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 .

For a given probability

p_{0}

, the deviate

q_{0}

is found as the solution to the equation

P (q_{0}; v, r) = p_{0},

(1)

using c05azf . Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.

4 References

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: $p$ – Real (Kind=nag_wp) Input: On entry: the lower tail probability for the Studentized range statistic, $p_{0}$ .

Constraint: $0.0 < p < 1.0$ .
2: $v$ – Real (Kind=nag_wp) Input: On entry: $v$ , the number of degrees of freedom.

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).

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 g01fmf may return useful information.

If on exit

ifail = 1

, then g01fmf returns

0.0

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

On entry, $p = ⟨ value ⟩$ .
Constraint: $0.0 < p < 1.0$ .

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

$ifail = 2$: The routine was unable to find an upper bound for the value of $q_{0}$ . This will be caused by $p_{0}$ being too close to $1.0$ .

$ifail = 3$: 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 solution,

q_{*}

, to equation (1) is determined so that at least one of the following criteria apply.

(a) $| P (q_{*}; v, r) - p_{0} | \leq 0.000005$
(b) $| q_{0} - q_{*} | \leq 0.000005 \times \max (1.0, | q_{*} |)$ .

8 Parallelism and Performance

g01fmf 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

To obtain the factors for Duncan's multiple-range test, equation (1) has to be solved for

p_{1}

, where

p_{1} = p_{0}^{r - 1}

, so on input p should be set to

p_{0}^{r - 1}

10 Example

Three values of

p

ν

and

r

are read in and the Studentized range deviates or quantiles are computed and printed.

g01fm: FL CL CPP AD

NAG FL Interfaceg01fmf (inv_​cdf_​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
g01fmf (inv_cdf_studentized_range)