G01FMF returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic, via the routine name.
The externally Studentized range,
q, for a sample,
x1,x2,…,xr, is defined as
where
σ^e is an independent estimate of the standard error of the
xi. 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,
T-1,T-2,…,T-r, the Studentized range statistic is defined to be the difference between the largest and smallest means,
T-largest and
T-smallest, divided by the square root of the mean-square experimental error,
MSerror, 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,
Pq;v,r, for
v degrees of freedom and
r groups, can be written as:
where
For a given probability
p0, the deviate
q0 is found as the solution to the equation
using
C05AZF
.
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
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The returned solution,
q*, to equation
(1) is determined so that at least one of the following criteria apply.
(a) |
Pq*;v,r-p0≤0.000005 |
(b) |
q0-q*≤0.000005×max1.0,q*. |
To obtain the factors for Duncan's multiple-range test, equation
(1) has to be solved for
p1, where
p1=p0r-1, so on input
P should be set to
p0r-1.