nag_stat_inv_cdf_studentized_range (g01fm) returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.

Syntax

[result, ifail] = g01fm(p, v, ir)

[result, ifail] = nag_stat_inv_cdf_studentized_range(p, v, ir)

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 nag_roots_contfn_brent_rcomm (c05az) . Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.

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

Parameters

Compulsory Input Parameters

1: $p$ – double scalar: The lower tail probability for the Studentized range statistic, $p_{0}$ .

Constraint: $0.0 < p < 1.0$ .
2: $v$ – double scalar: $v$ , the number of degrees of freedom.

Constraint: $v \geq 1.0$ .
3: $ir$ – int64int32nag_int scalar: $r$ , the number of groups.

Constraint: $ir \geq 2$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_stat_inv_cdf_studentized_range (g01fm) may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the function:

If on exit

ifail = 1

, then nag_stat_inv_cdf_studentized_range (g01fm) returns

0.0

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$ifail = 1$

On entry,	$p \leq 0.0$ ,
or	$p \geq 1.0$ ,
or	$v < 1.0$ ,
or	$ir < 2$ .

$ifail = 2$: 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 $1.0$ .

W $ifail = 3$: 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.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

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

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}

Example

Three values of

p

ν

and

r

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

Open in the MATLAB editor: g01fm_example

function g01fm_example


fprintf('g01fm example results\n\n');

p  = [    0.95     0.30    0.9];
v  = [   10       60       5  ];
ir = [int64(5)  12       4  ];
quantile  = p;

fprintf('     p       v    ir   quantile\n');
for j = 1:numel(p)
   [quantile(j), ifail] = g01fm( ...
				 p(j), v(j), ir(j));
end

fprintf('%8.3f%8.3f%4d %8.3f\n', [p; v; double(ir); quantile]);

g01fm example results

     p       v    ir   quantile
   0.950  10.000   5    4.654
   0.300  60.000  12    2.810
   0.900   5.000   4    4.264

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox