NAG C Library Function Document

nag_deviates_studentized_range (g01fmc) returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.

2

Specification

#include <nag.h>

#include <nagg01.h>

double

nag_deviates_studentized_range (double p, double v, Integer ir, NagError *fail)

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 a root-finding procedure. 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: $p$ – doubleInput: On entry: the lower tail probability for the Studentized range statistic, $p_{0}$ .

Constraint: $0.0 < p < 1.0$ .
2: $v$ – doubleInput: 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: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ACCURACY: There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INIT_ESTIMATE: The function was unable to find an upper bound for the value of $q_{0}$ . This will be caused by $p_{0}$ being too close to $1.0$ .
NE_INT: On entry, $ir = 〈value〉$ .
Constraint: $ir \geq 2$ .
NE_INTERNAL_ERROR: 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 for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL: On entry, $p = 〈value〉$ .
Constraint: $0.0 < p < 1.0$ .

On entry, $v = 〈value〉$ .
Constraint: $v \geq 1.0$ .

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

nag_deviates_studentized_range (g01fmc) 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 function. 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.

NAG C Library Function Document

nag_deviates_studentized_range (g01fmc)

▸▿ 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

1

Purpose