Extensions of empirical likelihood and chi-squared-based tests for ordered alternatives

Abstract

Several methods for comparing k populations have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. We focus on two important alternative hypotheses: monotone and umbrella ordering. Two new families of test statistics are proposed, including two known tests, as well as two new powerful tests under monotone ordering. Furthermore, these families are adapted for testing umbrella ordering. We compare some members of the families with respect to power and Type I errors under different simulation scenarios. Finally, the methods are illustrated in several applications to real data.

Keywords:

• Empirical likelihood test statistic
• chi-squared test statistic
• umbrella ordering
• stochastic ordering
• distribution-free

Acknowledgments

The authors gratefully acknowledge the constructive comments of two anonymous referees, which led to great improvements in the manuscript. The authors thank professor Christos Nakas for sending the data on ADC for HIV and non-HIV groups.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors gratefully acknowledge the constructive comments of two anonymous referees, which led to great improvements in the manuscript. This work was partially supported by grants MTM2016-75351-R and MTM2017-89422-P of the Spanish Ministry of Economy and Competitiveness, Beca del Amo grant, the Xunta de Galicia (Centro singular de investigación de Galicia accreditation 2016-2019), the European Union (European Regional Development Fund – ERDF) and US NIH 5UL1 TR001085 and 4P30CA124435.