Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

APPENDIX E | Hypothesis Tests for Ordinal Data: Mann-Whitney, Wilcoxon, Kruskal-Wallis, and Friedman Tests 693
■ Zero Differences and Tied Scores
For the Wilcoxon test, there are two possibilities for tied scores.
1. A participant may have the same score in treatment 1 and in treatment 2, resulting
in a difference score of zero.
2. Two (or more) participants may have identical difference scores (ignoring the sign
of the difference).
When the data include individuals with difference scores of zero, one strategy is to
discard these individuals from the analysis and reduce the sample size (n). However, this
procedure ignores the fact that a difference score of zero is evidence for retaining the null
hypothesis. A better procedure is to divide the zero differences evenly between the positives
and negatives. (With an odd number of zero differences, discard one and divide the rest
evenly.) When there are ties among the difference scores, each of the tied scores should be
assigned the average of the tied ranks. This procedure was presented in detail in an earlier
section of this appendix (see page 690).
■ Hypotheses for the Wilcoxon Test
The null hypothesis for the Wilcoxon test simply states that there is no consistent, systematic
difference between the two treatments.
If the null hypothesis is true, any differences that exist in the sample data must be due
to chance. Therefore, we would expect positive and negative differences to be intermixed
evenly. On the other hand, a consistent difference between the two treatments should cause
the scores in one treatment to be consistently larger than the scores in the other. This should
produce difference scores that tend to be consistently positive or consistently negative. The
Wilcoxon test uses the signs and the ranks of the difference scores to decide whether there
is a significant difference between the two treatments.
H 0
: There is no difference between the two treatments. Therefore, in the general
population there is no tendency for the difference scores to be either systematically
positive or systematically negative.
H 1
: There is a difference between the two treatments. Therefore, in the general population
the difference scores are systematically positive or systematically negative.
■ Calculation and Interpretation of the Wilcoxon T
After ranking the absolute values of the difference scores, the ranks are separated into two
groups: those associated with positive differences (increases) and those associated with
negative differences (decreases). Next, the sum of the ranks is computed for each group.
The smaller of the two sums is the test statistic for the Wilcoxon test and is identified by
the letter T. For the difference scores in Table E.1, the positive differences have ranks of 1
and 2, which add to SR 5 3 and the negative difference scores have ranks of 5, 7, 3, 6, 4,
and 8, which add up to SR 5 33. For these scores, T 5 3.
Table B10 in Appendix B lists critical values of T for a 5 .05 and a 5 .01. The null
hypothesis is rejected when the sample data produce a T that is less than or equal to
the table value. With n 5 8 and a 5 .05 for a two-tailed test, the table lists a critical
value of 3. For the data in Table E.1, we obtained T 5 3 so we would reject the null
hypothesis and conclude that there is a significant difference between the two treatments.
As with the Mann-Whitney U-test, there is no specified format for reporting the
results of a Wilcoxon T-test. It is suggested, however, that the report include a summary
of the data and the value obtained for the test statistic as well as the p value. If there are