NAG Library Function Document
nag_deviates_studentized_range (g01fmc) returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.
||nag_deviates_studentized_range (double p,
The externally Studentized range,
, for a sample,
, is defined as
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,
, divided by the square root of the mean-square experimental error,
, over the number of observations in each group,
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,
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
p – doubleInput
On entry: the lower tail probability for the Studentized range statistic, .
v – doubleInput
On entry: , the number of degrees of freedom.
ir – IntegerInput
On entry: , the number of groups.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Warning – There is some doubt as to whether full accuracy has been achieved.
Unable to find initial estimate.
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
On entry, .
On entry, .
The returned solution,
, to equation (1)
is determined so that at least one of the following criteria apply.
8 Parallelism and Performance
To obtain the factors for Duncan's multiple-range test, equation (1)
has to be solved for
, 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.
10.1 Program Text
Program Text (g01fmce.c)
10.2 Program Data
Program Data (g01fmce.d)
10.3 Program Results
Program Results (g01fmce.r)