Modified Wilcoxon Signed-Rank Test

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 Wilcoxon 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.


Introduction
Wilcoxon signed rank test is a rank based alternative to the parametric t test that assumes only that the distribution 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 population medians are equal that is M 1 = M 2 or that one population 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 hypothesis 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-homogeneity.If the necessary assumptions of the parametric ttest cannot be reasonably made, use of a non-parametric method that often readily suggests itself in these situations is the Wilcoxon signed rank sum test [2].
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.

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 distribution is equal to some value and can be used in place of a one sample t-test, a paired t-test or for ordered categorical 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 hypothesized median in the one sample problem or the difference between the paired observations in the paired sample problems.That is, in the one sample case, we find 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 requirement 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   , the absolute value of the ith difference ; for   1, if 0; 0, if 0.
  That is T is the sum of the ranks of the absolute values with positive differences.Now for simplicity but without loss of generality, we let   Hence Note that an estimate of θ namely  may be obtained from the expression The null hypothesis that is usually tested in the Wilcoxon signed rank sum approach is [2].0 versus either a two sided or an appropriate one sided alternative hypothesis.For the paired sample case, this null Similarly for the one sample case this null hypothesis is equivalent to 0 0 : H M M  .A large sample test statistic for any hypothesized value of 0, θ 0 say (0 < θ < 1) is given by This has approximately a standard normal distribution under the desired null hypothesis.But under the null hypothesis usually tested using the Wilcoxon signed rank test, that is,    and 0 1 2 1 Var 24 Hence the test statistics of Equation ( 9) becomes which under H 0 has a standard normal distribution for fairly large sample size n and may be used to test the null hypothesis of equal population medians.When all non zero values are of the same sign, the Wilcoxon signed rank test reduces to the sign test and the two tailed P- , where n is the number of non zero values [5].

The Proposed Modified Method
We here drop the requirement that the populations from which the samples are drawn are continuous.We now only require that the populations be quantitative data measured on at most the ordinal scale.The populations could be continuous or discrete.The requirements of no zero differences or tied absolute values are also no longer necessary; since these problems are taken care of structurally by following model specifications.We here continue to use d i to represent the difference between x i and the hypothesized population median M 0 , in the one-sample case and the differences x i -y i (or x i -y i -k) in the paired sample case for   .We also continue to use again without loss of generality   where   That is, T is the difference between the sum of ranks assigned to absolute values with positive differences and the sum of the ranks assigned to absolute values with negative differences. and Note that may be estimated as Note that in the case of one sample , and π  0 π π  are respectively on the average the probabilities that the population median is greater than, equal to, or less than the hypothesized median Mo while in the paired sample case, they are on the average the probabilities that one population median is greater than, equal to or less than the other population median.They are estimated respectively as the relative frequencies of occurrence of 1, 0 and -1 in the frequency distribution of the n elements of where f  0 , f and f  i are respectively the frequencies of occurrence of 1, 0 and -1 in the frequency distri-bution of Z .Often the null hypothesis required to be tested is or versus a two tailed or an appropriate one tailed alternative hypothesis.For the paired sample case, this null hypothesis is equivalent to For the one sample case, the null hypothesis is equivalent to The test statistic which under 0 H has a standard normal distribution for fairly large n.In practical application and π  π  of Equation ( 24) are usually replaced with their sample estimates of Equation ( 22).Note that as defined by Wilcoxon is the sum of the ranks of the absolute values of positive differences and that θ is the probability of the occurrence of only positive differences.Thus the specification of Wilcoxon's statistic does not explicitly provide for the possible occurrence of negative differences.So the T  π  in our proposed modification in Equation ( 14) is automatically set equal to zero in the Wilcoxon's approach and under 0 H the probability of the occurrence of positive differences is hypothesized as . Hence if we automatically set   π 0  and under the null hypothesis we set    in Equation ( 24), then the proposed test statistic is seen to coincide with Wilcoxon signed rank test statistic could be rewritten as while the test statistic for the proposed method could be rewritten as Copyright © 2012 SciRes.

OJS
The hypothesis tested under the Wilcoxon's app roach tistics are able hypothesis or reject a false null hypo be obtained by com uation (25) with the va H equivalent to : H tested under th Therefore the relative rate at which these two test stato accept a true null thesis can therefore paring the variance of + 4T of Eq riance of 2T of Equation (26) that is in terms of the relative efficiency of T compared with + T that is In other words

 
T  whenever 0 π is not ere are zero differences or equal to 0, that ver th serv ta.fficiency of T c 0 , eth re te the actual and the ideal number have.The results are preolute differences are assigned their mean ranks.The results are presented in Table 2. Ta is whene tied ob ations in the da The relative e ompared with T  increases as 0 π increases.The two methods are equally efficient when 0 π  that is when there are no ties in the data whatsoever.
Thus unless there are no ties whatsoever in the data, the proposed m od yields a mo powerful test criterion than the Wilcoxon signed rank sum test statistic for the same sample size.only

Illustrative Example
A random sample of twelve married women were selected and asked to sta of children they would like to sented in Table 1.
To apply the Wilcoxon signed rank test, we take and rank the differences between the actual and ideal number of children by the sample of married women shown in

Conclusion
s test and proposes method, the propo populations being studied be continuous.Also unlike the Wilcoxon 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.The proposed method shown to be 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.

Table 2 . Ranks of absolute diff ces eren   i r d , of e dif- ferences d i tween actual and ideal number of c dren in Table 1.
Also note from Equation (28) that the relative efficiency of the modified test statistic T to the Wilcoxon test statistic T  is estimated as fficient and hence more powerful than the Wilcoxon signed rank test statistic T  .Thi paper briefly reviews the Wilcoxon signed rank sum a modification.Unlike the Wilcoxon sed approach does not require that the [1] R. De-W.Derryberry, S. B. Schou and W. J. Conover,