3.5.3.3.15 Srangeinv

Definition:

\(q=srangeinv(p, v, ir)\) computes the deviate, \(x_p\), associated with the lower tail probability of the distribution of the Studentized range statistic.

The externally Studentized range,\( q\), for a sample,\(x_1,x_2 \cdots x_r\) is defined as:

\[q=\frac{\max (x_i)-\min (x_i)}{ \hat{\sigma _e} }\]

Where \(\hat{\sigma _e}\) is an independent estimate of the standard error of the \(x_i\) 's.

For a Studentized range statistic the probability integral,\(P(q)\) , for \(\nu\) degrees of freedom and \(r\) groups, can be written as: \( p(q)=C\int_0^\infty x^{\nu -1}e^{-\nu x^2/2}\{\Phi (y)[\Phi (y)-\Phi (y-qx)]^{r-1}dy\}dx \)

where \(p(q)C=\frac{\nu ^{\nu /2}}{\Gamma (\nu /2)2^{\nu /2-1}},\Phi (y)=\int_{-\infty }^y\frac 1{\sqrt{2\pi }}e^{-t^2/2}dt\)

For a given probability \( p_0\), the deviate \(q_0\) is found as the solution to the equation

\[ P(q_0)=p_0\]

Parameters:

p (output, double)

the probability.

v (input,double)

the number of degrees of freedom for the experimental error \(\nu\). \( \nu\) ≥ 1.0

ir (input, int)

the number of groups,\(r\) .\(ir \geq 2\)

q (output, double)

the Studentized range statistic,\(q\). \(q>0.0\)