In statistics, the Lepage test is an exactly distribution-free test (
nonparametric test) 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 ...
) and scale (
variability) in two-sample treatment versus control comparisons. This is one of the most famous
rank tests 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 a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
of standardized
Wilcoxon rank-sum Wilcoxon is a surname, and may refer to:
* Charles Wilcoxon, drum educator
* Henry Wilcoxon, an actor
* Frank Wilcoxon, chemist and statistician, inventor of two non-parametric tests for statistical significance:
** The Wilcoxon signed-rank test
...
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''. Kössler, W. in 2006 also introduced various Lepage type tests using some alternative score functions optimal for various distributions. Dr. Amitava Mukherjee and Dr. Marco Marozzi introduced a class of percentile modified version 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, an open-source software
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, Lepage statistic is widely used statistical process monitoring and quality control. In a classical development, in 2012, Amitava Mukherjee, an Indian statistician, and Subhabrata Chakraborti, an American statistician of Indian origin, 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, sometimes known as the ''father of statistical quality control'' and also related to the Shewhart cycl ...
-type Phase-II monitoring scheme
(
control chart
Control charts is a graph 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 abnormalit ...
) 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 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 detection.
CUSUM was announced in Bi ...
-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, an Italian statistician known for promoting the Cucconi test, came together to design 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.
This work recently emerge as the motivation behind the proposal of the Phase-I distribution-free Shewhart-type control chart for joint monitoring of location and scale.
See also
*
Cucconi test
References
{{Reflist
Statistical tests
Nonparametric statistics