Wilcoxon Signed Rank test

Compiled by Sreedevi Menon

As part of the efficacy analysis of a recent Phase III study , it was planned to determine the significant difference in efficacy of treatments over the weeks. Each treatment group was analyzed separately. The paired t-test was employed to carry out the analysis on the parametric data.The Wilcoxon Signed Rank test was used for analysis of efficacy of treatments with respect to non-parametric data. The null hypothesis was that in the underlying population of differences among pairs the median difference is equal to 0. The alternate hypothesis may be one sided or two sided.

In SAS, this test is performed using the proc univariate procedure. Before applying the procedure, the difference between the value of parameter at the two time points that are to be compared (usually pre and post dose values) has to be computed. This difference is then defined in the “var” statement of the procedure. The syntax is as followed

ods output BasicMeasures= _A TestsforLocation= _B
proc univariate data= eff_data;
var diff;
run;
ods output close;

The output will contain p-value corresponding to Student’s t, Sign and Signed Rank tests. The p-value corresponding to the signed rank test will be considered. Further, For a 1-sided p-value, we would divide the 2-sided p-value by 2.The ‘ods output statement’ is used to extract descriptive statistics and the p value to two different datasets by using the as shown in the above example

Hodges-Lehmann ∆

A parametric test, such as the t-test, compares the means of the two samples. A nonparametric method, such as the Wilcoxon Rank Sum Test compares the entire distributions of the two independent samples. The null hypothesis of the Wilcoxon Rank Sum test says the two samples can be viewed as a single sample from one population. The alternative hypothesis is that the first treatment group has a different different distribution (or location) than the second treatment group.

The treatment effect, denoted as ∆, is the difference between treatment groups. If parametric methods were used, means could be calculated for each treatment group, and a subtraction of the means can be used to estimate ∆. However, when the data are not normally distributed and the median value of the response variable of interest is calculated for each treatment group, the estimate of the difference in treatment groups is not as straightforward as subtracting one median from the other.

The difference in medians is estimated using the methodology of Hodges-Lehmann. It is a very simple approach. The following steps can be used to estimate ∆:

• form all possible differences between the first treatment group and the second treatment group, in the response variable of interest. For example, if there are 100 patients in each group then 10,000 (100*100) differences would be calculated.
• the estimator ∆ is the median of those 10,000 differences.

All this taken from this SUPER PAPER.

Wilcoxon rank-sum test

The Wilcoxon rank-sum test is a non-parametric analog of the two-sample t-test. It is based on ranks of the data, and is used to compare location parameters, such as the mean or median, between two independent populations without the assumption of normally distributed data.


DATA RNKSM;
INPUT TRT $ PAT SCORE @@;
DATALINES;
SER 2 0 SER 3 2 SER 5 3 SER 6 3
SER 8 -2 SER 10 1 SER 12 3 SER 14 3
SER 16 -1 SER 17 2 SER 20 -3 SER 21 3
SER 22 3 SER 24 0 SER 26 2 SER 27 -1
PBO 1 3 PBO 4 -1 PBO 7 2 PBO 9 3
PBO 11 -2 PBO 13 1 PBO 15 0 PBO 18 -1
PBO 19 -3 PBO 23 -2 PBO 25 1 PBO 28 0
;

PROC NPAR1WAY WILCOXON DATA = RNKSM;
CLASS TRT; VAR SCORE;
TITLE1 'The Wilcoxon Rank-Sum Test';
RUN;