Skip to main page content
Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
, 4, 1-39

Wilcoxon-Mann-Whitney or T-Test? On Assumptions for Hypothesis Tests and Multiple Interpretations of Decision Rules

Affiliations

Wilcoxon-Mann-Whitney or T-Test? On Assumptions for Hypothesis Tests and Multiple Interpretations of Decision Rules

Michael P Fay et al. Stat Surv.

Abstract

In a mathematical approach to hypothesis tests, we start with a clearly defined set of hypotheses and choose the test with the best properties for those hypotheses. In practice, we often start with less precise hypotheses. For example, often a researcher wants to know which of two groups generally has the larger responses, and either a t-test or a Wilcoxon-Mann-Whitney (WMW) test could be acceptable. Although both t-tests and WMW tests are usually associated with quite different hypotheses, the decision rule and p-value from either test could be associated with many different sets of assumptions, which we call perspectives. It is useful to have many of the different perspectives to which a decision rule may be applied collected in one place, since each perspective allows a different interpretation of the associated p-value. Here we collect many such perspectives for the two-sample t-test, the WMW test and other related tests. We discuss validity and consistency under each perspective and discuss recommendations between the tests in light of these many different perspectives. Finally, we briefly discuss a decision rule for testing genetic neutrality where knowledge of the many perspectives is vital to the proper interpretation of the decision rule.

Figures

Fig 1
Fig 1
Relationship between assumptions. Ai ← Aj denotes that Ai ⊏ Aj (i.e., Ai are more restrictive assumptions than Aj).
Fig 2
Fig 2
The probability density functions for some log transformed gamma distributions. All distributions are scaled and shifted to have mean 0 and variance 1. The value a is the shape parameter, and ARE is asymptotic relative efficiency. An ARE of 2 denotes that it will take twice as many observations to obtain the same asymptotic power for the t-test compared to the WMW-test.
Fig 3
Fig 3
Relative efficiency of WMW test to t-test for testing for a location shift in log-gamma distribution. The value a is the shape parameter. The solid black line is the ARE. The dotted grey horizontal line is at 1, and is where both tests are equally asymptotically efficient, which occurs at the dotted grey vertical line at a = 5.55. The solid grey horizontal lines are at 3 and 3/π = .955, which are the limits as a → 0 and a → ∞. Points are simulated relative efficiency for shifts which give about 80% power for the WMW DR when there are about 20 in each group.
Fig 4
Fig 4
Standard normal distribution (black dashed) and scaled t-distribution with 18 degrees of freedom (grey solid), and the distribution with the minimum ARE (black dotted), where all distributions have mean 0 and variance 1. The plots are the same except the right plot (b), has the f(x) plotted on the log scale to be able to see the difference in the extremities of the tails.
Fig 5
Fig 5
Relative efficiency of WMW test to t-test for testing for location shift in t-distributions. The dotted grey horizontal line is at 1, and is where both tests are equally asymptotically efficient, which occurs at the dotted grey vertical line at 18.76. The solid grey lines denote limits, the vertical line shows ARE goes to infinity at df = 2, the horizontal line shows ARE goes to 3/π = .955 as df → ∞. Points are simulated relative efficiency for shifts which give about 80% power for the WMW DR when there are about 20 in each group.

Similar articles

See all similar articles

Cited by 72 PubMed Central articles

See all "Cited by" articles

LinkOut - more resources

Feedback