nag_ranks_and_scores (g01dhc) : NAG Library, Mark 24

3 Description

nag_ranks_and_scores (g01dhc) 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 $s_{t (i)} = s_{t (i)}^{*}$ .

4 References

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

5 Arguments

1: scores – Nag_ScoresInput

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

$scores = Nag_RankScores$: The ranks.
$scores = Nag_NormalScores$: The Normal scores, that is the expected value of the Normal order statistics.
$scores = Nag_BlomScores$: The Blom version of the Normal scores.
$scores = Nag_TukeyScores$: The Tukey version of the Normal scores.
$scores = Nag_WaerdenScores$: The van der Waerden version of the Normal scores.
$scores = Nag_SavageScores$: The Savage scores, that is the expected value of the exponential order statistics.

Constraint:

scores = Nag_RankScores

Nag_NormalScores

Nag_BlomScores

Nag_TukeyScores

Nag_WaerdenScores

Nag_SavageScores

2: ties – Nag_TiesInput

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

$ties = Nag_AverageTies$: The average of the scores for tied observations is used.
$ties = Nag_LowestTies$: The lowest score in the group of ties is used.
$ties = Nag_HighestTies$: The highest score in the group of ties is used.
$ties = Nag_RandomTies$: The repeatable random number generator is used to randomly untie any group of tied observations.
$ties = Nag_IgnoreTies$: Any ties are ignored, that is the scores are assigned to tied observations in the order that they appear in the data.

Constraint:

ties = Nag_AverageTies

Nag_LowestTies

Nag_HighestTies

Nag_RandomTies

Nag_IgnoreTies

3: n – IntegerInput

On entry:

n

, the number of observations.

Constraint:

n \geq 1

4: x[n] – const doubleInput

On entry: the sample of observations,

x_{i}

, for

i = 1, 2, \dots, n

5: r[n] – doubleOutput

On exit: contains the scores,

s_{i}

, for

i = 1, 2, \dots, n

, as specified by scores.

6: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_INT_ARG_LT

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 1

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.

7 Accuracy

For

scores = Nag_RankScores

, the results should be accurate to machine precision.

For

scores = Nag_SavageScores

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

For

scores = Nag_NormalScores

, the results should have a relative accuracy of at least

\max (100 \times ε, 10^{- 8})

where

ε

is the machine precision.

For

scores = Nag_BlomScores

Nag_TukeyScores

Nag_WaerdenScores

, the results should have a relative accuracy of at least

\max (10 \times ε, 10^{- 12})

NAG Library Function Document

nag_ranks_and_scores (g01dhc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Documentnag_ranks_and_scores (g01dhc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 Function Document

nag_ranks_and_scores (g01dhc)