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 Zstatistic 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 Zstatistic
is:
min(Sp,Sn)  (n(n+1)/4)
Z = 
SQRT[n(n+1)(2n+1)/24SUM(Ti^3Ti)/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 SPSSXL
>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 signedranks 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 punchline: Which of the first two solutions is used by
>SPSS
>when calculating the Wilcoxon SignedRanks 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
>snailmail: School of Psychology,
> Deakin University, Geelong, Vic.,
> Australia 3217
>email: patwig@deakin.edu.au
>Phone: x71189 (613 5227 1189)
>
>
>
