G01DHF : NAG Library, Mark 25

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

G01DHF computes the ranks, Normal scores, an approximation to the Normal scores or the exponential scores as requested by you.

G01DHF computes one of the following scores for a sample of observations,

x_{1}, x_{2}, \dots, x_{n}

Rank Scores

The ranks are assigned to the data in ascending order, that is the

i

th observation has score

s_{i} = k

if it is the

k

th smallest observation in the sample.

Normal Scores

The Normal scores are the expected values of the Normal order statistics from a sample of size

n

. If

x_{i}

is the

k

th smallest observation in the sample, then the score for that observation,

s_{i}

, is

E (Z_{k})

where

Z_{k}

is the

k

th order statistic in a sample of size

n

from a standard Normal distribution and

E

is the expectation operator.

Blom, Tukey and van der Waerden Scores

These scores are approximations to the Normal scores. The scores are obtained by evaluating the inverse cumulative Normal distribution function,

Φ^{- 1} (\cdot)

, at the values of the ranks scaled into the interval

(0, 1)

using different scaling transformations.

The Blom scores use the scaling transformation

\frac{r_{i} - \frac{3}{8}}{n + \frac{1}{4}}

for the rank

r_{i}

, for

i = 1, 2, \dots, n

. Thus the Blom score corresponding to the observation

x_{i}

s_{i} = Φ^{- 1} (\frac{r_{i} - \frac{3}{8}}{n + \frac{1}{4}}) .

The Tukey scores use the scaling transformation

\frac{r_{i} - \frac{1}{3}}{n + \frac{1}{3}}

; the Tukey score corresponding to the observation

x_{i}

s_{i} = Φ^{- 1} (\frac{r_{i} - \frac{1}{3}}{n + \frac{1}{3}}) .

The van der Waerden scores use the scaling transformation

\frac{r_{i}}{n + 1}

; the van der Waerden score corresponding to the observation

x_{i}

s_{i} = Φ^{- 1} (\frac{r_{i}}{n + 1}) .

The van der Waerden scores may be used to carry out the van der Waerden test for testing for differences between several population distributions, see Conover (1980).

Savage Scores

The Savage scores are the expected values of the exponential order statistics from a sample of size

n

. They may be used in a test discussed by Savage (1956) and Lehmann (1975). If

x_{i}

is the

k

th smallest observation in the sample, then the score for that observation is

s_{i} = E (Y_{k}) = \frac{1}{n} + \frac{1}{n - 1} + \dots + \frac{1}{n - k + 1},

where

Y_{k}

is the

k

th order statistic in a sample of size

n

from a standard exponential distribution and

E

is the expectation operator.

Ties may be handled in one of five ways. Let

x_{t (i)}

, for

i = 1, 2, \dots, m

, denote

m

tied observations, that is

x_{t (1)} = x_{t (2)} = \dots = x_{t (m)}

with

t (1) < t (2) < \dots < t (m)

. If the rank of

x_{t (1)}

k

, then if ties are ignored the rank of

x_{t (j)}

will be

k + j - 1

. Let the scores ignoring ties be

s_{t (1)}^{*}, s_{t (2)}^{*}, \dots, s_{t (m)}^{*}

. Then the scores,

s_{t (i)}

, for

i = 1, 2, \dots, m

, may be calculated as follows:

– if averages are used, then $s_{t (i)} = \sum_{j = 1}^{m} s_{t (j)}^{*} / m$ ;
– if the lowest score is used, then $s_{t (i)} = s_{t (1)}^{*}$ ;
– if the highest score is used, then $s_{t (i)} = s_{t (m)}^{*}$ ;
– if ties are to be broken randomly, then $s_{t (i)} = s_{t (I)}^{*}$ where $I \in \{random permutation of 1, 2, \dots, m\}$ ;
– if ties are to be ignored, then $s_{t (i)} = s_{t (i)}^{*}$ .

Blom G (1958) Statistical Estimates and Transformed Beta-variables Wiley

Conover W J (1980) Practical Nonparametric Statistics Wiley

Lehmann E L (1975) Nonparametrics: Statistical Methods Based on Ranks Holden–Day

Savage I R (1956) Contributions to the theory of rank order statistics – the two-sample case Ann. Math. Statist. 27 590–615

Tukey J W (1962) The future of data analysis Ann. Math. Statist. 33 1–67

On entry: indicates which of the following scores are required.

$SCORES ='R'$: The ranks.
$SCORES ='N'$: The Normal scores, that is the expected value of the Normal order statistics.
$SCORES ='B'$: The Blom version of the Normal scores.
$SCORES ='T'$: The Tukey version of the Normal scores.
$SCORES ='V'$: The van der Waerden version of the Normal scores.
$SCORES ='S'$: The Savage scores, that is the expected value of the exponential order statistics.

Constraint:

SCORES ='R'

'N'

'B'

'T'

'V'

'S'

On entry: indicates which of the following methods is to be used to assign scores to tied observations.

$TIES ='A'$: The average of the scores for tied observations is used.
$TIES ='L'$: The lowest score in the group of ties is used.
$TIES ='H'$: The highest score in the group of ties is used.
$TIES ='N'$: The nonrepeatable random number generator is used to randomly untie any group of tied observations.
$TIES ='R'$: The repeatable random number generator is used to randomly untie any group of tied observations.
$TIES ='I'$: Any ties are ignored, that is the scores are assigned to tied observations in the order that they appear in the data.

Constraint:

TIES ='A'

'L'

'H'

'N'

'R'

'I'

On entry:

n

, the number of observations.

Constraint:

N \geq 1

On entry: the sample of observations,

x_{i}

, for

i = 1, 2, \dots, n

On exit: contains the scores,

s_{i}

, for

i = 1, 2, \dots, n

, as specified by SCORES.

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.8 in the Essential Introduction for further information.

Your licence key may have expired or may not have been installed correctly.

See Section 3.7 in the Essential Introduction for further information.

Dynamic memory allocation failed.

See Section 3.6 in the Essential Introduction for further information.

For

SCORES ='R'

, the results should be accurate to machine precision.

For

SCORES ='S'

, the results should be accurate to a small multiple of machine precision.

For

SCORES ='N'

, the results should have a relative accuracy of at least

\max (100 \times ε, 10^{- 8})

where

ε

is the machine precision.

For

SCORES ='B'

'T'

'V'

, the results should have a relative accuracy of at least

\max (10 \times ε, 10^{- 12})

G01DHF 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

If more accurate Normal scores are required G01DAF should be used with appropriate settings for the input parameter ETOL.

This example computes and prints the Savage scores for a sample of five observations. The average of the scores of any tied observations is used.

On entry,	$SCORES \neq'R'$ , $'N'$ , $'B'$ , $'T'$ , $'V'$ or $'S'$ ,
or	$TIES \neq'A'$ , $'L'$ , $'H'$ , $'N'$ , $'R'$ or $'I'$ ,
or	$N < 1$ .

NAG Library Routine Document

G01DHF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine DocumentG01DHF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

G01DHF