naginterfaces.library.stat.inv_​cdf_​studentized_​range

naginterfaces.library.stat.inv_cdf_studentized_range(p, v, ir)

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

For full information please refer to the NAG Library document for g01fm

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01fmf.html

Parameters

pfloat

The lower tail probability for the Studentized range statistic, $p_0$.

vfloat

$v$, the number of degrees of freedom.

irint

$r$, the number of groups.

Returns

xfloat

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

Raises

NagValueError

(errno $1$)

On entry, $p=⟨value⟩$.

Constraint: $0.0<p<1.0$.

(errno $1$)

On entry, $ir=⟨value⟩$.

Constraint: $ir\ge 2$.

(errno $1$)

On entry, $v=⟨value⟩$.

Constraint: $v\ge 1.0$.

(errno $2$)

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 $1.0$.

Warns

NagAlgorithmicWarning

(errno $3$)

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

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 $x_i$. 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, ${\overline{T}}_{\text{largest}}$ and ${\overline{T}}_{\text{smallest}}$, divided by the square root of the mean-square experimental error, $M{S}_{\text{error}}$, over the number of observations in each group, $n$, i.e.,

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

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

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

For a given probability $p_0$, the deviate $q_0$ is found as the solution to the equation

$$P(q_0\text{;}v,r)=p_0\text{,}$$

using roots.contfn_brent_rcomm. 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