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## Permutation, Parametric, and Bootstrap Tests of Hypotheses, Third Edition

 Author: Phillip Good Publisher: Springer Copyright: 2005 ISBN-13: 978-0-387-20279-2 Pages: 376; hardcover Price: \$79.00

### Comment from the Stata technical group

Today’s more powerful computers have yielded an increase in the popularity of data-resampling methods, such as permutation tests and the bootstrap. This text provides a practical guide to resampling methods for testing hypotheses.

A practical background in classical statistical methods is sufficient to pick up on the ideas and methods presented in this text.

The main body of the text includes discussions of hypothesis testing and experimental designs, including clinical trials, multivariate analysis, analysis of contingency tables, regression models, and clustering in time and space.

The author also provides practical advice for the analysis of real data, how to choose a test statistics, how to choose a statistical test, and what to remember when publishing results. There is also a chapter that gives an overview of computational efficiency. The advanced reader will appreciate the appendix, which serves as a mathematically rigorous foundation for the theory of permutation tests.

Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
1 A Wide Range of Applications
1.1 Basic Concepts
1.1.1 Stochastic Phenomena
1.1.2 Distribution Functions
1.1.3 Hypotheses
1.2 Applications
1.3 Testing a Hypothesis
1.3.1 Five Steps to a Test
1.3.2 Analyze the Experiment
1.3.3 Choose a Test Statistic
1.3.4 Compute the Test Statistic
1.3.5 Determine the Frequency Distribution of the Test Statistic
1.3.6 Make a Decision
1.3.7 Variations on a Theme
1.4 A Brief History of Statistics in Decision-Making
1.5 Exercises
2 Optimal Procedures
2.1 Defining Optimal
2.1.1 Trustworthy
2.1.2 Two Types of Error
2.1.3 Losses and Risk
2.1.4 Significance Level and Power
2.1.4.1 Power and the Magnitude of the Effect
2.1.4.2 Power and Sample Size
2.1.4.3 Power and the Alternative
2.1.5 Exact, Unbiased, Conservative
2.1.6 Impartial.
2.1.7 Most Stringent Tests
2.2 Basic Assumptions
2.2.1 Independent Observations
2.2.2 Exchangeable Observations
2.3 Decision Theory
2.3.1 Bayes’ Risk
2.3.2 Mini-Max
2.3.3 Generalized Decisions
2.4 Exercises
3 Testing Hypotheses
3.1 Testing a Simple Hypothesis
3.2 One-Sample Tests for a Location Parameter
3.2.1 A Permutation Test
3.2.2 A Parametric Test
3.2.3 Properties of the Parametric Test
3.2.4 Student’s t
3.2.5 Properties of the Permutation Test
3.2.6 Exact Significance Levels: A Digression
3.3 Confidence Intervals
3.3.1 Confidence Intervals Based on Permutation Tests
3.3.2 Confidence Intervals Based on Parametric Tests
3.3.3 Confidence Intervals Based on the Bootstrap
3.3.4 Parametric Bootstrap
3.3.5 Better Confidence Intervals
3.4 Comparison Among the Test Procedures
3.5 One-Sample Tests for a Scale Parameter
3.5.1 Semiparametric Tests
3.5.2 Parametric Tests: Sufficiency
3.5.3 Unbiased Tests
3.5.4 Comparison Among the Test Procedures
3.6 Comparing the Location Parameters of Two Populations
3.6.1 A UMPU Parametric Test: Student’s t
3.6.2 A UMPU Semiparametric Procedure
3.6.3 An Example
3.6.4 Comparison of the Tests: The Behrens–Fisher Problem
3.7 Comparing the Dispersions of Two Populations
3.7.1 The Parametric Approach
3.7.2 The Permutation Approach.
3.7.3 The Bootstrap Approach
3.8 Bivariate Correlation
3.9 Which Test?
3.10 Exercises
4 Distributions
4.1 Properties of Independent Observations
4.2 Binomial Distribution
4.3 Poisson: Events Rare in Time and Space
4.3.1 Applying the Poisson
4.3.2 A Poisson Distribution of Poisson Distributions
4.3.3 Comparing Two Poissons.
4.4 Time Between Events
4.5 The Uniform Distribution
4.6 The Exponential Family of Distributions
4.6.1 Proofs of the Properties
4.6.2 Normal Distribution
4.7 Which Distribution?
4.8 Exercises
5 Multiple Tests
5.1 Controlling the Overall Error Rate
5.1.1 Standardized Statistics
5.1.2 Paired Sample Tests
5.2 Combination of Independent Tests
5.2.1 Omnibus Statistics
5.2.2 Binomial Random Variables
5.2.3 Bayes’ Factor
5.3 Exercises
6 Experimental Designs
6.1 Invariance
6.1.1 Some Examples
6.2 k-Sample Comparisons: Least-Squares Loss Function
6.2.1 Linear Hypotheses
6.2.2 Large and Small Sample Properties of the F-ratio Test
6.2.3 Discrete Data and Time-to-Event Data
6.3 k-Sample Comparisons: Other Loss Functions
6.3.1 F-ratio
6.3.2 Pitman Correlation
6.3.3 Effect of Ties
6.3.4 Cochran–Armitage Test
6.3.5 Linear Estimation
6.3.6 A Unifying Theory
6.4 Four Ways to Control Variation
6.4.1 Control the Environment
6.4.2 Block the Experiment
6.4.2.1 Using Ranks
6.4.2.2 Matched Pairs
6.4.3 Measure Factors That Cannot Be Controlled
6.4.3.1 Eliminate the Functional Relationship
6.4.3.2 Selecting Variables
6.4.3.3 Restricted Randomization
6.4.4 Randomize
6.5 Latin Square
6.6 Very Large Samples
6.7 Sequential Analysis
6.7.1 A Vaccine Trial
6.7.2 Determining the Boundary Values
6.7.3 Power of a Sequential Analysis
6.7.4 Expected Sample Size
6.7.5 Curtailed Inspection
6.7.6 Restricted Sequential Sampling Schemes
6.8.1 Group Sequential Trials
6.8.2 Determining the Sampling Ratio
6.8.3 Exact Random Allocation Tests
6.9 Exercises
7 Multifactor Designs
7.1 Multifactor Models
7.2 Analysis of Variance
7.3 Permutation Methods: Main Effects
7.3.1 An Example
7.4 Permutation Methods: Interactions
7.5 Synchronized Rearrangements
7.5.1 Exchangeable and Weakly Exchangeable Variables
7.5.2 Two Factors
7.5.3 Three or More Factors
7.5.4 Similarities
7.5.5 Test for Interaction
7.6 Unbalanced Designs
7.6.1 Missing Combinations
7.6.2 The Boot-Perm Test
7.7 Which Test Should You Use?
7.8 Exercises
8 Categorical Data
8.1 Fisher’s Exact Test
8.1.1 Hypergeometric Distribution
8.1.2 One-Tailed and Two-Tailed Tests
8.1.3 The Two-Tailed Test
8.1.4 Determining the p-Value
8.1.5 What is the Alternative?
8.1.6 Increasing the Power
8.1.7 Ongoing Controversy
8.2 Odds Ratio
8.2.1 Stratified 2 × 2’s
8.3 Exact Significance Levels
8.4 Unordered r × c Contingency Tables
8.4.1 Agreement Between Observers
8.4.2 What Should We Randomize?
8.4.3 Underlying Assumptions
8.4.4 Symmetric Contingency Tables
8.5 Ordered Contingency Tables
8.5.1 Ordered 2 × c Tables
8.5.1.1 Alternative Hypotheses
8.5.1.2 Back-up Statistics
8.5.1.3 Directed Chi-Square
8.5.2 More Than Two Rows and Two Columns
8.5.2.1 Singly Ordered Tables
8.5.2.2 Doubly Ordered Tables
8.6 Covariates
8.6.1 Bross’ Method
8.6.2 Blocking
8.7 Exercises
9 Multivariate Analysis
9.1 Nonparametric Combination of Univariate Tests
9.2 Parametric Approach
9.2.1 Canonical For
m 9.2.2 Hotelling’s T2
9.2.3 Multivariate Analysis of Variance (MANOVA)
9.3 Permutation Methods
9.3.1 Which Test: Parametric or Permutation?
9.3.2 Interpreting the Results
9.4 Alternative Statistics
9.4.1 Maximum-t
9.4.2 Block Effects
9.4.3 Runs Test
9.4.4 Which Statistic?
9.5 Repeated Measures
9.5.1 An Example
9.5.2 Matched Pairs
9.5.3 Response Profiles
9.5.4 Missing Data
9.5.5 Bioequivalence
9.6 Exercises
10 Clustering in Time and Space
10.1.1 Mantel’s U
10.1.2 An Example
10.2 Applications
10.2.1 The MRPP Statistic
10.2.2 The BW Statistic of Cliff and Ord [1973]
10.2.3 Equivalances
10.2.4 Extensions
10.2.5 Another Dimension
10.3 Alternate Approaches
10.3.2 Nearest-Neighbor Analysis
10.3.3 Comparing Two Spatial Distributions
10.4 Exercises
11 Coping with Disaster
11.1 Missing Data
11.2 Covariates After the Fact
11.2.1 Observational Studies
11.3 Outliers
11.3.1 Original Data
11.3.2 Ranks
11.3.3 Scores
11.3.4 Robust Transformations.
11.3.5 Use an L1 Test
11.3.6 Censoring
11.4 Censored Data
11.4.1 GAMP Tests
11.4.2 Fishery and Animal Counts
11.5 Censored Match Pairs
11.5.1 GAMP Test for Matched Pairs
11.5.2 Rank
11.5.3 One-Sample: Bootstrap Estimates
11.7 Exercises
12 Solving the Unsolved and the Insolvable
12.1 Key Criteria
12.1.1 Sufficient Statistics
12.1.2 Three Stratagems
12.1.3 Restrict the Alternatives
12.1.4 Consider the Loss Function
12.1.5 Impartiality
12.2 The Permutation Distribution
12.2.1 Ensuring Exchangeability
12.2.1.1 Test for Parallelism
12.2.1.2 Linear Transforms That Preserve Exchangeability
12.3 New Statistics
12.3.1 Nonresponders
12.3.1.1 Extension to K-samples
12.3.2 Animal Movement
12.3.3 The Building Blocks of Life
12.3.4 Structured Exploratory Data Analysis
12.3.5 Comparing Multiple Methods of Assessment
12.4 Model Validation
12.4.1 Regression Models
12.4.1.1 Via the Bootstrap
12.4.1.2 Via Permutation Tests
12.4.2 Models With a Metric
12.5 Bootstrap Confidence Intervals
12.5.1 Hall–Wilson Criteria
12.5.2 Bias-Corrected Percentile
12.6 Exercises
13.1 Design Methodology
13.1.1 Randomization in Assignment
13.1.2 Choosing the Experimental Unit
13.1.3 Determining Sample Size
13.1.4 Power Comparisons
13.2 Preparing Manuscripts for Publication
13.2.1 Reportable Elements
13.2.2 Details of the Analysis
14 Increasing Computational Efficiency
14.1 Seven Techniques
14.2 Monte Carlo
14.2.1 Stopping Rules
14.2.2 Variance of the Result
14.2.3 Cutting the Computation Time
14.3 Rapid Enumeration and Selection Algorithms
14.3.1 Matched Pairs
14.4 Recursive Relationships
14.5 Focus on the Tails
14.5.1 Contingency Tables
14.5.1.1 Network Representation
14.5.1.2 The Network Algorithm
14.5.2 Play the Winner Allocation
14.5.3 Directed Vertex Peeling
14.6 Gibbs Sampling
14.6.1 Metropolis–Hastings Sampling Methods
14.7 Characteristic Functions.
14.8 Asymptotic Approximations
14.8.1 A Central Limit Theorem
14.8.2 Edgeworth Expansions
14.8.3 Generalized Correlation
14.9 Confidence Intervals
14.10 Sample Size and Power
14.10.1 Simulations
14.10.2 Network Algorithms
14.11 Some Conclusions
14.12 Software
14.12.1 Do-It-Yourself
14.12.2 Complete Packages
14.12.2.1 Freeware
14.12.2.2 Shareware
14.12.2.3 \$\$\$\$
14.13 Exercises
Appendix: Theory of Testing Hypotheses
A.1 Probability
A.2 The Fundamental Lemma
A.3 Two-Sided Tests
A.3.1 One-Parameter Exponential Families
A.4 Tests for Multiparameter Families
A.4.1 Basu’s Theorem
A.4.2 Conditional Probability and Expectation
A.4.3 Multiparameter Exponential Families
A.5 Exchangeable Observations
A.5.1 Order Statistics
A.5.2 Transformably Exchangeable
A.5.3 Exchangeability-Preserving Transforms
A.6 Confidence Intervals
A.7 Asymptotic Behavior
A.7.1 A Theorem on Linear Forms
A.7.2 Monte Carlo
A.7.3 Asymptotic Efficiency
A.7.4 Exchangeability
A.7.5 Improved Bootstrap Confidence Intervals
A.8 Exercises
Bibliography
Author Index
Subject Index