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,

P (q; v, r)

, for

v

degrees of freedom and

r

groups can be written as:

P (q; v, r) = C \int_{0}^{\infty} x^{v - 1} e^{- v x^{2} / 2} {r \int_{- \infty}^{\infty} ϕ (y) {[Φ (y) - Φ (y - q x)]}^{r - 1} d y} d x,

where

C = \frac{v^{v / 2}}{Γ (v / 2) 2^{v / 2 - 1}}, ϕ (y) = \frac{1}{\sqrt{2 π}} e^{- y^{2} / 2} and Φ (y) = \int_{- \infty}^{y} ϕ (t) d t .

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:

P (q; r) = r \int_{- \infty}^{\infty} ϕ (y) {[Φ (y) - Φ (y - q)]}^{r - 1} d y .

This integral is evaluated using d01amf.

4 References

NIST Digital Library of Mathematical Functions

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

5 Arguments

1: $q$ – Real (Kind=nag_wp) Input: On entry: $q$ , the Studentized range statistic.

Constraint: $q > 0.0$ .
2: $v$ – Real (Kind=nag_wp) Input: On entry: $v$ , the number of degrees of freedom for the experimental error.

Constraint: $v \geq 1.0$ .
3: $ir$ – Integer Input: On entry: $r$ , the number of groups.

Constraint: $ir \geq 2$ .
4: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $−1$ is recommended since useful values can be provided in some output arguments even when $ifail \neq 0$ on exit. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

If on exit $ifail = 1$ , then g01emf returns to $0.0$ .

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

Note: in some cases g01emf may return useful information.

$ifail = 1$: On entry, $ir = ⟨ value ⟩$ .
Constraint: $ir \geq 2$ .

On entry, $q = ⟨ value ⟩$ .
Constraint: $q > 0.0$ .

On entry, $v = ⟨ value ⟩$ .
Constraint: $v \geq 1.0$ .

$ifail = 2$: There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g01emf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

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

g01em: FL CL CPP AD PY MB

NAG FL Interfaceg01emf (prob_​studentized_​range)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g01emf (prob_studentized_range)