The algorithm for the Wilcoxon matched pairs signed rank test in SPSS works as follows:

Compute the differences between each pair. Sort nonzero absolute differences in order and assign ranks to them, giving the mean rank to tied sets. The sum of the positive ranks and the sum of the negative ranks is then computed. The Z-statistic is then computed as:

min(Sp,Sn) - (n(n+1)/4) Z = ----------------------- SQRT[n(n+1)(2n+1)/24]

in releases prior to 6.1.2. For release 6.1.2 and later, the Z-statistic is:

min(Sp,Sn) - (n(n+1)/4) Z = --------------------------------------- SQRT[n(n+1)(2n+1)/24-SUM(Ti^3-Ti)/48]

In each case, min(Sp,Sn) is the minimum of the sum of the positive or negative ranks. The n is the number of untied pairs. The Ti are the numbers of differences that are tied in the ith tied set.

So there are really two kinds of ties at issue. One is a tie between the members of a pair. These are eliminated from the analysis in all releases. The second kind are nonzero pair differences that are of the same magnitude. These are used in obtaining sums of positive and negative ranks of the differences. In the older releases, there is no correction to the standard error (denominator of the Z) for these ties. In 6.1.2 and later releases, there is a correction for such ties (the latter is preferable).

Incidentally, there are further assumptions for the Wilcoxon test in addition to continuity. See, e.g., Conover's _Practical Nonparametric Statistics_.

David Nichols Senior Support Statistician SPSS Inc. nichols@spss.com

>---------- >From: Patrick Wig[SMTP:patwig@DEAKIN.EDU.AU] >Sent: Thursday, November 14, 1996 9:03 PM >To: Multiple recipients of list SPSSX-L >Subject: Wilcoxon handling of "ties" or equal ranks > >Hello all, > >A lurker awakes... > >Because of site license restrictions, I don't have ready access to the >full >manual set for SPSS 6.0 for Windows 3.1. I've checked all the >information >provided in the online help, and scanned through 13.5 MB of back >messages in >this group dating back to June 1994, but not found an answer to the >following question. > >The Wilcoxon signed-ranks test specifies one assumption: That the >dependent >variable is continuous. One symptom of a discontinuous dependent >variable >is a large number of ties in the signed ranks. > >Two solutions are recommended: 1) Discard ties and reduce N; or 2) >Retain >ties, divide them equally among positive and negative signs, and >discard one >if there is an odd number. This second option is considered more >conservative and therefore superior (at least by Gravetter & Wallnau, >1992), >since ties can be interpreted as evidence in support of the null >hypothesis >and preserving them in this fashion increases the likelihood of >retaining >the null hypothesis. A third option is to find a better means of >analysing >the data *grin*, but I'm interested in the first two options >particularly. > >Here comes the punch-line: Which of the first two solutions is used by >SPSS >when calculating the Wilcoxon Signed-Ranks test for related samples? >The >output does not make the decision clear. > >Thanks in advance for your help, > >Pat > > >------------------- >Gravetter, F. J., & Wallnau, L. B. (1992). _Statistics for the > Behavioral Sciences_ (3rd ed.). St. Paul: West. > >Patrick Wig >snail-mail: School of Psychology, > Deakin University, Geelong, Vic., > Australia 3217 >e-mail: patwig@deakin.edu.au >Phone: x71189 (613 5227 1189) >------------------------------------------------------------------------ >---- >