naginterfaces.library.stat.prob_​studentized_​range

naginterfaces.library.stat.prob_studentized_range(q, v, ir)

prob_studentized_range returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

For full information please refer to the NAG Library document for g01em

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01emf.html

Parameters

qfloat

$q$, the Studentized range statistic.

vfloat

$v$, the number of degrees of freedom for the experimental error.

irint

$r$, the number of groups.

Returns

pfloat

The probability associated with the lower tail of the distribution of the Studentized range statistic.

Raises

NagValueError

(errno $1$)

On entry, $q=⟨value⟩$.

Constraint: $q>0.0$.

(errno $1$)

On entry, $ir=⟨value⟩$.

Constraint: $ir\ge 2$.

(errno $1$)

On entry, $v=⟨value⟩$.

Constraint: $v\ge 1.0$.

Warns

NagAlgorithmicWarning

(errno $2$)

There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.

Notes

The externally Studentized range, $q$, for a sample, $x_1,x_2,\dots ,x_r$, is defined as:

$$q=\frac{max\left(x_i\right)-min\left(x_i\right)}{{\hat{\sigma }}_e}\text{,}$$

where ${\hat{\sigma }}_e$ 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, ${\overline{T}}_1,{\overline{T}}_2,\dots ,{\overline{T}}_r$, the Studentized range statistic is defined to be the difference between the largest and smallest means, ${\overline{T}}_{largest}$ and ${\overline{T}}_{smallest}$, 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.,

$$q=\frac{{\overline{T}}_{largest}-{\overline{T}}_{smallest}}{\sqrt{M{S}_{error}/n}}\text{.}$$

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:

$$P(q;v,r)=C{\int }_0^{\infty }{x}^{v-1}{e}^{-v{x}^2/2}\left\{r{\int }_{-\infty }^{\infty }\varphi \left(y\right){[\Phi \left(y\right)-\Phi (y-qx)]}^{r-1}dy\right\}dx\text{,}$$

where

$$C=\frac{v^{v/2}}{\Gamma \left(v/2\right)2^{v/2-1}}\text{,}\varphi \left(y\right)=\frac{1}{\sqrt{2\pi }}{e}^{-y^2/2}\text{and}\Phi \left(y\right)={\int }_{-\infty }^y\varphi \left(t\right)dt\text{.}$$

The above two-dimensional integral is evaluated using quad.dim2_fin 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:

$$P(q;r)=r{\int }_{-\infty }^{\infty }\varphi \left(y\right){[\Phi \left(y\right)-\Phi (y-q)]}^{r-1}dy\text{.}$$

This integral is evaluated using quad.dim1_inf.

References

NIST Digital Library of Mathematical Functions

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