or	p≥1.0,
or	v<1.0,
or	ir<2.