nag_ranks_and_scores (g01dhc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_ranks_and_scores (g01dhc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_ranks_and_scores (g01dhc) computes the ranks, Normal scores, an approximation to the Normal scores or the exponential scores as requested by you.

2  Specification

#include <nag.h>
#include <nagg01.h>
void  nag_ranks_and_scores (Nag_Scores scores, Nag_Ties ties, Integer n, const double x[], double r[], NagError *fail)

3  Description

nag_ranks_and_scores (g01dhc) computes one of the following scores for a sample of observations, x1,x2,,xn.
  1. Rank Scores
    The ranks are assigned to the data in ascending order, that is the ith observation has score si=k if it is the kth smallest observation in the sample.
  2. Normal Scores
    The Normal scores are the expected values of the Normal order statistics from a sample of size n. If xi is the kth smallest observation in the sample, then the score for that observation, si, is EZk where Zk is the kth order statistic in a sample of size n from a standard Normal distribution and E is the expectation operator.
  3. 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·, at the values of the ranks scaled into the interval 0,1 using different scaling transformations.
    The Blom scores use the scaling transformation ri-38 n+14  for the rank ri, for i=1,2,,n. Thus the Blom score corresponding to the observation xi is
    si = Φ-1 ri - 38 n+14 .
    The Tukey scores use the scaling transformation ri-13 n+13 ; the Tukey score corresponding to the observation xi is
    si = Φ-1 ri - 13 n+13 .
    The van der Waerden scores use the scaling transformation rin+1; the van der Waerden score corresponding to the observation xi is
    si = Φ-1 ri 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).
  4. 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 xi is the kth smallest observation in the sample, then the score for that observation is
    si = EYk = 1n + 1n-1 + + 1n-k+1 ,
    where Yk is the kth 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 xti, for i=1,2,,m, denote m tied observations, that is xt1=xt2==xtm with t1<t2<<tm. If the rank of xt1 is k, then if ties are ignored the rank of xtj will be k+j-1. Let the scores ignoring ties be st1*,st2*,,stm*. Then the scores, sti, for i=1,2,,m, may be calculated as follows:

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:     scoresNag_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 or Nag_SavageScores.
2:     tiesNag_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 or Nag_IgnoreTies.
3:     nIntegerInput
On entry: n, the number of observations.
Constraint: n1.
4:     x[n]const doubleInput
On entry: the sample of observations, xi, for i=1,2,,n.
5:     r[n]doubleOutput
On exit: contains the scores, si, for i=1,2,,n, as specified by scores.
6:     failNagError *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: n1.
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 max100×ε,10-8 where ε is the machine precision.
For scores=Nag_BlomScores, Nag_TukeyScores or Nag_WaerdenScores, the results should have a relative accuracy of at least max10×ε,10-12.

8  Further Comments

If more accurate Normal scores are required nag_normal_scores_exact (g01dac) should be used with appropriate settings for the input argument etol.

9  Example

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.

9.1  Program Text

Program Text (g01dhce.c)

9.2  Program Data

Program Data (g01dhce.d)

9.3  Program Results

Program Results (g01dhce.r)


nag_ranks_and_scores (g01dhc) (PDF version)
g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012