nag_prob_studentized_range (g01emc) returns the probability associated with the lower tail of the distribution of the Studentized range statistic.
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:
where
The above two-dimensional integral is evaluated using
numerical quadrature
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:
This integral is evaluated using
nag_1d_quad_inf_1 (d01smc).
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
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.
nag_prob_studentized_range (g01emc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
Users' Note for your implementation for any additional implementation-specific information.
None.
The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of , and .