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A Chronicle of Permutation Statistical Methods: 1920-2000, and Beyond

Kenneth J. Berry, Janis E. Johnston and Paul W. Mielke Jr.
Publication Date: 
Number of Pages: 
[Reviewed by
Robert W. Hayden
, on

Most readers of this review will be familiar with statistical inference for random sampling based on the Central Limit Theorem and friends. While these techniques sometimes offer reasonable approximations for data from experiments, it is worth knowing what we are approximating. There are exact tests based on permuting the observations to mimic random assignment of treatments, which is the form of randomness relevant to experiments. This has been known since R. A. Fisher’s early work in the last century, but Fisher lacked the computing power to carry out such tests. Their theory was developed anyway, however, and now we so have the computing power to use them.

Today, many college introductory statistics courses include permutation methods. The Common Core Standards adopted for the schools in most states treat inference for experiments via “simulations” which some interpret to indicate some form of permutation test. We can expect these methods to be much more widely used and discussed in the near future.

The word “chronicle” in the title of the book is well-chosen. We find a broadly chronological account of the history of permutation methods. The emphasis is on volume of information rather than on identifying the most important contributions. For instance, the reference list cites 1498 publications.

The authors divide the time period covered into five intervals. Within each interval, they try to identify the main trends of research at that time, and group their reporting accordingly. This can be useful to the person seeking a thorough treatment of a particular research trend, but for individual topics there seems to be little consistency in what gets reported. In some cases there are detailed mathematical explanations and concrete examples, while elsewhere entire paragraphs are but lists of references. One consistency is that there are thumbnail biographies of many of the people who have done research in this area. Your reviewer found many of these interesting, but at times these do seem to take us far afield. Although the book wisely covers developments in computing that made permutation methods practical, it seems a bit of a stretch to include biographies of Bill Gates and the founders of Google. Perhaps the most extravagant digression is a five-page history of the property where Fisher did his early research. On the whole one gets the feeling the authors included everything they unearthed during their research. They are also very generous in citing their own work.

So, we have here a huge amount of information that does not appear to have been critically chosen or interpreted. Still the volume of information to be found here may trump that consideration. The information is not readily available elsewhere. This book would be a great asset to anyone about to do a literature search on some aspect of permutation methods. Despite the claim on the book’s cover, this is for the specialist and not the general reader. In short, a book of great value to a narrow audience.

After a few years in industry, Robert W. Hayden ( taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.



2.1.Overview of Chapter 2
2.2.Neyman–Fisher–Geary and the Beginning
2.3.Fisher and the Variance-ratio Statistic
2.4.Eden–Yates and Non-normal Data
2.5.Fisher and 2 by 2 Contingency Tables
2.6 Yates and the Chi-squared Test for Small Samples
2.7.Irwin and Fourfold Contingency Tables
2.8.The Rothamsted Manorial Estate
2.9.Fisher and the Analysis of Darwin’s Zea mays Data
2.10.Fisher and the Coefficient of Racial Likeness
2.11.Hotelling–Pabst and Simple Bivariate Correlation
2.12.Friedman and Analysis of Variance for Ranks
2.13.Welch’s Randomized Blocks and Latin Squares
2.14.Egon Pearson on Randomization
2.15.Pitman and Three Seminal Articles
2.16.Welch and the Correlation Ratio
2.17.Olds and Rank-order Correlation
2.18.Kendall and Rank Correlation
2.19.McCarthy and Randomized Blocks
2.20.Computing and Calculators
2.21.Looking Ahead

3.1.Overview of Chapter 3
3.2.Development of Computing
3.3.Kendall–Babington Smith and Paired Comparisons
3.4.Dixon and a Two-sample Rank Test
3.5.Swed-Eisenhart and Tables for the Runs Test
3.6.Scheff´e and Non-parametric Statistical Inference
3.7.Wald–Wolfowitz and Serial Correlation
3.8.Mann and a Test of Randomness Against Trend
3.9.Barnard and 2 by 2 Contingency Tables
3.10.Wilcoxon and the Two-sample Rank-sum Test
3.11.Festinger and the Two-sample Rank-sum Test
3.12.Mann–Whitney and a Two-sample Rank-sum Test
3.13.Whitfield and a Measure of Ranked Correlation
3.14.Olmstead–Tukey and the Quadrant-sum Test
3.15.Haldane–Smith and a Test for Birth-order Effects
3.16.Finney and the Fisher–Yates Test for 2 by 2 Tables
3.17.Lehmann–Stein and Non-parametric Tests
3.18 Rank-order Statistics
3.19.van der Reyden and a Two-sample Rank-sum Test
3.20.White and Tables for the Rank-sum Test
3.21.Other Results for the Two-sample Rank-sum Test
3.22.David–Kendall–Stuart and Rank-order Correlation
3.23.Freeman–Halton and an Exact Test of Contingency
3.24.Kruskal–Wallis and the C-sample Rank-sum Test
3.25.Box–Andersen and Permutation Theory
3.26.Leslie and Small Contingency Tables
3.27.A Two-sample Rank Test for Dispersion
3.28.Dwass and Modified Randomization Tests
3.29.Looking Ahead

4.1.Overview of Chapter 4
4.2.Development of Computing
4.3 Permutation Algorithms and Programs
4.4.Ghent and the Fisher–Yates Exact Test
4.5.Programs for Contingency Table Analysis
4.6.Siegel–Tukey and Tables for the Test of Variability
4.7 .Other Tables of Critical Values
4.8.Edgington and Randomization Tests
4.9.The Matrix Occupancy Problem
4.10.Kempthorne and Experimental Inference
4.11.Baker–Collier and Permutation F Tests 4.12.Permutation Tests in the 1970s
4.13.Feinstein and Randomization
4.14.The Mann–Whitney, Pitman, and Cochran Tests
4.15.Mielke–Berry–Johnson and MRPP
4.16.Determining the Number of Contingency Tables
4.17.Soms and the Fisher Exact Permutation Test
4.18.Baker–Hubert and Ordering Theory
4.19.Green and Two Permutation Tests for Location
4.20.Agresti–Wackerly–Boyett and Approximate Tests
4.21.Boyett and Random R by C Tables
4.22.Looking Ahead

5.1.Overview of Chapter 5
5.2.Development of Computing
5.3.Permutation Methods and Contingency Tables
5.4.Yates and 2 by 2 Contingency Tables
5.5.Mehta–Patel and a Network Algorithm
5.6.MRPP and the Pearson type III Distribution
5.7.MRPP and Commensuration
5.8.Tukey and Re randomization
5.9.Matched-pairs Permutation Analysis
5.10.Subroutine PERMUT
5.11.Moment Approximations and the F Test
5.12.Mielke–Iyer and MRBP
5.13.Relationships of MRBP to Other Tests
5.14.Kappa and the Measurement of Agreement
5.15.Basu and the Fisher Randomization Test
5.16.Still–White and Permutation Analysis of Variance
5.17.Walters and the Utility of Resampling Methods
5.18.Conover–Iman and Rank Transformations
5.19.Green and Randomization Tests
5.20.Gabriel–Hall and Re randomization Inference
5.21.Pagano–Tritchler and Polynomial-time Algorithms
5.22.Welch and a Median Permutation Test
5.23.Boik and the Fisher–Pitman Permutation Test
5.24.Mielke–Yao Empirical Coverage Tests
5.25.Randomization in Clinical Trials
5.26.The Period From 1990 to 2000
5.27.Algorithms and Programs
5.28.Page–Brin and Google
5.29.Spino–Pagano and Trimmed/Winsorized Means
5.30.May–Hunter and Advantages of Permutation Tests
5.31.Mielke–Berry and Tests for Common Locations
5.32.Kennedy–Cade and Multiple Regression
5.33.Blair et al. and Hotelling’s T2 Test
5.34.Mielke–Berry–Neidt and Hotelling’s T2 Test
5.35.Cade-Richards and Tests for LAD Regression
5.36.Walker–Loftis–Mielke and Spatial Dependence
  5.37.Frick on Process-based Testing
5.38.Ludbrook–Dudley and Biomedical Research
5.39.The Fisher Z Transformation
5.40.Looking Ahead

6.Beyond 2000
6.1.Overview of Chapter 6
6.2.Computing After Year 2000
6.3.Books on Permutation Methods
6.4.A Summary of Contributions by Publication Year
6.5.Agresti and Exact Inference for Categorical Data
6.6.The Unweighted Kappa Measure of Agreement
6.7.Mielke et al. and Combining Probability Values
6.8.Legendre and Kendall’s Coefficient of Concordance
6.9.The Weighted Kappa Measure of Agreement
6.10.Berry et al. and Measures of Ordinal Association
6.11.Resampling for Multi-way Contingency Tables
6.12.Mielke–Berry and a Multivariate Similarity Test
6.13.Cohen’s Weighted Kappa With Multiple Raters
6.14.Exact Variance of Weighted Kappa
6.15.Campbell and Two-by-two Contingency Tables
6.16.Permutation Tests and Robustness
6.17.Advantages of the Median for Analyzing Data
6.18.Consideration of Statistical Outliers
6.19.Multivariate Multiple Regression Analysis
6.20.O’Gorman and Multiple Linear Regression
6.21.Brusco–Stahl–Steinley and Weighted Kappa
6.22.Mielke et al. and Ridit Analysis
6.23.Knijnenburg et al. and Probability Values
6.24.Reiss et al. and Multivariate Analysis of Variance
6.25.A Permutation Analysis of Trend
6.26.Curran-Everett and Permutation Methods




Name Index

Subject Index.