On Mixture Alternatives and Wilcoxon’s Signed-Rank Test

ABSTRACT

The shift alternative model has been the canonical alternative hypothesis since the early days of statistics. This holds true both in parametric and nonparametric statistical testing. In this contribution, we argue that in several applications of interest, the shift alternative is dubious while a mixture alternative is more plausible, because the treatment is expected to affect only a subpopulation. When considering mixture hypotheses, classical tests may no longer enjoy their desirable properties. In particular, we show that the t-test may be underpowered compared to Wilcoxon’s signed-rank test, even under a Gaussian null. We consider implications to personalized medicine and medical imaging.

Keywords:

• Efficiency
• Hypothesis testing
• Mixture model

Acknowledgments

J. D. R. and Y. B. are very grateful to an anonymous associate editor for a meticulous review of the manuscript which revealed an error in an earlier version.

Additional information

Funding

The research by J. D. R. and Y. B. leading to these results has received funding from the European Research Council under the European Communitys Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. PSARPS- 294519. J.D.R. further acknowledges the support by ISF grant 900/16.