NAG Library Function Document

nag_prob_studentized_range (g01emc)

+− 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

nag_prob_studentized_range (g01emc) returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_prob_studentized_range (double q, 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}

's. 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 .

The above two-dimensional integral is evaluated using numerical quadrature 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 nag_1d_quad_inf_1 (d01smc).

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

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 – doubleInput: On entry: $q$ , the Studentized range statistic.
Constraint: $q > 0.0$ .
2: v – doubleInput: On entry: $v$ , the number of degrees of freedom for the experimental error.
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 3.6 in the Essential Introduction).

If on exit $fail . code =$ NE_INT or NE_REAL, then nag_prob_studentized_range (g01emc) returns to $0.0$ .

6 Error Indicators and Warnings

NE_ACCURACY: Warning – There is some doubt as to whether full accuracy has been achieved.
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.
NE_REAL: On entry, $q = ⟨value⟩$ .
Constraint: $q > 0.0$ .
On entry, $v = ⟨value⟩$ .
Constraint: $v \geq 1.0$ .

7 Accuracy

The returned value will have absolute accuracy to at least four decimal places (usually five), unless

fail . code =

NE_ACCURACY. When

fail . code =

NE_ACCURACY it is usual that the returned value will be a good estimate of the true value.

8 Parallelism and Performance

nag_prob_studentized_range (g01emc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please 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

NAG Library Function Documentnag_prob_studentized_range (g01emc)