I want to understand why the results of a t-test and Wilcoxon test are not similar? Are there any specific conditions where the tests should be applied? What care should be taken before applying these tests?
Nice summary paper, Pablo. I'd say, however, that it isn't the difference in the samples that is normally distributed although we usually take this as our rationale...it's difficult to assess normality in small samples and the tests are usually sensitive in large samples. Your decision, in theory, should be based on what you know about the underlying populations. In truth, we do usually look at the sample distributions. If those are relatively symmetric, the t-test usually performs okay and the mean is a reasonable parameter for reporting the central tendancy. If not symmetric but unimodal, the median might be "closer" to the center. If bimodal or multimodal, it's hard to report a measure of central tendancy but the Wilcoxon test still is useful for comparing the distributions.
Nice summary paper, Pablo. I'd say, however, that it isn't the difference in the samples that is normally distributed although we usually take this as our rationale...it's difficult to assess normality in small samples and the tests are usually sensitive in large samples. Your decision, in theory, should be based on what you know about the underlying populations. In truth, we do usually look at the sample distributions. If those are relatively symmetric, the t-test usually performs okay and the mean is a reasonable parameter for reporting the central tendancy. If not symmetric but unimodal, the median might be "closer" to the center. If bimodal or multimodal, it's hard to report a measure of central tendancy but the Wilcoxon test still is useful for comparing the distributions.
Dear May,
Both test will give the same conclusion, either rejection or acceptance of the hypothesis. But, these two tests depends on the underlying distribution of the population.
If you are using the Wilcoxon-Mann-Whitney test as a test of location, you may wish to take a look at this interesting simulation study by Fagerland & Sandvik (2009).
http://www.researchgate.net/publication/24045186_The_Wilcoxon-Mann-Whitney_test_under_scrutiny/file/9fcfd502aa62c9ed7b.pdf
Cheers!
Article The Wilcoxon-Mann-Whitney test under scrutiny
The Wilcoxon-Mann-Whitney test tests equality of medians ONLY when the population shapes are identical. This is what Fagerland & Sandvik refer to as the "pure shift" model (see the link I posted earlier in this thread). Under that condition, the WMW test also tests equality of any percentile point you might choose AND the means. See also:
http://www.graphpad.com/support/faqid/1327/
http://www.bmj.com/content/323/7309/391
I agree with some colleagues above that the difference between the results of the t-test and Wilcoxon test is that the t-test tests the significant differences between means and the Wilcoxon tests significant differences between Medians. If the data is close to normally distributed then no difference between the mean and median and hence the results of the two tests. Moreover, if the data involve one or more outliers or the data is highly skewed then the t-test and Wilcoxon test are not expected to give similar results.
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