NAG Library Routine Document

G01FMF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G01FMF returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic, via the routine name.

2 Specification

FUNCTION G01FMF (

P, V, IR, IFAIL)

REAL (KIND=nag_wp) G01FMF

INTEGER	IR, IFAIL
REAL (KIND=nag_wp)	P, V

3 Description

The externally Studentized range,

q

, for a sample,

x_{1}, x_{2}, \dots, x_{r}

, is defined as

q = \frac{\max (x_{i}) - \min (x_{i})}{{\hat{σ}}_{e}},

where

{\hat{σ}}_{e}

is an independent estimate of the standard error of the

x_{i}

. 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,

{\bar{T}}_{1}, {\bar{T}}_{2}, \dots, {\bar{T}}_{r}

, the Studentized range statistic is defined to be the difference between the largest and smallest means,

{\bar{T}}_{largest}

and

{\bar{T}}_{smallest}

, divided by the square root of the mean-square experimental error,

M S_{error}

, over the number of observations in each group,

n

, i.e.,

q = \frac{{\bar{T}}_{largest} - {\bar{T}}_{smallest}}{\sqrt{M S_{error} / n}} .

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 Parameters

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 – INTEGERInput: On entry: $r$ , the number of groups.

Constraint: $IR \geq 2$ .
4: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if $IFAIL \neq 0$ on exit, the recommended value is $- 1$ . 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).

Note: G01FMF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

If on exit

IFAIL = 1

, then G01FMF returns

0.0

$IFAIL = 1$

On entry,	$P \leq 0.0$ ,
or	$P \geq 1.0$ ,
or	$V < 1.0$ ,
or	$IR < 2$ .

$IFAIL = 2$: 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.

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 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}

9 Example

Three values of

p

ν

and

r

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

NAG Library Routine DocumentG01FMF