# NAG CL Interfaceg01fmc (inv_​cdf_​studentized_​range)

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## 1Purpose

g01fmc returns the deviate associated with the lower tail probability of the distribution of the Studentized range statistic.

## 2Specification

 #include
 double g01fmc (double p, double v, Integer ir, NagError *fail)
The function may be called by the names: g01fmc, nag_stat_inv_cdf_studentized_range or nag_deviates_studentized_range.

## 3Description

The externally Studentized range, $q$, for a sample, ${x}_{1},{x}_{2},\dots ,{x}_{r}$, is defined as
 $q = max(xi) - min(xi) σ^e ,$
where ${\stackrel{^}{\sigma }}_{e}$ 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.,
 $q=T¯largest-T¯smallest MSerror/n .$
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, $P\left(q;v,r\right)$, for $v$ degrees of freedom and $r$ groups, can be written as:
 $P(q;v,r)=C∫0∞xv-1e-vx2/2 (r∫-∞∞ϕ(y)(Φ(y)-Φ(y-qx)) r-1dy)dx,$
where
 $C=vv/2Γ (v/2)2v/2- 1 , ϕ (y)=12πe-y2/2 and Φ (y)=∫-∞yϕ (t)dt.$
For a given probability ${p}_{0}$, the deviate ${q}_{0}$ is found as the solution to the equation
 $P(q0;v,r)=p0,$ (1)
using a root-finding procedure. Initial estimates are found using the approximation given in Lund and Lund (1983) and a simple search procedure.

## 4References

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

## 5Arguments

1: $\mathbf{p}$double Input
On entry: the lower tail probability for the Studentized range statistic, ${p}_{0}$.
Constraint: $0.0<{\mathbf{p}}<1.0$.
2: $\mathbf{v}$double Input
On entry: $v$, the number of degrees of freedom.
Constraint: ${\mathbf{v}}\ge 1.0$.
3: $\mathbf{ir}$Integer Input
On entry: $r$, the number of groups.
Constraint: ${\mathbf{ir}}\ge 2$.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

If on exit ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT or NE_REAL, then g01fmc returns $0.0$.
NE_ACCURACY
There is some doubt as to whether full accuracy has been achieved. The returned value should be a reasonable estimate of the true value.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_INIT_ESTIMATE
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 $1.0$.
NE_INT
On entry, ${\mathbf{ir}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ir}}\ge 2$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0.0<{\mathbf{p}}<1.0$.
On entry, ${\mathbf{v}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{v}}\ge 1.0$.

## 7Accuracy

The returned solution, ${q}_{*}$, to equation (1) is determined so that at least one of the following criteria apply.
1. (a)$|P\left({q}_{*}\text{;}v,r\right)-{p}_{0}|\le 0.000005$
2. (b)$|{q}_{0}-{q}_{*}|\le 0.000005×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1.0,|{q}_{*}|\right)$.

## 8Parallelism and Performance

g01fmc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 ${p}_{0}^{r-1}$.

## 10Example

Three values of $p$, $\nu$ and $r$ are read in and the Studentized range deviates or quantiles are computed and printed.

### 10.1Program Text

Program Text (g01fmce.c)

### 10.2Program Data

Program Data (g01fmce.d)

### 10.3Program Results

Program Results (g01fmce.r)