# NAG CL Interfaceg01emc (prob_​studentized_​range)

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## 1Purpose

g01emc returns the probability associated with the lower tail of the distribution of the Studentized range statistic.

## 2Specification

 #include
 double g01emc (double q, double v, Integer ir, NagError *fail)
The function may be called by the names: g01emc, nag_stat_prob_studentized_range or nag_prob_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 ${x}_{i}$'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, ${\overline{T}}_{\mathrm{largest}}$ and ${\overline{T}}_{\mathrm{smallest}}$, divided by the square root of the mean-square experimental error, $M{S}_{\mathrm{error}}$, 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
 $C=vv/2Γ (v/2)2v/2- 1 , ϕ (y)=12π e-y2/2 and Φ (y)=∫-∞yϕ (t) dt.$
The above two-dimensional integral is evaluated using numerical quadrature with the upper and lower limits computed to give stated accuracy (see Section 7).
If the degrees of freedom $v$ are greater than $2000$ the probability integral can be approximated by its asymptotic form:
 $P(q;r)=r∫-∞∞ϕ(y)[Φ(y)-Φ(y-q)] r-1dy.$
This integral is evaluated using d01smc.

## 4References

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

## 5Arguments

1: $\mathbf{q}$double Input
On entry: $q$, the Studentized range statistic.
Constraint: ${\mathbf{q}}>0.0$.
2: $\mathbf{v}$double Input
On entry: $v$, the number of degrees of freedom for the experimental error.
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).
If on exit ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT or NE_REAL, then g01emc returns to $0.0$.

## 6Error Indicators and Warnings

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_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, ${\mathbf{q}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{q}}>0.0$.
On entry, ${\mathbf{v}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{v}}\ge 1.0$.

## 7Accuracy

The returned value will have absolute accuracy to at least four decimal places (usually five), unless ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_ACCURACY. When ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_ACCURACY it is usual that the returned value will be a good estimate of the true value.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g01emc 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.

None.

## 10Example

The lower tail probabilities for the distribution of the Studentized range statistic are computed and printed for a range of values of $q$, $\nu$ and $r$.

### 10.1Program Text

Program Text (g01emce.c)

### 10.2Program Data

Program Data (g01emce.d)

### 10.3Program Results

Program Results (g01emce.r)