public static double g01em(
	double q,
	double v,
	int ir,
	out int ifail
)

Public Shared Function g01em ( _
	q As Double, _
	v As Double, _
	ir As Integer, _
	<OutAttribute> ByRef ifail As Integer _
) As Double

public:
static double g01em(
	double q, 
	double v, 
	int ir, 
	[OutAttribute] int% ifail
)

static member g01em : 
        q : float * 
        v : float * 
        ir : int * 
        ifail : int byref -> float 

g01em returns the probability associated with the lower tail 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 $x_i$'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, ${\stackrel{-}{T}}_1,{\stackrel{-}{T}}_2,\dots ,{\stackrel{-}{T}}_r$, the Studentized range statistic is defined to be the difference between the largest and smallest means, ${\stackrel{-}{T}}_{\mathrm{largest}}$ and ${\stackrel{-}{T}}_{\mathrm{smallest}}$, divided by the square root of the mean-square experimental error, $M{S}_{\mathrm{error}}$, 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 The above two-dimensional integral is evaluated using d01da with the upper and lower limits computed to give stated accuracy (see [Accuracy]).

If the degrees of freedom $v$ are greater than $2000$ the probability integral can be approximated by its asymptotic form: This integral is evaluated using d01am.

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

Note: g01em may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

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

None.

None.

The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of $q$, $\nu$ and $r$.