Wilcoxon signed rank test (exact distributions)

trharris14 — Wed, 30 May 2012 19:31:51 GMT

For n <= 20, proc univariate uses the exact distribution to compute the significance of S, but what exactly happens when there are tied ranks within the small dataset? I've read all 3 references (Iman, Conover, and Lehmann), but none of them really explain what happens when ties occur with a small sample (n <= 20) with the exact distribution. An example is below...

Consider the data:

Grp1 Grp2 Diff Rank

.32 .39 -0.07 3.5

.4 .47 -0.07 3.5

.11 .11 0.00 ---

.47 .43 0.04 1

.32 .42 -0.10 5

.35 .3 0.05 2

.32 .43 -0.11 6

.63 .98 -0.35 8

.5 .86 -0.36 9

.6 .79 -0.19 7

Sum of ranks for positive differences (ri+) = 3

Given that the ranks are {1, 2, 3.5, 3.5, 5, 6, 7, 8, 9}, the only ways to get a sum of ranks that is less than or equal to 3 is for the set of positive ranks to be one of:

Set Sum

{} = 0

{1} = 1

{2} = 2

{1,2} = 3

So, there are 4 configurations on the left-hand side extreme and 4 on the right. Thus, the p-value should be 8/2^9 = 8/512 = 0.0156. However, SAS reports p=10/512 = 0.0195.

Perhaps, SAS is saying that {3.5} is either {3} or {4} with ½ probability. Thus, this would be ½ more cases. Since there are two ranks of 3.5, each could be {3} with ½ probability and thus there would be a total of 5 configurations on the left extreme and 5 on the right?

Set Sum

{} = 0 with 100% probability = 1.0 case

{1} = 1 with 100% probability = 1.0 case

{2} = 2 with 100% probability = 1.0 case

{1,2} = 3 with 100% probability = 1.0 case

{3.5} = 3 with 50% probability = 0.5 case

{3.5} = 3 with 50% probability = 0.5 case
                                                      ---------
                                                      5.0 cases

If you know how SAS is computing the exact p-value with ties (all possible combinations of the sum of ranks less than or equal to the sum of positive ranks) or know where a 'useful' reference might be, please let me know. Thanks in advance!!

Re: Wilcoxon signed rank test (exact distributions)

Rick_SAS — Wed, 30 May 2012 23:14:15 GMT

This SUGI 1994 paper contains a macro that uses the DATA step to reproduce the WSR test, so you ought to be able to see exactly what happens for tied values.

http://www.sascommunity.org/sugi/SUGI94/Sugi-94-172%20Tian.pdf

Re: Wilcoxon signed rank test (exact distributions)

trharris14 — Thu, 31 May 2012 16:27:48 GMT

Thanks for the paper Rick, but it doesn't really tell how SAS calculates the exact p-values, just shows an attached file in the macro code (WSRtable.dat). Thanks for your help though!! Have a great day!

Re: Wilcoxon signed rank test (exact distributions)

PGStats — Thu, 31 May 2012 18:01:54 GMT

For details of exact test calculation methods, you might have to look in the references within :

http://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#statug_npar1way_a0000000203.htm

topic Wilcoxon signed rank test (exact distributions) in Statistical Procedures

Wilcoxon signed rank test (exact distributions)

Re: Wilcoxon signed rank test (exact distributions)

Re: Wilcoxon signed rank test (exact distributions)

Re: Wilcoxon signed rank test (exact distributions)