In statistics, the Lepage test is an exact distribution-free test (
nonparametric test
Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric sta ...
) for jointly monitoring the location (
central tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...
) and scale (
variability) in two-sample treatment versus control comparisons. It is a
rank test {{Short description, Type of statistical test
In statistics, a rank test is any test involving ranks. Rank tests are related to permutation tests.
Motivation
The motivation to test differences between samples is that ranks are in some sense maxima ...
for the
two-sample location-scale problem. The Lepage test statistic is the squared
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
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''. 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''. Wolfgang Kössler 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.
An alternative to the Lepage-type tests is known as the
Cucconi test proposed by Odoardo Cucconi in 1968.
Conducting the Lepage test with R
Practitioners can apply the Lepage test using the pLepage function of the contributory package NSM3, 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 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
Walter Andrew Shewhart (pronounced like "shoe-heart";
March 18, 1891 – March 11, 1967) was an American physicist, engineer and statistician. He is sometimes also known as the ''grandfather of statistical quality control'' and also related to th ...
-type Phase-II monitoring scheme
(
control chart
Control charts are graphical plots used in production control to determine whether quality and manufacturing processes are being controlled under stable conditions. (ISO 7870-1)
The hourly status is arranged on the graph, and the occurrence of ...
) 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
In statistical process control, statistical quality control, the CUSUM (or cumulative sum control chart) is a sequential analysis technique developed by E. S. Page of the University of Cambridge. It is typically used for monitoring change detecti ...
-type Phase-II monitoring scheme based on the Lepage statistic. In 2017, Mukherjee further designed an EWMA-type distribution-free Phase-II monitoring scheme 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.
Multisample version of the Lepage test
In 2005, František Rublìk introduced the multisample version of the original two-sample Lepage test.
See also
*
Cucconi test
*
permutation test
A permutation test (also called re-randomization test or shuffle test) is an exact statistical hypothesis test.
A permutation test involves two or more samples. The (possibly counterfactual) null hypothesis is that all samples come from the same ...
References
{{Reflist
Statistical tests
Nonparametric statistics