Computes the ranks, normal scores, or exponential scores for a vector of observations.

Synopsis

#include <imsls.h>

float *imsls_f_ranks (int n_observations, float x[], ..., 0)

The type double function is imsls_d_ranks.

Required Arguments

int n_observations   (Input)
Number of observations.

float x[]   (Input)
Array of length n_observations containing the observations to be ranked.

Return Value

A pointer to a vector of length n_observations containing the rank (or optionally, a transformation of the rank) of each observation.

Synopsis with Optional Arguments

#include <imsl.h>

float* imsls_f_ranks (int n_observations, float x[],
IMSLS_AVERAGE_TIE, or
IMSLS_HIGHEST, or
IMSLS_LOWEST, or
IMSLS_RANDOM_SPLIT,
IMSLS_FUZZ, float fuzz_value,
IMSLS_RANKS, or
IMSLS_BLOM_SCORES, or
IMSLS_TUKEY_SCORES, or
IMSLS_VAN_DER_WAERDEN_SCORES, or
IMSLS_EXPECTED_NORMAL_SCORES, or
IMSLS_SAVAGE_SCORES,
IMSLS_RETURN_USER, float ranks[],
0)

Optional Arguments

IMSLS_AVERAGE_TIE, or

IMSLS_HIGHEST, or

IMSLS_LOWEST, or

IMSLS_RANDOM_SPLIT
Exactly one of these optional arguments can be used to change the method used to assign a score to tied observations.

 

Argument	Method
IMSLS_AVERAGE_TIE	average of the scores of the tied observations (default)
IMSLS_HIGHEST	highest score in the group of ties
IMSLS_LOWEST	lowest score in the group of ties
IMSLS_RANDOM_SPLIT	tied observations are randomly split using a random number generator

IMSLS_FUZZ, float fuzz_value   (Input)
Value used to determine when two items are tied. If abs(x [i] − x [j]) is less than or equal to fuzz_value, then x[i] and x[j] are said to be tied.
Default: fuzz_value = 0.0

IMSLS_RANKS, or

IMSLS_BLOM_SCORES, or

IMSLS_TUKEY_SCORES, or

IMSLS_VAN_DER_WAERDEN_SCORES, or

IMSLS_EXPECTED_NORMAL_SCORES, or

IMSLS_SAVAGE_SCORES
Exactly one of these optional arguments can be used to specify the type of values returned.

Argument	Result
IMSLS_RANKS	ranks (default)
IMSLS_BLOM_SCORES	Blom version of normal scores
IMSLS_TUKEY_SCORES	Tukey version of normal scores
IMSLS_VAN_DER_WAERDEN_SCORES	Van der Waerden version of normal scores
IMSLS_EXPECTED_NORMAL_SCORES	expected value of normal order statistics (for tied observations, the average of the expected normal scores)
IMSLS_SAVAGE_SCORES	Savage scores (the expected value of exponential order statistics)

IMSLS_RETURN_USER, float ranks[]   (Output)
If specified, the ranks are returned in the user-supplied array ranks.

Description

Ties

In data without ties, the output values are the ordinary ranks (or a transformation of the ranks) of the data in x. If x[i] has the smallest value among the values in x and there is no other element in x with this value, then ranks [i] = 1. If both x[i] and x[j] have the same smallest value, the output value depends on the option used to break ties.

When the ties are resolved randomly, function imsls_f_random_uniform (Chapter 12;) is used to generate random numbers. Different results may occur from different executions of the program unless the “seed” of the random number generator is set explicitly by use of the function imsls_f_random_seed_set (Chapter 12;).

Scores

As an option, normal and other functions of the ranks can be returned. Normal scores can be defined as the 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, function imsls_f_normal_inverse_cdf (Chapter 11;), at the ranks scaled into the open interval (0, 1). In the Blom version (see Blom 1958), the scaling transformation for the rank ri (1 ≤ ri ≤ n, where n is the sample size, n_observations) is (ri − 3/8)/(n + 1/4). The Blom normal score corresponding to the observation with rank ri is

where Φ(·) 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_value) 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. Then, these normal scores are averaged or selected in the manner specified. (Whether the transformations are made first or ties are resolved first makes no difference except when IMSLS_AVERAGE_TIE is specified.)

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 as follows:

Ties are handled in the same way as for the Blom normal scores.

In the Van der Waerden version (see Lehmann 1975, p. 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 as follows:

Ties are handled in the same way as for the Blom normal scores.

When option IMSLS_EXPECTED_NORMAL_SCORES is used, the output values are the expected values of the normal order statistics from a sample of size n_observations. If the value in x[i] is the k-th smallest, the value output in ranks [i] is E(zk), where E(·) is the expectation operator and zk is the k-th order statistic in a sample of size n_observations from a standard normal distribution. Ties are handled in the same way as for the Blom normal scores.

Savage scores are the expected values of the exponential order statistics from a sample of size n_observations. These values are called Savage scores because of their use in a test discussed by Savage 1956 (see also Lehmann 1975). If the value in x[i] is the k-th smallest, the value output in ranks [i] is E(yk), where yk is the k-th order statistic in a sample of size n_observations from a standard exponential distribution. The expected value of the k-th order statistic from an exponential sample of size n (n_observations) is as follows:

Ties are handled in the same way as for the Blom normal scores.

Examples

Example 1

The data for this example, from Hinkley (1977), contains 30 observations. Note that the fourth and sixth observations are tied and that the third and twentieth observations are tied.

#include <imsls.h>

#define N_OBSERVATIONS          30

main()
{
    float       *ranks;
    float       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};

    ranks = imsls_f_ranks(N_OBSERVATIONS, x, 0);
    imsls_f_write_matrix("Ranks", 1, N_OBSERVATIONS, ranks, 0);
}

Output

                                 Ranks
         1           2           3           4           5           6
       5.0        18.0         6.5        11.5        21.0        11.5
 
         7           8           9          10          11          12
       2.0        15.0        29.0        24.0        27.0        28.0
 
        13          14          15          16          17          18
      16.0        23.0         3.0        17.0        13.0         1.0
 
        19          20          21          22          23          24
       4.0         6.5        26.0        19.0        10.0        14.0
 
        25          26          27          28          29          30
      30.0        25.0         9.0        20.0         8.0        22.0

 

Example 2

This example uses all the score options with the same data set, which contains some ties. Ties are handled in several different ways in this example.

#include <imsls.h>

#define N_OBSERVATIONS          30

void main()
{
    float       fuzz_value=0.0, score[4][N_OBSERVATIONS], *ranks;
    float       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};
    char        *row_labels[] = {"Blom", "Tukey", "Van der Waerden",
                                 "Expected Value"};

                                /* Blom scores using largest ranks */
                                /* for ties */
    imsls_f_ranks(N_OBSERVATIONS, x,
                 IMSLS_HIGHEST,
                 IMSLS_BLOM_SCORES,
                 IMSLS_RETURN_USER,   &score[0][0],
                 0);
                                /* Tukey normal scores using smallest */
                                /* ranks for ties */
    imsls_f_ranks(N_OBSERVATIONS, x,
                 IMSLS_LOWEST,
                 IMSLS_TUKEY_SCORES,
                 IMSLS_RETURN_USER,  &score[1][0],
                 0);
                                /* Van der Waerden scores using */
                                /* randomly resolved ties */
    imsls_random_seed_set(123457);
    imsls_f_ranks(N_OBSERVATIONS, x,
                 IMSLS_RANDOM_SPLIT,
                 IMSLS_VAN_DER_WAERDEN_SCORES,
                 IMSLS_RETURN_USER, &score[2][0],
                 0);
                                /* Expected value of normal order */
                                /* statistics using averaging to */
                                /* break ties */
    imsls_f_ranks(N_OBSERVATIONS, x,
                 IMSLS_EXPECTED_NORMAL_SCORES,
                 IMSLS_RETURN_USER, &score[3][0],
                 0);
    imsls_f_write_matrix("Normal Order Statistics", 4, N_OBSERVATIONS,
                  (float *)score,
                 IMSLS_ROW_LABELS,   row_labels,
                 IMSLS_WRITE_FORMAT, "%9.3f",
                 0);
                                /* Savage scores using averaging */
                                /* to break ties */
    ranks = imsls_f_ranks(N_OBSERVATIONS, x,
                 IMSLS_SAVAGE_SCORES,
                 0);
    imsls_f_write_matrix("Expected values of exponential order "
                 "statistics", 1,
                 N_OBSERVATIONS, ranks,
                 0);
}

Output

                        Normal Order Statistics
                         1          2          3          4          5
Blom                -1.024      0.209     -0.776     -0.294      0.473
Tukey               -1.020      0.208     -0.890     -0.381      0.471
Van der Waerden     -0.989      0.204     -0.753     -0.287      0.460
Expected Value      -1.026      0.209     -0.836     -0.338      0.473
 
                         6          7          8          9         10
Blom                -0.294     -1.610     -0.041      1.610      0.776
Tukey               -0.381     -1.599     -0.041      1.599      0.773
Van der Waerden     -0.372     -1.518     -0.040      1.518      0.753
Expected Value      -0.338     -1.616     -0.041      1.616      0.777
 
                        11         12         13         14         15
Blom                 1.176      1.361      0.041      0.668     -1.361
Tukey                1.171      1.354      0.041      0.666     -1.354
Van der Waerden      1.131      1.300      0.040      0.649     -1.300
Expected Value       1.179      1.365      0.041      0.669     -1.365
 
                        16         17         18         19         20
Blom                 0.125     -0.209     -2.040     -1.176     -0.776
Tukey                0.124     -0.208     -2.015     -1.171     -0.890
Van der Waerden      0.122     -0.204     -1.849     -1.131     -0.865
Expected Value       0.125     -0.209     -2.043     -1.179     -0.836
 
                        21         22         23         24         25
Blom                 1.024      0.294     -0.473     -0.125      2.040
Tukey                1.020      0.293     -0.471     -0.124      2.015
Van der Waerden      0.989      0.287     -0.460     -0.122      1.849
Expected Value       1.026      0.294     -0.473     -0.125      2.043
 
                        26         27         28         29         30
Blom                 0.893     -0.568      0.382     -0.668      0.568
Tukey                0.890     -0.566      0.381     -0.666      0.566
Van der Waerden      0.865     -0.552      0.372     -0.649      0.552
Expected Value       0.894     -0.568      0.382     -0.669      0.568
 
            Expected values of exponential order statistics
         1           2           3           4           5           6
     0.179       0.892       0.240       0.474       1.166       0.474
 
         7           8           9          10          11          12
     0.068       0.677       2.995       1.545       2.162       2.495
 
        13          14          15          16          17          18
     0.743       1.402       0.104       0.815       0.555       0.033
 
        19          20          21          22          23          24
     0.141       0.240       1.912       0.975       0.397       0.614
 
        25          26          27          28          29          30
     3.995       1.712       0.350       1.066       0.304       1.277

 

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