nag_deviates_studentized_range (g01fmc) : NAG Library, Mark 24

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

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.

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

On entry: the lower tail probability for the Studentized range statistic,

p_{0}

Constraint:

0.0 < p < 1.0

On entry:

v

, the number of degrees of freedom.

Constraint:

v \geq 1.0

On entry:

r

, the number of groups.

Constraint:

ir \geq 2

The NAG error argument (see Section 3.6 in the Essential Introduction).

Warning – There is some doubt as to whether full accuracy has been achieved.

Unable to find initial estimate.

On entry,

ir = ⟨value⟩

.
Constraint:

ir \geq 2

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.

On entry,

p = ⟨value⟩

.
Constraint:

0.0 < p < 1.0

On entry,

v = ⟨value⟩

.
Constraint:

v \geq 1.0

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)	$\|q_{0} - q_{}\| \leq 0.000005 \times \max (1.0, \|q_{}\|)$ .

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}

Three values of

p

ν

and

r

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

NAG 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

NAG Library Function Documentnag_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

NAG Library Function Document

nag_deviates_studentized_range (g01fmc)