Tied values occur when two or more observations are equal, whether the observations occur in the same sample or in different samples. In theory, nonparametric tests were developed for continuous distributions where the probability of a tie is zero. In practice, however, ties often occur. PROC NPAR1WAY uses the same method to handle ties for all score types. The procedure computes the scores as if there were no ties, averages the scores for tied observations, and assigns this average score to each observation with the same value.

When there are tied values, PROC NPAR1WAY first sorts the observations in ascending order and assigns ranks as if there were no ties. Then the procedure computes the scores based on these ranks by using the formula for the specified score type. The procedure averages the scores for tied observations and assigns this average score to each of the tied observations. Thus, all equal data values have the same score value. PROC NPAR1WAY then computes the test statistic from these scores.

Note that the asymptotic tests might be less accurate when the distribution of the data is heavily tied. For such data, it might be appropriate to use the exact tests provided by PROC NPAR1WAY as described in the section Exact Tests.

When computing empirical distribution function statistics for data with ties, PROC NPAR1WAY uses the formulas given in the section Tests Based on the Empirical Distribution Function. No special handling of ties is necessary.

Note that PROC NPAR1WAY 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 NPAR1WAY 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 NPAR1WAY. For information about the ROUND function, see the discussion in SAS Language Reference: Dictionary.