26 · Chapter 1: Basic Statistics IMSL STAT/LIBRARY 1 5.0000 24.0000 3.0000 9.0000 3.0000 2 4.0000 6.0000 2.0000 5.0000 3.0000 Intermediate Statistics from UVSTA
Variable Mean Sum Sqs. Minimum Maximum Count 1 5.5000 25.5000 3.0000 9.0000 6.0000 2 3.5000 7.5000 2.0000 5.0000 6.0000 Intermediate Statistics from UVSTA
Variable Mean Sum Sqs. Minimum Maximum Count 1 5.0000 24.0000 3.0000 9.0000 3.0000 2 4.0000 6.0000 2.0000 5.0000 3.0000 Univariate Statistics from UVSTA
Variable Mean Variance Std. Dev. Skewness Kurtosis 1 3.0000 9.6000 3.0984 1.4142 0.5000 2 4.0000 3.0000 1.7321 -0.7071 -1.5000 Variable Minimum Maximum Range Coef. Var. Count 1 1.0000 9.0000 8.0000 1.0328 6.0000 2 2.0000 5.0000 3.0000 0.4330 3.0000 Variable Lower CLM Upper CLM Lower CLV Upper CLV
1 -0.2516 6.2516 3.7405 57.7470 2 -0.3027 8.3027 0.8133 118.4935
IMSL STAT/LIBRARY Chapter 1: Basic Statistics · 27 ITIE — Option for determining the method used to assign a score to tied observations.
(Input)
Default: ITIE = 0.
ITIE Method
0 The average of the scores of the tied observations is used.
1 The highest score in the group of ties is used.
2 The lowest score in the group of ties is used.
3 The tied observations are to be randomly untied using an IMSL random number generator.
ISCORE — Option for specifying the type of values returned in SCORE. (Input) Default: ISCORE = 0.
ISCORE Type 0 Ranks
1 Blom version of normal scores 2 Tukey version of normal scores
3 Van der Waerdan version of normal scores
4 Expected value of normal order statistics (For tied observations, the average of the expected normal scores are used.)
5 Savage scores (the expected value of exponential order statistics) FORTRAN 90 Interface
Generic: CALL RANKS(X, SCORE [,…])
Specific: The specific interface names are S_RANKS and D_RANKS. FORTRAN 77 Interface
Single: CALL RANKS (NOBS, X, FUZZ, ITIE, ISCORE, SCORE) Double: The double precision name is DRANKS.
28 · Chapter 1: Basic Statistics IMSL STAT/LIBRARY
Example
The data for this example, from Hinkley (1977), are the same used in several examples in this chapter. There are 30 observations. Note that the fourth and sixth observations are tied and that the third and twentieth are tied.
USE RANKS_INT USE UMACH_INT USE RNSET_INT INTEGER NOBS PARAMETER (NOBS=30)
!
INTEGER ISCORE, ISEED, ITIE, NOUT REAL FUZZ, SCORE(NOBS), X(NOBS)
!
DATA X/0.77, 1.74, 0.81, 1.20, 1.95, 1.20, 0.47, 1.43, 3.37,&
2.20, 3.00, 3.09, 1.51, 2.10, 0.52, 1.62, 1.31, 0.32, 0.59, &
0.81, 2.81, 1.87, 1.18, 1.35, 4.75, 2.48, 0.96, 1.89, 0.90, &
2.05/
!
CALL UMACH (2, NOUT)
! Ranks.
ISCORE = 0
! Average ties.
ITIE = 0 FUZZ = 0.0
!
CALL RANKS (X, SCORE, ISCORE=ISCORE, ITIE=ITIE, FUZZ=FUZZ) WRITE (NOUT,99994) SCORE
99994 FORMAT (’ Ranks’, /, (1X,10F7.1))
! Blom normal scores.
ISCORE = 1
! Take largest ranks for ties.
ITIE = 1 FUZZ = 0.0
!
CALL RANKS (X, SCORE, ISCORE=ISCORE, ITIE=ITIE, FUZZ=FUZZ) WRITE (NOUT,99995) SCORE
99995 FORMAT (/, ’ Blom normal scores’, /, (1X,10F7.3))
! Tukey normal scores.
ISCORE = 2
! Take smallest ranks for ties.
ITIE = 2 FUZZ = 0.0
!
CALL RANKS (X, SCORE, ISCORE=ISCORE, ITIE=ITIE, FUZZ=FUZZ) WRITE (NOUT,99996) SCORE
99996 FORMAT (/, ’ Tukey normal scores’, /, (1X,10F7.3))
! Van der Waerden scores.
ISCORE = 3
! Randomly resolve ties.
ISEED = 123457 CALL RNSET (ISEED) ITIE = 3
FUZZ = 0.0
IMSL STAT/LIBRARY Chapter 1: Basic Statistics · 29
!
CALL RANKS (X, SCORE, ISCORE=ISCORE, ITIE=ITIE, FUZZ=FUZZ) WRITE (NOUT,99997) SCORE
99997 FORMAT (/, ’ Van der Waerden scores’, /, (1X,10F7.3))
! Expected value of normal O. S.
ISCORE = 4
! Average ties.
ITIE = 0 FUZZ = 0.0
!
CALL RANKS (X, SCORE, ISCORE=ISCORE, ITIE=ITIE, FUZZ=FUZZ) WRITE (NOUT,99998) SCORE
99998 FORMAT (/, ’ Expected values of normal order statistics’, /,&
(1X,10F7.3))
! Savage scores.
ISCORE = 5
! Average ties.
ITIE = 0 FUZZ = 0.0
!
CALL RANKS (X, SCORE, ISCORE=ISCORE, ITIE=ITIE, FUZZ=FUZZ) WRITE (NOUT,99999) SCORE
99999 FORMAT (/, ’ Expected values of exponential order statistics’, &
/, (1X,10F7.2)) END
Output
Ranks
5.0 18.0 6.5 11.5 21.0 11.5 2.0 15.0 29.0 24.0 27.0 28.0 16.0 23.0 3.0 17.0 13.0 1.0 4.0 6.5 26.0 19.0 10.0 14.0 30.0 25.0 9.0 20.0 8.0 22.0 Blom normal scores
-1.024 0.209 -0.776 -0.294 0.473 -0.294 -1.610 -0.041 1.610 0.776 1.176 1.361 0.041 0.668 -1.361 0.125 -0.209 -2.040 -1.176 -0.776 1.024 0.294 -0.473 -0.125 2.040 0.893 -0.568 0.382 -0.668 0.568 Tukey normal scores
-1.020 0.208 -0.890 -0.381 0.471 -0.381 -1.599 -0.041 1.599 0.773 1.171 1.354 0.041 0.666 -1.354 0.124 -0.208 -2.015 -1.171 -0.890 1.020 0.293 -0.471 -0.124 2.015 0.890 -0.566 0.381 -0.666 0.566 Van der Waerden scores
-0.989 0.204 -0.753 -0.287 0.460 -0.372 -1.518 -0.040 1.518 0.753 1.131 1.300 0.040 0.649 -1.300 0.122 -0.204 -1.849 -1.131 -0.865 0.989 0.287 -0.460 -0.122 1.849 0.865 -0.552 0.372 -0.649 0.552 Expected values of normal order statistics
-1.026 0.209 -0.836 -0.338 0.473 -0.338 -1.616 -0.041 1.616 0.777 1.179 1.365 0.041 0.669 -1.365 0.125 -0.209 -2.043 -1.179 -0.836 1.026 0.294 -0.473 -0.125 2.043 0.894 -0.568 0.382 -0.669 0.568 Expected values of exponential order statistics
0 18 0.89 0.24 0.47 1.17 0.47 0.07 0.68 2.99 1.54 2.16 2.49 0.74 1.40 0.10 0.81 0.56 0.03 0.14 0.24
30 · Chapter 1: Basic Statistics IMSL STAT/LIBRARY 1.91 0.98 0.40 0.61 3.99 1.71 0.35 1.07 0.30 1.28
Comments
1. Workspace may be explicitly provided, if desired, by use R2NKS/DR2NKS. The reference is:
CALL R2NKS (NOBS, X, FUZZ, ITIE, ISCORE, SCORE, IWK) The additional argument is:
IWK — Integer work vector of length NOBS.
2. The routine RNSET (see Chapter 18, Random Number Generation) can be used to initialize the seed of the random number generator used to break ties. If the seed is not initialized by RNSET; different runs of the same program can yield different results if there are tied observations and ITIE = 3.
Description
The routine RANKS determines the ranks, or various transformations of the ranks of the data in X. Ties in the data can be resolved in four different ways, as specified in ITIE.
ISCORE = 0: Ranks
For this option, the values output in SCORE are the ordinary ranks of the data in X. If X(I) has the smallest value among those in X and there is no other element in X with this value, then SCORE(I) = 1. If both X(I) and X(J) have the same smallest value, then
if ITIE = 0, SCORE(I) = SCORE(J) = 1.5 if ITIE = 1, SCORE(I) = SCORE(J) = 2.0 if ITIE = 2, SCORE(I) = SCORE(J) = 1.0 if ITIE = 3, SCORE(I) = 1.0 and SCORE(J) = 2.0
or SCORE(I) = 2.0 and SCORE(J) = 1.0.
When the ties are resolved by use of routine RNUNF (see Chapter 18, Random Number Generation) to generate random numbers, different results may occur when running the same program at different times unless the “seed” of the random number generator is set explicitly by use of the routine RNSET (see Chapter 18, Random Number Generation). Ordinarily, there is no need to call the routine to set the seed, even if there are ties in the data.
ISCORE = 1: Normal Scores, Blom Version
Normal scores are expected values, or approximations to the expected values, of order statistics from a normal distribution. The simplest approximations are obtained by evaluating the inverse cumulative normal distribution function (routine ANORIN, see Chapter 18, Random Number Generation) at the ranks scaled into the open interval (0, 1). In the Blom version (see
IMSL STAT/LIBRARY Chapter 1: Basic Statistics · 31 Blom 1958), the scaling transformation for the rank ri(1 £ ri £ n, where n is the sample size, NOBS) is (ri - 3/8)/(n + 1/4). The Blom normal score corresponding to the observation with rank ri is
1 3 / 8 1/ 4 ri
n
- æ - ö
F çè + ÷ø where F(×) is the normal cumulative distribution function.
Adjustments for ties are made after the normal score transformation. That is, if X(I) equals X(J) (within FUZZ) and their value is the k-th smallest in the data set, the Blom normal scores are determined for ranks of k and k + 1, and then these normal scores are averaged or selected in the manner specified by ITIE. (Whether the transformations are made first or ties are resolved first makes no difference except when averaging is done.)
ISCORE = 2: Normal Scores, Tukey Version
In the Tukey version (see Tukey 1962), the scaling transformation for the rank ri is
(ri - 1/3)/(n + 1/3). The Tukey normal score corresponding to the observation with rank ri is
1 1/ 3
1/ 3 ri
n
- æ - ö
F çè + ÷ø
Ties are handled in the same way as discussed above for the Blom normal scores.
ISCORE = 3: Normal Scores, Van der Waerden Version
In the Van der Waerden version (see Lehmann 1975, page 97), the scaling transformation for the rank ri is ri/(n + 1). The Van der Waerden normal score corresponding to the observation with rank ri is
1
1 ri
n
- æ ö
F çè + ÷ø
Ties are handled in the same way as discussed above for the Blom normal scores.
ISCORE = 4: Expected Value of Normal Order Statistics
For this option, the values output in SCORE are the expected values of the normal order statistics from a sample of size NOBS. If the value in X(I) is the k-th smallest, then the value output in SCORE(I) is E(Zk), where E(×) is the expectation operator and Zk is the k-th order statistic in a sample of size NOBS from a standard normal distribution. Such expected values are computed by the routine ENOS (see Chapter 20, Mathematical Support). Ties are handled in the same way as discussed above for the Blom normal scores.
ISCORE = 5: Savage Scores
For this option, the values output in SCORE are the expected values of the exponential order statistics from a sample of size NOBS. These values are called Savage scores because of their use
32 · Chapter 1: Basic Statistics IMSL STAT/LIBRARY in a test discussed by Savage (1956) (see Lehman 1975). If the value in X(I) is the k-th smallest, then the value output in SCORE(I) is E(Yk), where Yk is the k-th order statistic in a sample of size NOBS from a standard exponential distribution. The expected value of the k-th order statistic from an exponential sample of size n (NOBS) is
1 1 1
1 1
n n+ + ××× +n k
- - +
Ties are handled in the same way as discussed above for the Blom normal scores.
The example uses all of these options with the same data set, which contains some ties. The ties are handled different ways in this example.