NAG Library Function Document
nag_prob_studentized_range (g01emc) returns the probability associated with the lower tail of the distribution of the Studentized range statistic.
||nag_prob_studentized_range (double q,
The externally Studentized range,
, for a sample,
, is defined as:
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,
, 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:
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:
This integral is evaluated using
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
Montgomery D C (1984) Design and Analysis of Experiments Wiley
Winer B J (1970) Statistical Principles in Experimental Design McGraw–Hill
q – doubleInput
On entry: , the Studentized range statistic.
v – doubleInput
On entry: , the number of degrees of freedom for the experimental error.
ir – IntegerInput
On entry: , the number of groups.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
If on exit NE_INT
, then nag_prob_studentized_range (g01emc) returns to
6 Error Indicators and Warnings
Warning – There is some doubt as to whether full accuracy has been achieved.
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 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.
The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of , and .
9.1 Program Text
Program Text (g01emce.c)
9.2 Program Data
Program Data (g01emce.d)
9.3 Program Results
Program Results (g01emce.r)