G01EMF returns the probability associated with the lower tail 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'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,
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
The above two-dimensional integral is evaluated using
D01DAF
with the upper and lower limits computed to give stated accuracy (see
Section 7).
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
D01AMF.
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
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
The returned value will have absolute accuracy to at least four decimal places (usually five), unless IFAIL=2. When IFAIL=2 it is usual that the returned value will be a good estimate of the true value.
None.
The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of q, ν and r.