Multiple comparison procedures.

*(English)*Zbl 0731.62125
Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. New York etc.: John Wiley & Sons, Inc. xxiv, 450 p. (1987).

The authors succeed in collating the diverse literature on multiple comparisons, thereby providing an invaluable guide to this special topic. It will probably find a greater audience among researchers in the field than among practitioners, although there are many worked-through examples for detailed guidance. To indicate the magnitude of the effort, the reference list contains 455 entries, of which 41% appeared since 1976, the year in which Miller last updated the material in his classic monograph. The literature of multiple comparisons has been dominated by attempts to guarantee control of various error rates, extending Neyman- Pearson ideas from two-decision to multiple-decision cases. The book reflects this dominance. An introductory chapter contains definitions of per-comparison, per-family and familywise error rates and descriptions of a few realistic situations in which each would deserve primary consideration.

The most challenging theoretical material, establishing general conditions for control of error rates, has been relegated to an appendix of 20 pages. Part I of the book itself contains five chapters devoted to results for fixed-effects linear models with normal, homoscedastic, independent errors. There is one chapter (2) for theory and general concepts such as the union-intersection method and simultaneous test procedures. Succeeding chapters give details and examples: single-step procedures for comparisons among treatments (3), stepwise procedures for comparisons among treatments (4), and procedures for comparisons in some nonhierarchical families, such as orthogonal comparisons and comparisons with the best (5). Chapter 6 contains a summary of work on experimental design, providing a link to the ranking and selection literature.

In Part II the authors discuss attempts to control error rates under broader assumptions, including heteroscedastic errors (7) and mixed models (8). There are also chapters on distribution-free and robust procedures (9) and procedures for categorical data, variances and means under order restrictions (10). Material on efforts to attack the multiple comparisons problem directly as a multiple decision problem is contained in Chapter 11. This consists primarily of a summary of work by Duncan and others to develop adaptive Bayesian procedures for comparing normal means with conjugate priors and additive linear losses, the so-called k-ratio methods.

In addition to the appendix on general theory mentioned above, there are also appendices on probability inequalities and on tables needed to apply the procedures. Author and subject indices are also provided.

The most challenging theoretical material, establishing general conditions for control of error rates, has been relegated to an appendix of 20 pages. Part I of the book itself contains five chapters devoted to results for fixed-effects linear models with normal, homoscedastic, independent errors. There is one chapter (2) for theory and general concepts such as the union-intersection method and simultaneous test procedures. Succeeding chapters give details and examples: single-step procedures for comparisons among treatments (3), stepwise procedures for comparisons among treatments (4), and procedures for comparisons in some nonhierarchical families, such as orthogonal comparisons and comparisons with the best (5). Chapter 6 contains a summary of work on experimental design, providing a link to the ranking and selection literature.

In Part II the authors discuss attempts to control error rates under broader assumptions, including heteroscedastic errors (7) and mixed models (8). There are also chapters on distribution-free and robust procedures (9) and procedures for categorical data, variances and means under order restrictions (10). Material on efforts to attack the multiple comparisons problem directly as a multiple decision problem is contained in Chapter 11. This consists primarily of a summary of work by Duncan and others to develop adaptive Bayesian procedures for comparing normal means with conjugate priors and additive linear losses, the so-called k-ratio methods.

In addition to the appendix on general theory mentioned above, there are also appendices on probability inequalities and on tables needed to apply the procedures. Author and subject indices are also provided.

##### MSC:

62J15 | Paired and multiple comparisons; multiple testing |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |