Fluorite | Level 6

## Wilcox Signed Rank Test Statistic W (not S)

I'm doing non-parametric tests on paired (pre/post) samples that are ordinal (7-point Likert) and non-normally distributed, therefore using PROC UNIVARIATE with the difference of Time1 and Time2.

I know that the test statistic in the SAS output (Singed Ranked, S) is the Wilcox Signed Rank test with its corresponding P value. But it seems like, for the sake of reporting in a manuscript, the literature uses "W" which doesn't seem to be the same test statistic. Do you know how to get this W or do you think I should report the S? (S=-387.5, p<.0001).

Thanks!

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## Re: Wilcox Signed Rank Test Statistic W (not S)

I think PROC UNIVARIATE is the correct procedure to use here, since you have matched-pairs data rather than two independent samples. The only concern I'd have is that your variables are not continuous but only ordinally scaled. It appears, however, that "many researchers do treat Likert scale response data as if it were interval data" (quote from http://pages.cpsc.ucalgary.ca/~saul/wiki/uploads/CPSC681/topic-dane-likert.doc, p. 2, including italics; please see also the caveat described there!).

So, strictly speaking, the sign test would be more appropriate -- but less powerful than the Wilcoxon signed-rank test. Both tests are computed by PROC UNIVARIATE (based on the differences of the paired values), as you can see in your SAS output.

If you go for the Wilcoxon signed-rank test (in spite of the concern), the test statistic S provided by PROC UNIVARIATE differs in fact from the more common test statistic, which is denoted by T+ in several standard textbooks on nonparametric statistics. But also the notation Wn+ can be found for it (e.g. in the German standard reference by Büning and Trenkler). The conversion between the two is quite easy: S = T+ − n(n+1)/4, where n denotes the number of matched pairs minus the number of pairs with difference zero (according to another German textbook I have in front of me; cf. also the SAS documentation: http://documentation.sas.com/?docsetId=procstat&docsetTarget=procstat_univariate_details17.htm&docse...). Since n(n+1)/4 = E(T+) [edit:] under the test's null hypothesis, S is just the centered version of random variable T+ and has in particular expectation zero.

I think you can report either test statistic as long as it is clear which one you mean (and as you don't misspell "Wilcoxon" ;-)).

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## Re: Wilcox Signed Rank Test Statistic W (not S)

Fluorite | Level 6

## Re: Wilcox Signed Rank Test Statistic W (not S)

Can you explain why I can't use PROC UNIVARIATE? I'm not sure how I would go about using PROC NPAR1WAY since my variables are arranged as such:

(Pre)                                        (Post)

ID               Knowledge1_i                         Knowledge1_ii

1                          3                                               4

2                          4                                               5

3                          5                                               4

...etc

Would I need to recode these variables? Many things I've read say to just take the difference between these two knowledge variables and use PROC UNIVARIATE. Its just I'm not really sure what "S" is. Clear?

## Re: Wilcox Signed Rank Test Statistic W (not S)

I think PROC UNIVARIATE is the correct procedure to use here, since you have matched-pairs data rather than two independent samples. The only concern I'd have is that your variables are not continuous but only ordinally scaled. It appears, however, that "many researchers do treat Likert scale response data as if it were interval data" (quote from http://pages.cpsc.ucalgary.ca/~saul/wiki/uploads/CPSC681/topic-dane-likert.doc, p. 2, including italics; please see also the caveat described there!).

So, strictly speaking, the sign test would be more appropriate -- but less powerful than the Wilcoxon signed-rank test. Both tests are computed by PROC UNIVARIATE (based on the differences of the paired values), as you can see in your SAS output.

If you go for the Wilcoxon signed-rank test (in spite of the concern), the test statistic S provided by PROC UNIVARIATE differs in fact from the more common test statistic, which is denoted by T+ in several standard textbooks on nonparametric statistics. But also the notation Wn+ can be found for it (e.g. in the German standard reference by Büning and Trenkler). The conversion between the two is quite easy: S = T+ − n(n+1)/4, where n denotes the number of matched pairs minus the number of pairs with difference zero (according to another German textbook I have in front of me; cf. also the SAS documentation: http://documentation.sas.com/?docsetId=procstat&docsetTarget=procstat_univariate_details17.htm&docse...). Since n(n+1)/4 = E(T+) [edit:] under the test's null hypothesis, S is just the centered version of random variable T+ and has in particular expectation zero.

I think you can report either test statistic as long as it is clear which one you mean (and as you don't misspell "Wilcoxon" ;-)).

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