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# Wilcoxon signed rank test (exact distributions)

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!!

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‎05-31-2012 02:01 PM
Posts: 5,052

## Re: Wilcoxon signed rank test (exact distributions)

Posted in reply to trharris14

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_a00...

PG

PG

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SAS Super FREQ
Posts: 3,839

## Re: Wilcoxon signed rank test (exact distributions)

Posted in reply to trharris14

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

New Contributor
Posts: 3

## Re: Wilcoxon signed rank test (exact distributions)

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!

Solution
‎05-31-2012 02:01 PM
Posts: 5,052

## Re: Wilcoxon signed rank test (exact distributions)

Posted in reply to trharris14

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_a00...

PG

PG
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