The function may be called by the names: g01emc, nag_stat_prob_studentized_range or nag_prob_studentized_range.
The externally Studentized range, , for a sample, , is defined as:
where is an independent estimate of the standard error of the '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, , 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:
The above two-dimensional integral is evaluated using
with the upper and lower limits computed to give stated accuracy (see Section 7).
If the degrees of freedom are greater than the probability integral can be approximated by its asymptotic form:
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.
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 value will have absolute accuracy to at least four decimal places (usually five), unless NE_ACCURACY. When NE_ACCURACY it is usual that the returned value will be a good estimate of the true value.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g01emc 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.
The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of , and .