Remarks on Zeros and Ties in the Wilcoxon Signed Rank Procedures

Abstract A Wilcoxon one-sample signed rank test may be made when some of the observations are 0 by dropping the 0's before ranking. However, a sample can be not significantly positive while a more negative sample (obtained by decreasing each observation equally), is significantly positive by the ordinary Wilcoxon test. The reverse is also possible. Two-piece confidence regions result. A procedure for avoiding these difficulties is proposed, namely to rank the observations including the 0's, drop the ranks of the 0's, and reject the null hypothesis if the sum of the remaining negative (or positive) ranks falls in the tail of its null distribution (given the number of 0's). If observations are tied in absolute value, their ranks may be averaged before attaching signs. This changes the null distribution. A sample may be significantly positive which is not significant if the observations are increased (unequally), or if the ties are broken in any way.