g01 Chapter Contents
g01 Chapter Introduction
NAG 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:
 $q = maxxi - minxi σ^e ,$
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:    $\mathbf{q}$doubleInput
On entry: $q$, the Studentized range statistic.
Constraint: ${\mathbf{q}}>0.0$.
2:    $\mathbf{v}$doubleInput
On entry: $v$, the number of degrees of freedom for the experimental error.
Constraint: ${\mathbf{v}}\ge 1.0$.
3:    $\mathbf{ir}$IntegerInput
On entry: $r$, the number of groups.
Constraint: ${\mathbf{ir}}\ge 2$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).
If on exit ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ 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_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction 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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction 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: ${\mathbf{v}}\ge 1.0$.

7  Accuracy

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.

8  Parallelism and Performance

nag_prob_studentized_range (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.

10  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 $r$.

10.1  Program Text

Program Text (g01emce.c)

10.2  Program Data

Program Data (g01emce.d)

10.3  Program Results

Program Results (g01emce.r)