Rank-based multiple test procedures and simultaneous confidence intervals

We study simultaneous rank procedures for unbalanced designs with independent observations. The hypotheses are formulated in terms of purely nonparametric treatment effects. In this context, we derive rankbased multiple contrast test procedures and simultaneous confidence intervals which take the correlation between the test statistics into account. Hereby, the individual test decisions and the simultaneous confidence intervals are compatible. This means, whenever an individual hypothesis has been rejected by the multiple contrast test, the corresponding simultaneous confidence interval does not include the null, i.e. the hypothetical value of no treatment effect. The procedures allow for testing arbitrary purely nonparametric multiple linear hypotheses (e.g. many-to-one, all-pairs, changepoint, or even average comparisons). We do not assume homogeneous variances of the data; in particular, the distributions can have different shapes even under the null hypothesis. Thus, a solution to the multiple nonparametric Behrens-Fisher problem is presented in this unified framework.

[1]  G. Box.  Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification , 1954 .

[2]  Robert G. D. Steel,et al.  A multiple comparison rank sum test: Treatments versus control. , 1959 .

[3]  Robert G. D. Steel,et al.  A Rank Sum Test for Comparing All Pairs of Treatments , 1960 .

[4]  O. J. Dunn.  Multiple Comparisons Using Rank Sums , 1964 .

[5]  K. Gabriel,et al.  SIMULTANEOUS TEST PROCEDURES-SOME THEORY OF MULTIPLE COMPARISONS' , 1969 .

[6]  Statistique non Paramétrique Asymptotique , 1980 .

[7]  Frits H. Ruymgaart,et al.  A unified approach to the asymptotic distribution theory of certain midrank statistics , 1980 .

[8]  Douglas A. Wolfe,et al.  A Class of Distribution-Free Two-Sample Tests Based on Placements , 1982 .

[9]  S. Rust,et al.  A modification of the kruskal-wallis statistic for the generalized behrens-fisher problem , 1984 .

[10]  F. T. Wright,et al.  Comparison of Several Treatments with a Control Using Multiple Contrasts , 1987 .

[11]  A. Tamhane,et al.  Multiple Comparison Procedures , 1989 .

[12]  Holger Dette,et al.  Box-Type Approximations in Nonparametric Factorial Designs , 1997 .

[13]  Edgar Brunner,et al.  Nonparametric Hypotheses and Rank Statistics for Unbalanced Factorial Designs , 1997 .

[14]  R. Kay.  Statistical Principles for Clinical Trials , 1998, The Journal of international medical research.

[15]  U. Munzel,et al.  Linear rank score statistics when ties are present , 1999 .

[16]  E. Brunner,et al.  The Nonparametric Behrens‐Fisher Problem: Asymptotic Theory and a Small‐Sample Approximation , 2000 .

[17]  L. Hothorn,et al.  A Unified Approach to Simultaneous Rank Test Procedures in the Unbalanced One-way Layout , 2001 .

[18]  A. Genz,et al.  On the Numerical Availability of Multiple Comparison Procedures , 2001 .

[19]  Edgar Brunner,et al.  Nonparametric methods in factorial designs , 2001 .

[20]  M. Puri,et al.  The multivariate nonparametric Behrens–Fisher problem , 2002 .

[21]  B. M. Brown,et al.  Kruskal–Wallis, Multiple Comparisons and Efron Dice , 2002 .

[22]  Subir Ghosh,et al.  Nonparametric Analysis of Longitudinal Data in Factorial Experiments , 2003, Technometrics.

[23]  Mayer Alvo,et al.  A Unified Nonparametric Approach for Unbalanced Factorial Designs , 2005 .

[24]  H. Keselman,et al.  Multiple Comparison Procedures , 2005 .

[25]  Thomas Jaki,et al.  Simultaneous confidence intervals by iteratively adjusted alpha for relative effects in the one-way layout , 2006, Stat. Comput..

[26]  E. Brunner,et al.  Wilcoxon–Mann–Whitney test for stratified samples and Efron's paradox dice , 2007 .

[27]  M. Kenward,et al.  An Introduction to the Bootstrap , 2007 .

[28]  Mayer Alvo,et al.  Nonparametric multiple comparison procedures for unbalanced two-way layouts , 2008 .

[29]  Alan Agresti,et al.  Modeling and inference for an ordinal effect size measure , 2008, Statistics in medicine.

[30]  Ludwig A Hothorn,et al.  Multiple Contrast Tests in the Presence of Heteroscedasticity , 2008, Biometrical journal. Biometrische Zeitschrift.

[31]  T. Hothorn,et al.  Simultaneous Inference in General Parametric Models , 2008, Biometrical journal. Biometrische Zeitschrift.

[32]  Gang Li,et al.  Nonparametric multiple comparison procedures for unbalanced one-way factorial designs , 2008 .

[33]  Euijung Ryu,et al.  Simultaneous confidence intervals using ordinal effect measures for ordered categorical outcomes , 2009, Statistics in medicine.

[34]  A. Genz,et al.  Computation of Multivariate Normal and t Probabilities , 2009 .

[35]  T. Hothorn,et al.  Multiple Comparisons Using R , 2010 .

[36]  T. Hothorn,et al.  A Robust Procedure for Comparing Multiple Means under Heteroscedasticity in Unbalanced Designs , 2010, PloS one.

[37]  Alan C. Elliott,et al.  A SAS® macro implementation of a multiple comparison post hoc test for a Kruskal-Wallis analysis , 2011, Comput. Methods Programs Biomed..

[38]  Ludwig A. Hothorn,et al.  Nonparametric Evaluation of Quantitative Traits in Population-Based Association Studies when the Genetic Model is Unknown , 2012, PloS one.

[39]  Ludwig A. Hothorn,et al.  Evaluation of Toxicological Studies Using a Nonparametric Shirley-Type Trend Test for Comparing Several Dose Levels with a Control Group , 2012 .