Open Journal of Statistics, 2012, 2, 172-176 doi:10.4236/ojs.2012.22019 Published Online April 2012 (http://www.SciRP.org/journal/ojs) Modified Wilcoxon Signed-Rank Test Ikewelugo Cyprian Anaene Oyeka, Godday Uwawunkonye Ebuh * Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Nigeria Email: * ablegod007@yahoo.com Received January 6, 2012; revised February 10, 2012; accepted February 19, 2012 ABSTRACT This paper briefly reviews the Wilcoxon signed rank sum test and proposes a modification. Unlike the Wilcoxon method, the proposed approach does not require that the populations being studied be continuous. Also unlike the Wil- coxon signed rank test the proposed method, does not require the absence of zero differences or tied absolute values of differences. Rather the proposed method structurally makes provisions for these possibilities. The proposed test statistic also enables the estimation of the probabilities of positive, zero or tied and negative differences within the data. This was illustrated with an example and the proposed method was generally more efficient and hence more powerful than the Wilcoxon test statistic with the power increasing as the number of tied observations or zero differences increases. Keywords: Proposed Method; Probabilities; Positive; Zero; Tied; Negative; Frequencies 1. Introduction Wilcoxon signed rank test is a rank based alternative to the parametric t test that assumes only that the distribu- tion of differences within pairs be symmetric without requiring normality [1]. Let X i be the ith observation, in a random sample of size n drawn from population X with unknown median M; or let (X i , Y i ) be the ith pair in a paired random sample of size n drawn from population X and Y with unknown M 1 and M 2 respectively. For the moment, we assume that X and Y are continuous. In the one sample case, interest may be in testing that the unknown population median M is equal to some specified value, M 0 . In the paired sample case interest may be in testing that the unknown popula- tion medians are equal that is M 1 = M 2 or that one popu- lation median is equal to at least some multiple of other population median, that is M 1 = c·M 2 + k say, where c (c > 0) and k are real numbers versus appropriate two-sided or one sided alternative hypotheses. If the assumption of parametric test are satisfied, the first hypothesis may be tested using the one sample t-test while the second hy- pothesis may be tested using the paired sample t test. The third hypothesis may however be readily tested using the parametric method because of problems of non-homo- geneity. If the necessary assumptions of the parametric t- test cannot be reasonably made, use of a non-parametric method that often readily suggests itself in these situa- tions is the Wilcoxon signed rank sum test [2]. 1, 2, , i   n n n 1, 2, , i   In this paper, we briefly discuss the Wilcoxon method and then proceed to present a modified version of the method that may be appropriate for testing the above hypotheses. 2. The Wilcoxon Signed Rank Sum Test According to [3,4], the Wilcoxon signed rank test is used to test the null hypothesis that the median of a distribu- tion is equal to some value and can be used in place of a one sample t-test, a paired t-test or for ordered categori- cal data where a numerical scale is inappropriate but where it is possible to rank the observations. To use the Wilcoxon signed rank sum test, we first find the difference between the observation and the hy- pothesized median in the one sample problem or the dif- ference between the paired observations in the paired sample problems. That is, in the one sample case, we find d i = x i or in the two sample case (d i = x i – cy i – k) for 1, 2, , i   . We then take the absolute values of these differences and rank them either from the smallest to the largest or from the largest to the smallest, always taking note of the ranks of the absolute values with positive differences and those with negative differences. The re- quirement that the populations from which the samples are drawn are continuous makes it possible to state at least theoretically that the probability of obtaining zero differences or tied absolute values of the differences is zero. Now, let   i r d be the rank assigned to i d , the absolute value of the ith difference ; for i d 1, 2, , i n   . Let 1, if 0; 0, if 0. i i i d Z d       (1) Let   1 i PZ    (2) * Corresponding author. Copyright © 2012 SciRes. OJS