The function may be called by the names: g01fmc, nag_stat_inv_cdf_studentized_range or nag_deviates_studentized_range.
The externally Studentized range, , for a sample, , is defined as
where is an independent estimate of the standard error of the . 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, , the Studentized range statistic is defined to be the difference between the largest and smallest means, and , divided by the square root of the mean-square experimental error, , over the number of observations in each group, , 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, , for degrees of freedom and groups, can be written as:
For a given probability , the deviate is found as the solution to the equation
a root-finding procedure.
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
1: – doubleInput
On entry: the lower tail probability for the Studentized range statistic, .
2: – doubleInput
On entry: , the number of degrees of freedom.
3: – IntegerInput
On entry: , the number of groups.
4: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
The function was unable to find an upper bound for the value of . This will be caused by being too close to .
On entry, .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, .
On entry, .
The returned solution, , to equation (1) is determined so that at least one of the following criteria apply.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g01fmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
To obtain the factors for Duncan's multiple-range test, equation (1) has to be solved for , where , so on input p should be set to .
Three values of , and are read in and the Studentized range deviates or quantiles are computed and printed.