Multivariate Permutation Tests With Applications in Biostatistics

by Pesarin, Fortunato

• ISBN13: 9780471496700
• ISBN10: 0471496707
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-06-08
• Publisher: WILEY

Summary

Complex multivariate problems are frequently encountered in many scientific disciplines and it can be very difficult to obtain meaningful results. Permutation and nonparametric combination methods provide flexible solutions to complex problems by reducing the problem down to a set of simpler sub-problems. The author presents a novel but well tested approach using real examples taken from biomedical research. Statistical analyses are performed in a nonparametric setting, so that no assumptions need be made about the underlying distribution and the dependence relations between variables. ? Provides a clear exposition of the use of multivariate permutation testing, with emphasis on the use of nonparametric combination methodology. ? Growing area of research with many practical applications, notably in biostatistics. ? Numerous case studies and examples help to illustrate the theory. ? Provides solutions to multi-aspect problems, to problems with missing data, analysis of factorial designs and repeated measures. ? Explains the analysis of categorical, ordered categorical, binary, continuous, and mixed variables in both an experimental and an observational context. ? NPC-Test? software (demo copy), SAS macros, S-Plus code and datasets are available on the Web at http://www.stat.unipd.it/~pesarin/ For researchers and practitioners in a number of scientific disciplines, particularly biostatistics, the vast collection of techniques, examples and case studies will be an invaluable resource. Graduate students of applied statistics and nonparametric methods will find the book provides an accessible introduction to multivariate permutation testing.

Author Biography

Fortunato Pesarin is with the Department of Statistics, University of Padova, Italy. Professor Pesarin has been actively involved in the areas of multidimensional testing and permutation for almost 40 years, and is the author of dozens of publications in numerous international journals.

Table of Contents

Preface	xv
Notation and Abbreviations	xxiii
Introduction	1	(16)
On Permutation Analysis	1	(2)
Conditionality and Exchangeability	3	(3)
The Permutation Testing Principle	6	(3)
Permutation Approaches	9	(1)
Randomization and Permutation	10	(1)
When Conditioning Is Appropriate	11	(2)
Computational Aspects	13	(2)
Basic Notation	15	(2)
Discussion of a Simple Testing Problem	17	(20)
A Problem with Paired Observations	17	(5)
Introduction	17	(1)
A Motivating Example	17	(2)
Modelling Responses	19	(1)
Symmetry Induced by Exchangeability	20	(1)
Further General Aspects	21	(1)
The Student t Paired Solution	22	(1)
Paired Normal Data	22	(1)
An Extension	23	(1)
The Signed Rank Test Solution	23	(1)
Paired Continuous Data	23	(1)
Generalized Scores	24	(1)
The McNemar Solution	24	(2)
Test on Signs of Differences	24	(2)
An Extension	26	(1)
The Permutation Solution	26	(2)
General Aspects	26	(1)
The Permutation Sample Space	27	(1)
The Conditional Monte Carlo Method	28	(3)
A Simulation Algorithm for Inspecting Permutation Sample Spaces	28	(1)
A Routine for Random Permutations	29	(1)
Approximating the Permutation Distribution	30	(1)
Permutation Confidence Interval for δ	31	(3)
Introductory Aspects	31	(1)
An Approximate Solution	32	(1)
A Conditional Monte Carlo Algorithm	33	(1)
Problems and Exercises	34	(3)
Theory of Permutation Tests for One-Sample Problems	37	(36)
Introduction	37	(6)
Basic Concepts	37	(4)
On Sufficient Statistics for Symmetric Data	41	(2)
Equivalence of Permutation Statistics	43	(1)
Formal Definition of Permutation Tests	44	(4)
Randomized Permutation Tests	44	(2)
Non-randomized Permutation Tests	46	(2)
Arguments Related to Permutation Tests	48	(4)
General Aspects	48	(1)
Arguments for Selecting a Test Statistic T	49	(3)
Examples of One-Sample Problems	52	(3)
Other Properties of Permutation Tests	55	(12)
Stochastic Ordering of p-values	55	(2)
Conditional and Unconditional Properties	57	(2)
Further Unconditional Properties	59	(1)
Some Consequences of the Theorems	60	(3)
Comments on Power Functions	63	(1)
An Algorithm for Evaluating the Conditional Power	64	(2)
Permutation Testing for Composite Hypotheses	66	(1)
Optimal Properties	67	(2)
Introductory Remarks	67	(1)
Further Remarks	68	(1)
Problems and Exercises	69	(4)
Examples of Univariate Multi-Sample Problems	73	(28)
Introduction	73	(2)
General Aspects	73	(1)
On Permutation Distributions	73	(2)
Inspection of the Permutation Sample Space	75	(9)
Conditional Monte Carlo Method	75	(1)
An Example	76	(5)
Rank Solutions	81	(1)
A Simple routine for Random Permutations	81	(1)
Permutation Confidence Interval for δ	82	(1)
Problems and Exercises	83	(1)
Permutation One-Way ANOVA	84	(5)
An Example	84	(1)
Modelling Responses	85	(1)
Permutation Solutions	86	(2)
Problems and Exercises	88	(1)
Goodness-of-Fit for Ordered Categorical Variables	89	(7)
Introduction	89	(1)
Goodness-of-Fit Tests for Ordered Categorical Variables	90	(1)
A Solution Based on Score Transformations	91	(1)
Typical Goodness-of-Fit Solutions	92	(1)
Extension to Non-dominance Alternatives and C Groups	93	(2)
Problems and Exercises	95	(1)
A Problem with Repeated Observations	96	(5)
Introduction	96	(1)
Friedman's Rank Test	97	(1)
A Permutation Solution	98	(1)
An Example	99	(1)
Problems and Exercises	99	(2)
Theory of Permutation Tests for Multi-Sample Problems	101	(32)
Introduction	101	(1)
Recalling Basic Notions	101	(1)
Recalling the Non-randomized Version	102	(1)
Examples of Multi-Sample Problems	102	(9)
Unbiasedness and Power of Some Multi-Sample Tests	111	(10)
Unbiasedness and Power for Two-Sample Location Problems	111	(5)
Some Corollaries	116	(2)
Unbiasedness of One-Way ANOVA	118	(3)
Some Asymptotic Properties	121	(2)
Introduction	121	(1)
Two Basic Theorems	121	(2)
Permutation Central Limit Theorems	123	(4)
Basic Notions	123	(1)
Permutation Central Limit Theorems	124	(3)
Problems and Exercises	127	(6)
Nonparametric Combination Methodology	133	(48)
Introduction	133	(7)
General Aspects	133	(2)
Bibliographic Notes	135	(1)
Main Assumptions and Notation	136	(3)
Some Comments	139	(1)
The Nonparametric Combination Methodology	140	(25)
Assumptions on Partial Tests	140	(1)
Desirable Properties of Combining Functions	141	(2)
A Two-Phase Algorithm for Nonparametric Combination	143	(4)
Some Useful Combining Functions	147	(8)
Four Examples of Nonparametric Combination	155	(8)
Problems and Exercises	163	(2)
Consistency and Unbiasedness of Combined Tests	165	(7)
Consistency	165	(1)
Unbiasedness	166	(2)
A Not Consistent Combining Function	168	(2)
Problems and Exercises	170	(2)
Some Further Asymptotic Properties	172	(5)
General Conditions	172	(1)
Further Asymptotic Properties	173	(4)
Comments of Nonparametric Combination	177	(4)
General Comments	177	(2)
Final Remarks	179	(2)
Examples of Nonparametric Combination	181	(48)
Introduction	181	(1)
Permutation Testing with Multivariate Paired Observations	182	(11)
Formal Description of Testing Problems	182	(4)
An Example	186	(2)
A Multivariate Extension of McNemar's Test	188	(3)
Asymptotic Behaviour of Multivariate Tests with Paired Observations	191	(1)
Problems and Exercises	192	(1)
Permutation MANOVA with Mixed Data	193	(13)
A Formal Description	193	(3)
An Example with Four Categorical Variables	196	(2)
A Two-Sample Multivariate Test	198	(3)
A Multivariate Extension of Fisher's Exact Probability Test	201	(1)
A Cross-over Design	202	(2)
Problems and Exercises	204	(2)
Goodness-of-Fit Tests for Ordered Categorical Variables	206	(7)
Revisiting the Problem	206	(2)
Some Extensions	208	(2)
Two Examples	210	(1)
Some Power Evaluations	211	(2)
A Problem of Isotonic Inference	213	(6)
Introduction	213	(1)
Conditions for Nonparametric Combination	214	(2)
An Application from Genetics	216	(1)
An Example	217	(1)
A Multivariate Extension	217	(2)
A Problem with Multivariate Homoscedastic Repeated Responses	219	(5)
A Formal Description	219	(3)
An Algorithm for Conditional Simulation	222	(1)
An Example	222	(1)
Problems and Exercises	223	(1)
Power Behaviour and Remarks on Restricted Alternatives	224	(5)
Power Behaviour of combined Tests	224	(2)
Remarks on Restricted Alternatives	226	(2)
A Few Simulation Results	228	(1)
Permutation Analysis in Factorial Designs	229	(50)
Introduction	229	(3)
General Aspects	229	(1)
Solutions Based on Residuals	230	(2)
Exact Separate Tests for Replicated 22 Factorial Designs	232	(5)
Separate Sets of Hypotheses	232	(1)
Synchronized Permutations	233	(4)
Exact Tests for 22 Unbalanced Designs	237	(4)
Weighting Intermediate Statistics for Factor A	237	(2)
Weighting Intermediate Statistics for Factors B and AB	239	(1)
An Extension of Welch's Test for Heteroscedastic Models	239	(2)
Exact Tests in I x J Balanced Designs	241	(5)
Balanced Two-Way Layout	241	(1)
Separate Testing	241	(2)
Some Comments	243	(1)
Multivariate Extension I x J Designs	243	(1)
Extension to a Kind of Unbalanced Heteroscedastic Model	244	(2)
Synchronized Tests in Replicated 2k Factorials	246	(9)
Extension to 2k Factorial Designs	246	(1)
Realignments	247	(3)
Construction of Test Statistics	250	(5)
General Characterization of Synchronized Tests	255	(4)
Characterizing Synchronized Permutations	255	(1)
Synchronized Permutation Tests for Fractional Designs of Different Resolution	256	(3)
A Comparing Simulation Study	259	(3)
Comparing Solutions in H0	259	(2)
Power Results	261	(1)
An Application	262	(2)
Permutation Tests in Unreplicated Factorials	264	(7)
Introduction	264	(3)
Paired Permutation Tests	267	(1)
Paired Permutation Testing Algorithm	268	(1)
Paired Permutation Testing Algorithm	268	(3)
Problems and Exercises	271	(1)
Appendices	272	(7)
Sufficient Statistics for Replicated 22 Factorial Designs	272	(2)
A Brief Review of Two-Level Factorial Designs	274	(5)
Permutation Testing with Missing Data	279	(26)
Introduction	279	(2)
General Aspects	279	(1)
Bibliographic Notes	280	(1)
On Missing Data Processes	281	(1)
Introduction	281	(1)
Data Missing completely at Random	281	(1)
Data Missing Not at Random	282	(1)
The Permutation Approach	282	(3)
Introduction	282	(2)
Breaking Down the Hypotheses	284	(1)
The Structure of Testing Problems	285	(3)
Hypotheses for Non-MAR Models	285	(1)
Hypotheses for MCAR Models	286	(1)
Permutation Structure with Missing Values	287	(1)
Permutation Analysis of Missing Values	288	(7)
Partitioning the Permutation Sample Space	288	(2)
Solution for Two-Sample MCAR Problems	290	(2)
Extensions to Multivariate C-sample Problems	292	(1)
Extension to Non-MAR Models	293	(1)
Some Comments	293	(2)
An Example of a Non-MAR Model	295	(4)
Introduction	295	(1)
The Example	296	(1)
The Permutation Solution	297	(2)
An Example with Multivariate Paired Observations	299	(2)
The Problem	299	(1)
An Example with Fictitious Data	300	(1)
Power Behaviour of Some Tests with Missing Values	301	(2)
Introduction	301	(1)
Simulation Results and Comments	302	(1)
Problems and Exercises	303	(2)
The Behrens-Fisher Permutation Problem	305	(24)
Introduction	305	(3)
General Aspects	305	(1)
Bibliographic Notes	306	(1)
Formal Description of the Problem	307	(1)
Parametric Solutions	308	(2)
Basic Statistics	308	(1)
Known Covariance Matrices	309	(1)
Unknown Proportional Covariance Matrices	309	(1)
Unknown Covariance Matrices	309	(1)
A First Approximate Permutation Solution	310	(3)
Approximate Solution Based on Aspin and Welch Statistics	310	(1)
An Example	311	(1)
Extension of Approximate Solution to Multivariate Situations	312	(1)
An Almost Exact Permutation Solution	313	(4)
A Univariate Solution by Testing for Symmetry	313	(2)
Combining Two Tests of Symmetry	315	(1)
Asymptotic Behaviour	316	(1)
Extension to C > 2 Groups	317	(1)
Multivariate Permutation Solutions	318	(3)
Multivariate Solutions for Symmetric Distributions	318	(1)
Three Multivariate Testing Problems	319	(2)
Distributional Behaviour	321	(3)
A Simulation Study in H0	321	(1)
A Simulation Study in H1	322	(2)
Permutation Testing for Location and Scale Coefficients	324	(2)
An Approximate Solution for Scale Coefficients	324	(1)
Joint Permutation Testing for Location and Scale Coefficients	325	(1)
Problems and Exercises	326	(3)
Permutation Testing for Repeated Measurements	329	(18)
Introduction	329	(1)
Modelling Repeated Measurements	330	(3)
A General Additive Model	330	(2)
The Hypotheses of interest	332	(1)
Testing Solutions	333	(5)
Solutions by the Nonparametric Combination Approach	333	(2)
Analysis of Two-Sample Dominance Problems	335	(1)
An Example from the Literature	335	(2)
Some Power Evaluations	337	(1)
Testing for Repeated Measurements with Missing Data	338	(5)
Introduction	338	(1)
A Formal Description of the Problem	338	(2)
An Example	340	(1)
A Comparative Simulation Study	341	(2)
Tests for Balanced and Unbalanced Repeated Measures Designs	343	(2)
Repeated Measures on 22 Unbalanced Factorials	343	(1)
On I X J Balanced Designs	344	(1)
Problems and Exercises	345	(2)
Further Applications	347	(24)
Introduction	347	(1)
Morphological Differences in Biting Files	348	(2)
Description of the Problem	348	(1)
Results of Analysis	349	(1)
A Clinical Trial on a Respiratory Drug	350	(2)
Description of the Problem	350	(1)
Results of Analysis	351	(1)
A Medical Experiment on Diabetic Patients	352	(3)
Description of the Problem	352	(1)
Results of Analysis	353	(2)
A Case Study of Recovery Wards	355	(3)
Description of the Problem	355	(2)
Results of Analysis	357	(1)
Analysis of Experimental Tumour Growth Curves	358	(3)
Description of the Problem	358	(2)
Results of Analysis	360	(1)
An Epidemiological Survey	361	(4)
Description of the Problem	361	(2)
Results of Analysis	363	(2)
Appendix: NPC Test 2.0 Software for Multivariate and Multivariate Permutation Analysis	365	(6)
References	371	(28)
Index	399