NAG CL Interface
g01dhc (ranks_​and_​scores)

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1 Purpose

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>
void  g01dhc (Nag_Scores scores, Nag_Ties ties, Integer n, const double x[], double r[], NagError *fail)
The function may be called by the names: g01dhc, nag_stat_ranks_and_scores or nag_ranks_and_scores.

3 Description

g01dhc computes one of the following scores for a sample of observations, x1,x2,,xn.
  1. 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. 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 E(Zk) 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. 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. 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 = E(Yk) = 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 xt(i), for i=1,2,,m, denote m tied observations, that is xt(1)=xt(2)==xt(m) with t(1)<t(2)<<t(m). If the rank of xt(1) is k, then if ties are ignored the rank of xt(j) will be k+j-1. Let the scores ignoring ties be st(1)*,st(2)*,,st(m)*. Then the scores, st(i), 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: scores Nag_Scores Input
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: ties Nag_Ties Input
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: n Integer Input
On entry: n, the number of observations.
Constraint: n1.
4: x[n] const double Input
On entry: the sample of observations, xi, for i=1,2,,n.
5: r[n] double Output
On exit: contains the scores, si, for i=1,2,,n, as specified by scores.
6: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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×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 max(10×ε,10−12).

8 Parallelism and Performance

g01dhc 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.

9 Further Comments

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

10 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.

10.1 Program Text

Program Text (g01dhce.c)

10.2 Program Data

Program Data (g01dhce.d)

10.3 Program Results

Program Results (g01dhce.r)