## Tied Values

*Tied values* occur when two are 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,
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 may be less accurate
when the distribution of the data is heavily tied.
For such data, it may be appropriate to use the exact
tests provided by PROC NPAR1WAY, as described in
the "Exact Tests" section.

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

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.