A class of percentile modified Lepage-type tests

AbstractThe two-sample problem usually tests for a difference in location. However, there are many situations, for example in biomedicine, where jointly testing for difference in location and variability may be more appropriate. Moreover, heavy-tailed data, outliers and small-sample sizes are common in biomedicine and in other fields. These considerations make the use of nonparametric methods more appealing than parametric ones. The aim of the paper is to contribute to the literature about nonparametric simultaneous location and scale testing. More precisely, several existing tests are generalized and unified, and a new class of tests based on the Mahalanobis distance between the percentile modified test statistics for location and scale differences is introduced. The asymptotic distributions of the test statistics are obtained, and small-sample size behaviour of the tests is studied and compared to other tests via Monte Carlo simulations. It is shown that the proposed class of tests performs well when there are differences in both location and variability. A practical application is presented.

[1]  Marco Marozzi,et al.  Nonparametric Simultaneous Tests for Location and Scale Testing: A Comparison of Several Methods , 2013, Commun. Stat. Simul. Comput..

[2]  Lixing Zhu,et al.  Testing equality of shape parameters in several inverse Gaussian populations , 2014 .

[3]  Evangelos Kontopantelis,et al.  Performance of statistical methods for meta-analysis when true study effects are non-normally distributed: A simulation study , 2012, Statistical methods in medical research.

[4]  Luigi Salmaso,et al.  Permutation Tests for Complex Data , 2010 .

[5]  H. Byun,et al.  The significant increase of summer rainfall occurring in Korea from 1998 , 2010 .

[6]  Robert V. Hogg,et al.  Certain Uncorrelated and Independent Rank Statistics , 1971 .

[7]  A. Janssen,et al.  Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem , 1997 .

[8]  Markus Neuhäuser,et al.  The Baumgartner-Wei?-Schindler test for the detection of differentially expressed genes in replicated microarray experiments , 2004, Bioinform..

[9]  Y. Lepage.  A combination of Wilcoxon's and Ansari-Bradley's statistics , 1971 .

[10]  H. Büning,et al.  An adaptive two-sample location-scale test of lepage type for symmetric distributions , 2000 .

[11]  Edgar Brunner,et al.  Permuting longitudinal data in spite of the dependencies , 2017, J. Multivar. Anal..

[12]  John Ludbrook,et al.  Why Permutation Tests are Superior to t and F Tests in Biomedical Research , 1998 .

[13]  Amitava Mukherjee,et al.  Distribution-free phase-II exponentially weighted moving average schemes for joint monitoring of location and scale based on subgroup samples , 2017 .

[14]  Richard A. Groeneveld,et al.  Measuring Skewness and Kurtosis , 1984 .

[15]  Heqing Shen,et al.  Metabolomic analysis reveals a unique urinary pattern in normozoospermic infertile men. , 2014, Journal of proteome research.

[16]  T. O. Lewis,et al.  A class of location-scale nonparametric tests , 1976 .

[17]  Frank Konietschke,et al.  Permutation‐based inference for the AUC: A unified approach for continuous and discontinuous data , 2016, Biometrical journal. Biometrische Zeitschrift.

[18]  Amitava Mukherjee,et al.  A New Distribution‐free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions , 2014, Qual. Reliab. Eng. Int..

[19]  A. K. McCracken,et al.  Control Charts for Joint Monitoring of Mean and Variance: An Overview , 2013 .

[20]  Parametric versus nonparametrics: two alternative methodologies , 2009 .

[21]  Jaroslav Hájek,et al.  Theory of rank tests , 1969 .

[22]  M J Podgor,et al.  On non-parametric and generalized tests for the two-sample problem with location and scale change alternatives. , 1994, Statistics in medicine.

[23]  Moti Lal Tiku,et al.  Robust Inference , 1986 .

[24]  Marco Marozzi,et al.  The multisample Cucconi test , 2014, Stat. Methods Appl..

[25]  Markus Neuhäuser,et al.  An exact two-sample test based on the baumgartner-weiss-schindler statistic and a modification of lepage's test , 2000 .

[26]  Markus Pauly,et al.  Rank‐based permutation approaches for non‐parametric factorial designs , 2017, The British journal of mathematical and statistical psychology.

[27]  J. Sheil,et al.  The Distribution of Non‐Negative Quadratic Forms in Normal Variables , 1977 .

[28]  Smiley W. Cheng,et al.  Single Variables Control Charts: an Overview , 2006, Qual. Reliab. Eng. Int..

[29]  Marco Marozzi,et al.  Multivariate multidistance tests for high‐dimensional low sample size case‐control studies , 2015, Statistics in medicine.

[30]  Subhabrata Chakraborti,et al.  Nonparametric Statistical Inference , 2011, International Encyclopedia of Statistical Science.

[31]  Joseph L. Gastwirth,et al.  Percentile Modifications of Two Sample Rank Tests , 1965 .

[32]  R. Nussenblatt,et al.  The Diagnosis of Intraocular Inflammation and Cytomegalovirus Retinitis in HIV-Infected Patients by Laser Flare Photometry Laser Flare Photometry in HIV-Infected Patients. , 1996, Ocular immunology and inflammation.

[33]  Michael A. Kuskowski,et al.  Withdrawal of chronic systemic corticosteroids in patients with COPD: a randomized trial. , 2000, American journal of respiratory and critical care medicine.

[34]  A. R. Ansari,et al.  Rank-Sum Tests for Dispersions , 1960 .

[35]  Amitava Mukherjee,et al.  Distribution‐free Phase II CUSUM Control Chart for Joint Monitoring of Location and Scale , 2015, Qual. Reliab. Eng. Int..

[36]  Hidetoshi Murakami,et al.  Lepage type statistic based on the modified Baumgartner statistic , 2007, Comput. Stat. Data Anal..

[37]  Joseph P. Romano,et al.  Subsampling inference for the mean in the heavy-tailed case , 1999 .

[38]  A. Janssen,et al.  Testing nonparametric statistical functionals with applications to rank tests , 1999 .

[39]  P. Sen,et al.  Nonparametric methods in multivariate analysis , 1974 .

[40]  Herman Chernoff,et al.  ASYMPTOTIC NORMALITY AND EFFICIENCY OF CERTAIN NONPARAMETRIC TEST STATISTICS , 1958 .

[41]  Markus Pauly.  Discussion about the quality of F-ratio resampling tests for comparing variances , 2011 .

[42]  Herbert Büning.  Robuste und adaptive Tests , 1991 .

[43]  E. L. Lehmann,et al.  Parametric versus nonparametrics: two alternative methodologies , 2009 .

[44]  Marco Marozzi.  A bi-aspect nonparametric test for the two-sample location problem , 2004, Comput. Stat. Data Anal..

[45]  Wolfgang Kössler,et al.  An Adaptive Test for the Two-Sample Scale Problem Based on U-Statistics , 2008, Commun. Stat. Simul. Comput..

[46]  Wolfgang Kössler,et al.  Theory & Methods: The Asymptotic Power Of Jonckheere‐Type Tests For Ordered Alternatives , 1999 .

[47]  Dennis D. Boos,et al.  Modifying the t and ANOVA F Tests When Treatment Is Expected to Increase Variability Relative to Controls , 1990 .

[48]  G. Box.  NON-NORMALITY AND TESTS ON VARIANCES , 1953 .