The RANK Procedure

Concepts: RANK Procedure

PROC RANK stores all values in memory of the variables for which it computes ranks.

Ranks are useful for investigating the distribution of values for a variable. The ranks divided by n or n+1 form values in the range 0 to 1, and these values estimate the cumulative distribution function. You can apply inverse cumulative distribution functions to these fractional ranks to obtain probability quantile scores, which you can compare to the original values to judge the fit to the distribution. For example, if a set of data has a normal distribution, the normal scores should be a linear function of the original values, and a plot of scores versus original values should be a straight line.

Many nonparametric methods are based on analyzing ranks of a variable:

A two-sample t-test applied to the ranks is equivalent to a Wilcoxon rank sum test using the t approximation for the significance level. If you apply the t-test to the normal scores rather than to the ranks, the test is equivalent to the van der Waerden test. If you apply the t-test to median scores (GROUPS=2), the test is equivalent to the median test.
A one-way analysis of variance applied to ranks is equivalent to the Kruskal-Wallis k-sample test; the F-test generated by the parametric procedure applied to the ranks is often better than the approximation used by Kruskal-Wallis. This test can be extended to other rank scores (Quade 1966).
You can obtain a Friedman's two-way analysis for block designs by ranking within BY groups and then performing a main-effects analysis of variance on these ranks (Conover 1998).
You can investigate regression relationships by using rank transformations with a method described by Iman and Conover (1979).

Treatment of Tied Values

When PROC RANK ranks values, if two or more values of an analysis variable that are within a BY group are equal, then tied values are present in the data. Because the values are indistinguishable and there is usually no further obvious information on which the ranks can reasonably be based, PROC RANK does not assign different ranks to the values. Tied values could be arbitrarily assigned different ranks. But in statistical applications such as nonparametric statistical tests employing ranks, it is conventional to assign the same rank to tied values.

These statistical tests commonly assume that the data is from a continuous distribution, in which the probability of a tie is theoretically zero. In practice, whether because of inaccuracies in measurement, the finite accuracy of representation within a digital computer, or other reasons, tied values often occur. It is also conventional in these statistical tests to assign the average rank to a group of tied values. Assignment of the average rank is preferred because it preserves the sum of the ranks and, therefore, does not distort the estimate of the cumulative distribution function.

For applications within and outside of statistics, the RANK procedure provides the TIES= option to control the treatment of tied values. The default value for this option depends on the specified ranking or scoring method, which you can specify with the options of the PROC RANK statement. For ranking and scoring methods, when TIES=LOW, TIES=HIGH, or TIES=MEAN, tied values are initially treated as though they are distinguishable. These methods all begin by sorting the values of the analysis variable within a BY group, and then assigning to each nonmissing value an ordinal number that indicates its position in the sequence.

Subsequently, for non-scoring methods, PROC RANK resolves tied values by selecting the minimum with TIES-LOW, selecting the maximum with TIES=HIGH, or calculating the average of the ordinals in a group of tied values with TIES=MEAN. PROC RANK then obtains the rank from this value through one or more further transformations such as scaling, translation, and truncation.

Scoring methods include normal and Savage scoring, which are requested by the NORMAL= and SAVAGE options. Non-scoring methods include ordinal ranking, the default, and those methods that are requested by the FRACTION, NPLUS1, GROUPS=, and PERCENT options. For the scoring methods NORMAL= and SAVAGE, PROC RANK obtains the probability quantile scores with the appropriate formulas as if no tied values were present within the data. PROC RANK then resolves tied values by selecting the minimum, selecting the maximum, or calculating the average of all scores within a tied group.

For all ranking and scoring methods, when TIES=DENSE, tied values are treated as indistinguishable, and each value within a tied group is assigned the same ordinal. As with the other TIES= resolution methods, all ranking and scoring methods begin by sorting the values of the analysis variable and then assigning ordinals. However, a group of tied values is treated as a single value. The ordinal assigned to the group differs by only +1 from the ordinal that is assigned to the value just prior to the group, if there is one. The ordinal differs by only -1 from the ordinal assigned to the value just after the group, if there is one. Therefore, the smallest ordinal within a BY group is 1, and the largest ordinal is the number of unique, nonmissing values in the BY group.

After the ordinals are assigned, PROC RANK calculates ranks and scores using the number of unique, nonmissing values instead of the number of nonmissing values for scaling. Because of its tendency to distort the cumulative distribution function estimate, dense ranking is not generally acceptable for use in nonparametric statistical tests.

Note that PROC RANK bases its computations on the internal numeric values of the analysis variables. The procedure does not format or round these values before analysis. When values differ in their internal representation, even slightly, PROC RANK does not treat them as tied values. If this is a concern for your data, then round the analysis variables by an appropriate amount before invoking PROC RANK. For information about the ROUND function, see ROUND Function in SAS Language Reference: Dictionary.

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