g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_prob_studentized_range (g01emc)

## 1  Purpose

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

## 2  Specification

 #include #include
 double nag_prob_studentized_range (double q, double v, Integer ir, NagError *fail)

## 3  Description

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, ${\stackrel{-}{T}}_{1},{\stackrel{-}{T}}_{2},\dots ,{\stackrel{-}{T}}_{r}$, the Studentized range statistic is defined to be the difference between the largest and smallest means, ${\stackrel{-}{T}}_{\mathrm{largest}}$ and ${\stackrel{-}{T}}_{\mathrm{smallest}}$, 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:
 $Pq;v,r=C∫0∞xv-1e-vx2/2 r∫-∞∞ϕyΦy-Φy-qx r-1dydx,$
where
 $C=vv/2Γ v/22v/2- 1 , ϕ y=12π e-y2/2 and Φ y=∫-∞yϕ t dt.$
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:
 $Pq;r=r∫-∞∞ϕyΦy-Φy-q r-1dy.$
This integral is evaluated using nag_1d_quad_inf_1 (d01smc).

## 4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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

## 5  Arguments

1:     qdoubleInput
On entry: $q$, the Studentized range statistic.
Constraint: ${\mathbf{q}}>0.0$.
2:     vdoubleInput
On entry: $v$, the number of degrees of freedom for the experimental error.
Constraint: ${\mathbf{v}}\ge 1.0$.
3:     irIntegerInput
On entry: $r$, the number of groups.
Constraint: ${\mathbf{ir}}\ge 2$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
If on exit NE_INT or NE_REAL, then nag_prob_studentized_range (g01emc) returns to $0.0$.

## 6  Error Indicators and Warnings

NE_ACCURACY
Warning – There is some doubt as to whether full accuracy has been achieved.
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.
NE_REAL
On entry, ${\mathbf{q}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{q}}>0.0$.
On entry, ${\mathbf{v}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{v}}\ge 1.0$.

## 7  Accuracy

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

None.

## 9  Example

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 $r$.

### 9.1  Program Text

Program Text (g01emce.c)

### 9.2  Program Data

Program Data (g01emce.d)

### 9.3  Program Results

Program Results (g01emce.r)