g01dh:: Simple Calculations on Statistical Data (NAG Toolbox)

nag_stat_ranks_and_scores (g01dh) computes the ranks, Normal scores, an approximation to the Normal scores or the exponential scores as requested by you.

nag_stat_ranks_and_scores (g01dh) 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

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'

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'

The sample of observations,

x_{i}

, for

i = 1, 2, \dots, n

Default: the dimension of the array x.

n

, the number of observations.

Constraint:

n \geq 1

Contains the scores,

s_{i}

, for

i = 1, 2, \dots, n

, as specified by scores.

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Errors or warnings detected by the function:

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

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

Dynamic memory allocation failed.

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})

If more accurate Normal scores are required nag_stat_normal_scores_exact (g01da) should be used with appropriate settings for the input argument 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 Toolbox: nag_stat_ranks_and_scores (g01dh)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example