nag_stat_prob_studentized_range (g01em) returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

Syntax

[result, ifail] = g01em(q, v, ir)

[result, ifail] = nag_stat_prob_studentized_range(q, 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}

'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 nag_quad_2d_fin (d01da) with the upper and lower limits computed to give stated accuracy (see Accuracy).

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_quad_1d_inf (d01am).

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

Parameters

Compulsory Input Parameters

1: $q$ – double scalar: $q$ , the Studentized range statistic.

Constraint: $q > 0.0$ .
2: $v$ – double scalar: $v$ , the number of degrees of freedom for the experimental error.

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).
If on exit $ifail = 1$ , then nag_stat_prob_studentized_range (g01em) returns to $0.0$ .

Error Indicators and Warnings

Note: nag_stat_prob_studentized_range (g01em) may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the function:

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,	$q \leq 0.0$ ,
or	$v < 1.0$ ,
or	$ir < 2$ .

W $ifail = 2$: There is some doubt as to whether full accuracy has been achieved.

$ifail = - 99$: 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 value will have absolute accuracy to at least four decimal places (usually five), unless

ifail = 2

. When

ifail = 2

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

Further Comments

None.

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

Open in the MATLAB editor: g01em_example

function g01em_example


fprintf('g01em example results\n\n');

% Probability for Studentized range statistic distribution
q  = [ 4.6543;  2.8099; 4.2636];
v  = [10;      60.0;    5.0];
ir = [int64(5); 12;   4];

fprintf('   q       v     ir    probability\n');
for j = 1:numel(q);

  [p, ifail] = g01em( ...
                      q(j) , v(j),  ir(j));

  fprintf('%8.4f%6.1f%4d%12.4f\n', q(j), v(j), ir(j), p);
end

g01em example results

   q       v     ir    probability
  4.6543  10.0   5      0.9500
  2.8099  60.0  12      0.3000
  4.2636   5.0   4      0.9000

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

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