Versions of the sign test in the presence of ties

The question of how to treat tied observations is often a problem in nonparametric tests. In this article the many different procedures that have been suggested for dealing with ties in the sign test are reviewed and compared. We discuss a new version of the sign test that is especially well-suited for situations in which the probability of an observation equalling the hypothesized median is known to be very high. This test is based on the test statistic formed by adding the number of positive observations and two-thirds of the number of zeroes. Studies of the power and significance level of many tests that have been proposed for this problem indicate that the asymptotic uniformly most powerful (UMP) test performs very well for almost all settings considered, even when the sample size is small. This asymptotic UMP test, which has been little used to date, is recommended for use in practice. A particular application of the use of the sign test in the study of fluctuating asymmetry of meristic traits in ornamental goldfish is presented.