public static double g01fm(
	double p,
	double v,
	int ir,
	out int ifail
)

Public Shared Function g01fm ( _
	p As Double, _
	v As Double, _
	ir As Integer, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double g01fm(
	double p, 
	double v, 
	int ir, 
	[OutAttribute] int% ifail
)

static member g01fm : 
        p : float * 
        v : float * 
        ir : int * 
        ifail : int byref -> float 

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

The externally Studentized range, $q$, for a sample, $x_1,x_2,\dots ,x_r$, is defined as where ${\hat{\sigma }}_e$ is an independent estimate of the standard error of the $x_i$. 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, ${\stackrel{-}{T}}_1,{\stackrel{-}{T}}_2,\dots ,{\stackrel{-}{T}}_r$, the Studentized range statistic is defined to be the difference between the largest and smallest means, ${\stackrel{-}{T}}_{\text{largest}}$ and ${\stackrel{-}{T}}_{\text{smallest}}$, divided by the square root of the mean-square experimental error, $M{S}_{\text{error}}$, over the number of observations in each group, $n$, i.e., 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, $P\left(q;v,r\right)$, for $v$ degrees of freedom and $r$ groups, can be written as: where For a given probability $p_0$, the deviate $q_0$ is found as the solution to the equation using c05az . Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.

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

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 $\mathbf{ifail}=1$, then g01fm returns $0.0$.

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

(a)	$\left|P\left(q_{*}\text{;}v,r\right)-p_0\right|\le 0.000005$
(b)	$\left|q_0-q_{*}\right|\le 0.000005\times \max \left(1.0,\left|q_{*}\right|\right)$.

None.

To obtain the factors for Duncan's multiple-range test, equation (1) has to be solved for $p_1$, where $p_1=p_0^{r-1}$, so on input p should be set to $p_0^{r-1}$.

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