Multivariate Permutation Tests: With Applications in Biostatistics

Transcription

1 Multivariate Permutation Tests: With Applications in Biostatistics Fortunato Pesarin University ofpadova, Italy JOHN WILEY & SONS, LTD Chichester New York Weinheim Brisbane Singapore Toronto

2 Contents Preface Notation and Abbreviations xv xxiii 1 Introduction On Permutation Analysis Conditionality and Exchangeability The Permutation Testing Principle Permutation Approaches Randomization and Permutation When Conditioning Is Appropriate Computational Aspects Basic Notation 15 2 Discussion of a Simple Testing Problem A Problem with Paired Observations Introduction A Motivating Example Modelling Responses Symmetry Induced by Exchangeability Further General Aspects The Student t Paired Solution Paired Normal Data An Extension The Signed Rank Test Solution Paired Continuous Data Generalized Scores The McNemar Solution Test on Signs of Differences An Extension The Permutation Solution General Aspects 26

3 viii CONTENTS The Permutation Sample Space The Conditional Monte Carlo Method A Simulation Algorithm for Inspecting Permutation Sample Spaces A Routine for Random Permutations Approximating the Permutation Distribution Permutation Confidence Interval for Introductory Aspects An Approximate Solution A Conditional Monte Carlo Algorithm Problems and Exercises 34 3 Theory of Permutation Tests for One-Sample Problems Introduction Basic Concepts On Sufficient Statistics for Symmetric Data Equivalence of Permutation Statistics Formal Definition of Permutation Tests Randomized Permutation Tests Non-randomized Permutation Tests Arguments Related to Permutation Tests General Aspects Arguments for Selecting a Test Statistic T Examples of One-Sample Problems Other Properties of Permutation Tests Stochastic Ordering of p-values Conditional and Unconditional Properties Further Unconditional Properties Some Consequences of the Theorems Comments on Power Functions An Algorithm for Evaluating the Conditional Power Permutation Testing for Composite Hypotheses Optimal Properties Introductory Remarks Further Remarks Problems and Exercises 69 4 Examples of Univariate Multi-Sample Problems Introduction General Aspects On Permutation Distributions Inspection of the Permutation Sample Space Conditional Monte Carlo Method An Example 76

4 CONTENTS ix Rank Solutions A Simple Routine for Random Permutations Permutation Confidence Interval for Problems and Exercises Permutation One-Way ANOVA An Example Modelling Responses Permutation Solutions Problems and Exercises Goodness-of-Fit for Ordered Categorical Variables Introduction Goodness-of-Fit Tests for Ordered Categorical Variables A Solution Based on Score Transformations Typical Goodness-of-Fit Solutions Extension to Non-dominance Alternatives and C Groups Problems and Exercises A Problem with Repeated Observations Introduction Friedman's Rank Test A Permutation Solution An Example Problems and Exercises 99 5 Theory of Permutation Tests for Multi-Sample Problems Introduction Recalling Basic Notions Recalling the Non-randomized Version Examples of Multi-Sample Problems Unbiasedness and Power of Some Multi-Sample Tests Ill Unbiasedness and Power for Two-Sample Location Problems Ill Some Corollaries Unbiasedness of One-Way ANOVA Some Asymptotic Properties Introduction Two Basic Theorems Permutation Central Limit Theorems Basic Notions Permutation Central Limit Theorems Problems and Exercises Nonparametric Combination Methodology Introduction General Aspects 133

5 CONTENTS Bibliographic Notes Main Assumptions and Notation Some Comments The Nonparametric Combination Methodology Assumptions on Partial Tests Desirable Properties of Combining Functions A Two-Phase Algorithm for Nonparametric Combination Some Useful Combining Functions Four Examples of Nonparametric Combination Problems and Exercises Consistency and Unbiasedness of Combined Tests Consistency Unbiasedness A Not Consistent Combining Function Problems and Exercises Some Further Asymptotic Properties General Conditions Further Asymptotic Properties Comments on Nonparametric Combination General Comments Final Remarks 179 Examples of Nonparametric Combination Introduction Permutation Testing with Multivariate Paired Observations Formal Description of Testing Problem An Example A Multivariate Extension of McNemar's Test Asymptotic Behaviour of Multivariate Tests with Paired Observations Problems and Exercises Permutation MANOVA with Mixed Data A Formal Description An Example with Four Categorical Variables A Two-Sample Multivariate Test A Multivariate Extension of Fisher's Exact Probability Test A Cross-over Design Problems and Exercises Goodness-of-Fit Tests for Ordered Categorical Variables Revisiting the Problem Some Extensions Two Examples Some Power Evaluations 211

6 CONTENTS xi 7.5 A Problem of Isotonic Inference Introduction Conditions for Nonparametric Combination An Application from Genetics An Example A Multivariate Extension A Problem with Multivariate Homoscedastic Repeated Responses A Formal Description An Algorithm for Conditional Simulation An Example Problems and Exercises Power Behaviour and Remarks on Restricted Alternatives Power Behaviour of Combined Tests Remarks on Restricted Alternatives A Few Simulation Results Permutation Analysis in Factorial Designs Introduction General Aspects Solutions Based on Residuals Exact Separate Tests for Replicated 2 2 Factorial Designs Separate Sets of Hypotheses Synchronized Permutations Exact Tests for 2 2 Unbalanced Designs Weighting Intermediate Statistics for Factor A Weighting Intermediate Statistics for Factors B and AB An Extension of Welch's Test for Heteroscedastic Models Exact Tests in / x J Balanced Designs Balanced Two-Way Layout Separate Testing Some Comments Multivariate Extension of / x J Designs Extension to a Kind of Unbalanced Heteroscedastic Model Synchronized Tests in Replicated 2 k Factorials Extension to 2 k Factorial Designs Realignments Construction of Test Statistics General Characterization of Synchronized Tests Characterizing Synchronized Permutations Synchronized Permutation Tests for Fractional Designs of Different Resolution A Comparative Simulation Study Comparing Solutions in Ho Power Results 261

7 xii CONTENTS 8.8 An Application Permutation Tests in Unreplicated Factorials Introduction Paired Permutation Tests Paired Permutation Testing Algorithm A Simulation Study Problems and Exercises Appendices Sufficient Statistics for Replicated 2 2 Factorial Designs A Brief Review of Two-Level Factorial Designs Permutation Testing with Missing Data Introduction General Aspects Bibliographic Notes On Missing Data Processes Introduction Data Missing Completely at Random Data Missing Not at Random The Permutation Approach Introduction Breaking Down the Hypotheses The Structure of Testing Problems Hypotheses for Non-MAR Models Hypotheses for MCAR Models Permutation Structure with Missing Values Permutation Analysis of Missing Values Partitioning the Permutation Sample Space Solution for Two-Sample MCAR Problems Extensions to Multivariate C-sample Problems Extension to Non-MAR Models Some Comments An Example of a Non-MAR Model Introduction The Example The Permutation Solution An Example with Multivariate Paired Observations The Problem An Example with Fictitious Data Power Behaviour of Some Tests with Missing Values Introduction Simulation Results and Comments Problems and Exercises 303

8 CONTENTS xiii 10 The Behrens Fisher Permutation Problem Introduction General Aspects Bibliographic Notes Formal Description of the Problem Parametric Solutions Basic Statistics Known Covariance Matrices Unknown Proportional Covariance Matrices Unknown Covariance Matrices A First Approximate Permutation Solution Approximate Solution Based on Aspin and Welch Statistics An Example Extension of Approximate Solution to Multivariate Situations An Almost Exact Permutation Solution A Univariate Solution by Testing for Symmetry Combining Two Tests of Symmetry Asymptotic Behaviour Extension to C > 2 Groups Multivariate Permutation Solutions Multivariate Solutions for Symmetric Distributions Three Multivariate Testing Problems Distributional Behaviour A Simulation Study in H o A Simulation Study in Hi Permutation Testing for Location and Scale Coefficients An Approximate Solution for Scale Coefficients Joint Permutation Testing for Location and Scale Coefficients Problems and Exercises Permutation Testing for Repeated Measurements Introduction Modelling Repeated Measurements A General Additive Model The Hypotheses of Interest Testing Solutions Solutions by the Nonparametric Combination Approach Analysis of Two-Sample Dominance Problems An Example from the Literature Some Power Evaluations Testing for Repeated Measurements with Missing Data 338

9 xiv CONTENTS Introduction A Formal Description of the Problem An Example A Comparative Simulation Study Tests for Balanced and Unbalanced Repeated Measures Designs Repeated Measures on 2 2 Unbalanced Factorials On I x J Balanced Designs Problems and Exercises Further Applications Introduction Morphological Differences in Biting Flies Description of the Problem Results of Analysis A Clinical Trial on a Respiratory Drug Description of the Problem Results of Analysis A Medical Experiment on Diabetic Patients Description of the Problem Results of Analysis A Case Study of Recovery Wards Description of the Problem Results of Analysis Analysis of Experimental Tumour Growth Curves Description of the Problem Results of Analysis An Epidemiological Survey Description of the Problem Results of Analysis Appendix: NPC Test 2.0 Software for Multivariate and Multistrata Permutation Analysis 365 References 371 Index 399