Lepage test
In statistics, the Lepage test is an exact distribution-free test (nonparametric test) for jointly monitoring the location (central tendency) and scale (variability) in two-sample treatment versus control comparisons. It is a rank test for the two-sample location-scale problem. The Lepage test statistic is the squared Euclidean distance of the standardized Wilcoxon rank-sum test for location and the standardized Ansari–Bradley test for scale. The Lepage test was first introduced by Yves Lepage in 1971 in a paper in Biometrika.{{Cite journal|last=Lepage|first=Yves|date=April 1971|title=A Combination of Wilcoxon's and Ansari-Bradley's Statistics|journal=Biometrika|volume=58|issue=1|pages=213–217|doi=10.2307/2334333|issn=0006-3444|jstor=2334333}} A large number of Lepage-type tests exists in statistical literature for simultaneously testing location and scale shifts in case-control studies. The details may be found in the book: Nonparametric statistical tests: A computational approach.{{Cite book|title=Nonparametric Statistical Tests|last=Neuhäuser|first=Markus|date=2011-12-19|publisher=Chapman and Hall/CRC|isbn=9781439867037|doi = 10.1201/b11427}} Wolfgang Kössler{{Cite book|title=Asymptotic power and efficiency of lepage-type tests for the treatment of combined location-scale alternatives|last=Kössler, W. (Wolfgang)|date=2006|publisher=Humboldt-Universität zu Berlin|doi=10.18452/2462|hdl=18452/3114|oclc=243600853}} in 2006 also introduced various Lepage type tests using some alternative score functions optimal for various distributions. Amitava Mukherjee and Marco Marozzi introduced a class of percentile modified versions of the Lepage test.{{Cite journal|last1=Mukherjee|first1=Amitava|last2=Marozzi|first2=Marco|date=2019-08-01|title=A class of percentile modified Lepage-type tests|journal=Metrika|language=en|volume=82|issue=6|pages=657–689|doi=10.1007/s00184-018-0700-1|issn=1435-926X}} An alternative to the Lepage-type tests is known as the Cucconi test proposed by Odoardo Cucconi in 1968.{{cite journal|jstor=23241361|title=Un Nuovo Test non Parametrico per Il Confronto Fra Due Gruppi di Valori Campionari|journal=Giornale Degli Economisti e Annali di Economia|volume=27|issue=3/4|pages=225–248|last1=Cucconi|first1=Odoardo|year=1968}}
Conducting the Lepage test with R
Practitioners can apply the Lepage test using the pLepage function of the contributory package NSM3,{{Citation|last1=Schneider|first1=Grant|title=NSM3: Functions and Datasets to Accompany Hollander, Wolfe, and Chicken – Nonparametric Statistical Methods, Third Edition|date=2018-05-16|url=https://cran.r-project.org/package=NSM3|access-date=2019-09-17|last2=Chicken|first2=Eric|last3=Becvarik|first3=Rachel}} built under R software. Andreas Schulz and Markus Neuhäuser also provided detailed R code for computation of test statistic and p-value of the Lepage test{{Cite web|url=https://www.hs-koblenz.de/fileadmin/media/profiles/mathematik_und_technik/neuhaeuser/Lepage_Test_R.pdf|title=R Programme for Lepage Test|last=Schulz|first=Andreas|date=|website=|archive-url=|archive-date=|access-date=}} for the users.
Application in statistical process monitoring
In recent years, the Lepage statistic is a widely used statistical process for monitoring and quality control. In 2012, Amitava Mukherjee and Subhabrata Chakraborti introduced a distribution-free Shewhart-type Phase-II monitoring scheme{{Cite journal|last1=Mukherjee|first1=A.|last2=Chakraborti|first2=S.|date=2011-09-26|title=A Distribution-free Control Chart for the Joint Monitoring of Location and Scale|journal=Quality and Reliability Engineering International|volume=28|issue=3|pages=335–352|doi=10.1002/qre.1249|issn=0748-8017}} (control chart) for simultaneously monitoring of location and scale parameter of a process using a test sample of fixed size, when a reference sample of sufficiently large size is available from an in-control population. Later in 2015, the same statisticians along with Shovan Chowdhury, proposed a distribution-free CUSUM-type Phase-II monitoring scheme{{Cite journal|last1=Chowdhury|first1=Shovan|last2=Mukherjee|first2=Amitava|last3=Chakraborti|first3=Subhabrata|date=2014-11-07|title=Distribution-free Phase II CUSUM Control Chart for Joint Monitoring of Location and Scale|journal=Quality and Reliability Engineering International|volume=31|issue=1|pages=135–151|doi=10.1002/qre.1677|issn=0748-8017|hdl=2263/50153|url=https://repository.up.ac.za/bitstream/2263/50153/1/Chowdhury_Distribution_2015.pdf|hdl-access=free}} based on the Lepage statistic. In 2017, Mukherjee further designed an EWMA-type distribution-free Phase-II monitoring scheme{{Cite journal|last=Mukherjee|first=Amitava|date=2017-02-18|title=Distribution-free phase-II exponentially weighted moving average schemes for joint monitoring of location and scale based on subgroup samples|journal=The International Journal of Advanced Manufacturing Technology|volume=92|issue=1–4|pages=101–116|doi=10.1007/s00170-016-9977-2|issn=0268-3768}} for joint monitoring of location and scale. In the same year, Mukherjee, with Marco Marozzi, known for promoting the Cucconi test, came together to design the Circular-Grid Lepage chart – a new type of joint monitoring scheme.{{Cite journal|last1=Mukherjee|first1=Amitava|last2=Marozzi|first2=Marco|date=2016-05-17|title=Distribution-free Lepage Type Circular-grid Charts for Joint Monitoring of Location and Scale Parameters of a Process|journal=Quality and Reliability Engineering International|volume=33|issue=2|pages=241–274|doi=10.1002/qre.2002|issn=0748-8017}}
Multisample version of the Lepage test
In 2005, František Rublìk introduced the multisample version of the original two-sample Lepage test.{{Cite journal|last=Rublík|first=František|date=2005|title=The multisample version of the Lepage test|hdl=10338.dmlcz/135688|journal=Kybernetika|volume=41|issue=6|pages=[713]–733|issn=0023-5954}}