Animal Behavior Laboratory ManualWILCOXON TEST
The Wilcoxon Test, like the Sign Test, requires paired observations. However, it takes into account the magnitude of the differences between paired measurements as well as the directions of the differences. When your measurements permit this refinement, the Wilcoxon Test has greater power than the Sign Test. By power, we mean a statistical test is less likely to make mistakes. In particular, there is less chance of accepting the null hypothesis when it is in fact wrong. In this case, for every pair of measurements, you can calculate the magnitude of the difference between the measurements as well as the appropriate sign. For instance, in the example of 10 fish tested for their schooling preferences, you can subtract the number of minutes each fish spends nearest test species B from the number of minutes it spends nearest test species A. The result is a difference score for each trial. In 7 cases, the difference score would be a positive number (more time near A than near B), while in 2 cases the difference score would be a negative number (more time near B than near A), and in one case the difference score would be zero. Just as in the Sign Test, we discard the tie (difference score = zero) from any further consideration. Say the remaining 9 difference scores are
The next step is to rank these difference scores without reference to their signs. The number with the smallest absolute value gets the rank of 1, the next larger in absolute value gets the rank of 2, and so forth. What happens when two scores have the same absolute value, such as 2 and -2 or 4 and 4 in the above example? In these cases, each number gets the average of the ranks that would have been assigned to them if they had been slightly different. Thus -2 and 2 both get the average of ranks 2 and 3, which equals 2.5. Then the next larger number in absolute value gets the rank of 4, the next unused rank. Your result looks like this:
The final step is to add up the ranks corresponding to the negative and the positive difference scores. The sum of ranks for the negative difference scores is 3.5; that for the positive difference scores is 41.5. The smaller of these two sums is represented by T. In this case, T = 3.5. Now the probability of obtaining a value of T as low as 3.5 for a sample size of N = 9 has been calculated by statisticians, on the assumption that the null hypothesis is true. The null hypothesis in this case asserts, remember, that the two kinds of test fish do not differ in attractiveness. As we did before, we can take 5% or less as a small probability, one that would reasonably permit us to reject the null hypothesis. We thus do not care so much about the exact probability of getting T = 3.5 in a sample of N = 9. Instead, we would prefer to know the critical value of T for a probability of 5% or less. Reference to the accompanying table shows that, for a sample size of N = 9, the critical value of T is 6. In the case of our fish, our value of T is less than 6, so we can, with reasonable justification, reject the null hypothesis. Note how, in this case, the Wilcoxon Test permits us to reject the null hypothesis while the Sign Test does not. The Wilcoxon Test took into account the information about the magnitudes of the differences between pairs of subjects as well as the information about the direction of the differences. In many cases, the two tests will give the same result, however. You should use the most powerful one suitable for your data.
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