Configural Polysampling. by Stephan Morganthaler; John W. TukeyReview by: Karen KafadarJournal of the American Statistical Association, Vol. 89, No. 425 (Mar., 1994), pp. 355-356Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2291235 .

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Book Reviews 355

Re.samnpling-Based Mutltiple Testing is well written and very readable. I found only a few errors, all of them minor. Although a Masters-level knowl- edge of statistics is required for a full appreciation of the material, the book's basic gist can be grasped by a less-sophisticated reader. The authors did not intend this book to be used as a course text, and I would not recommend it as such, except possibly for a special topics course. Readers involved in methodological research will find this book a valuable resource that com- plements rather than replicates material in other books on the subject, such as Hochberg and Tamhane (1987). The book will also be extremely useful to researchers seeking practical advice on how to deal with multiplicity issues in real data applications. For my own part, besides learning a great deal about an important topic on which I am by no means an expert, I found the book interesting because it addresses an important class of problems for which bootstrap methods often provide the only method of solution. I com- pliment the authors on a job well done.

James G. BOOTH University of Florida

REFERENCE Hochberg, Y., and Tamhane, A. C. (1987). Mulltiple, CoTnparison Procedur(es, New

York: John Wiley.

Configural Polysampling. Stephan MORGANTHALER and John W. TUKEY (eds.). New York: John Wiley, 1992. xiv + 218 pp. $44.95.

Configural polysampling is a tool that relies on numerical and compu- tational methods to solve problems that are analytically intractable. This book focuses on the problem of deriving and solving for the value of an optimal equivariant location estimator or its variance in a small-sample situation. Classical robustness theory, with its reliance on asymptotic results, is inadequate for small samples, particularly those that arise as mixtures of several distributions. This book is the first to address issues in finite sample robustness, through configural polysampling, and further describes other problems to which this methodology may be applied.

An example of an important yet very common problem that can be solved by configural polysampling will help illustrate this book's purpose. Suppose that we want to estimate the location parameter in a random sample y = (Yi, * - . yn) from a distribution F(y - ji), and , is a statistic satisfying 4(y + r) = ,u(y) + r(rany real number). The sample y can be represented as (4, c), where c = (y, - ,, . . ., Yn - 4); c is called a location configuration. Then all samples y + r (r any real number) are essentially equivalent in the sense that they can be obtained from one another by translation; all have the same value of c. So if we have a "sensible" estimator of location, say T (namely, one that satisfies T(y + r) = T(y) + r), then we need to know only how it behaves on c to know how it behaves on c + r (for any r). Because the distribution of c does not depend on (i.e., is ancillary for) the location parameter, an optimal T may be found by minimizing the mean squared error of T = min EF* { E4( T(y) - k)2 I C] }; that is, by conditioning on the configuration and then averaging with respect to F*, the distribution of configurations. The first step, conditioning on the configuration, reduces to a one-dimensional integral that can be solved by quadrature; the second step, averaging over configurations, relies on sampling to handle the higher- dimensional integral. Historical references to this approach are given in the book's Section 3; a useful and classical reference for the theory of equivariant estimators is Lehmann (1983 chap. 3).

The preceding description of the book's title makes it clear why the editors chose a book format rather than separate journal articles for the different concepts. There are so many novel approaches, techniques, and definitions that the philosophy underlying configural polysampling would be lost in a series of fragmented articles. With a book format, the basic techniques (such as conditioning and sampling) and definitions (such as configuration, slash, and confrontation) need be presented only once. An edited book with chapters written by six authors runs the risk of reading like a series ofjournal articles. (This can happen with even fewer than six authors.) But in this case, the editors have taken care to achieve unification. Chapters flow logically, ter- minology is consistent, and frequent cross-references to other chapters all contribute to the book's achievement of its goals, among which are "to develop the machinery needed to attain optimum estimators for [small sam- ples from] confrontations," and "to use similar machinery to compare simple estimators with formally optimal ones" (p. 7).

One particularly useful feature is the discussion of current themes in sta- tistical practice (Chaps. 1 and 2). I quite enjoyed these chapters, because they force us to challenge fundamental statistical concepts. For example,

why does the problem of location estimation receive so much attention? What role does conditional inference play? Why do we tend to concentrate on symmetric distributions? Why do we rely on asymptotic results? What is the value of estimators in "pure" situations (e.g., Gaussian)? Why should we use "Gaussian" instead of "Normal" for the name of the distribution? Why is the slash distribution (Gaussian/Uniform) an extreme but more useful distribution for robustness studies than are members of the Student's t family? Such issues are at the heart of statistical theory, and it is useful as well as interesting to have them placed in the context of practical data analysis. By challenging classical statistical concepts, the authors motivate their phi- losophy of statistics and their theory of finite sample robustness.

Issues in robustness are likewise challenged (in Chap. 2, "The Background of Configural Polysampling: An Historical Review"). Although certain ap- proaches in classical (asymptotic) robustness are useful in discussing finite samples (e.g., robustness of validity), others need to be broadened (e.g., central mnodel confrontation, optimality in single situations - bioptimality for two situations). To elaborate, classical robustness evaluates performance of estimators at a central model (e.g., Gaussian) and aims for good perfor- mance in a neighborhood of that central model; but here the authors argue that finite sample robustness is better measured by evaluating performance at a serieAs of distributions, or a controntation, which represents extremes of what we are likely to observe in practice. The Gaussian and slash distributions represent two such extremes in terms of symmetric tail behavior; other extremes may be appropriate when focusing on other distributional char- acteristics such as skewness or kurtosis. This shift in focus from a central model to a confrontation is an essential component of the authors' theory of finite sample robustness.

After these first two chapters lay the ground rules for the approaches to be discussed (e.g., "Estimators that do well both at the Gaussian and the slash will be called robust in this book"; p. 6), subsequent chapters extend the theory to the problem of optimal equivariant estimators in the location and scale model for specific distributions (Chap. 4), for confrontations of distributions (Chap. 5), and for regression settings (Chap. 10). Other chapters present conditional confidence intervals for these estimators (Chaps. 8 and 9) and other Pitman-type estimators (Chap. I 1) and discuss numerical con- siderations (Ch. 7, 13). A particularly useful chapter is Chapter 12, "Ex- amples": On the theory that one does not truly understand something unless one can compute it, these tools are applied to actual data sets, with calcu- lations spelled out and results discussed in detail.

For the most part, new ideas are presented first intuitively with little formal mathematics; these explanations will suffice for most readers. Those who are comfortable with abstract algebra and functional analysis will appreciate the formal presentation; e.g., configuration is defined formally as one of the orbits in the group of transformations (p. 22; an example would be the maximal invariant for location and scale, namely (jy - j)/sy); approximate conditional maximum likelihood estimators are obtained using Laplace in- tegral approximations to the multiple integrals that define optimal compro- mise estimators of location for the Gaussian and slash distributions (Chap. I 1).

Although any statistics graduate student who feels comfortable in math- ematical analysis will be able to follow this book, those who have studied or at least are familiar with the robustness literature stand to gain the most. The book's structure allows it to be offered as part of a graduate course in robustness, although of course it is quite suitable for independent study as well. Reminders of purpose and goals appear frequently, serving as road maps to the reader who pursues the study of this somewhat unfamiliar topic. Each chapter starts with an outline ("what we will do"), ends with a summary ("what we have done"), and points to coming attractions. Frequent cross- references to other chapters provide indications of how the ideas all fit to- gether, and most (not all) chapters end with study problems of various types (i.e., review, application, additional challenge). The inevitable typographical errors occur, but most are grammatical; the few technical errors are easy to spot and to correct (e.g., on p. 165, line -14, "a(y)/&(y)" should be crG(YV)IF(YY))- Configutral Polvsampling represents a novel and important contribution

to the robustness literature. It is difficult to compare the book with others in this field, because its topic is quite specialized. Moreover, unlike most books that consolidate ideas and/or techniques that have been discussed and debated for awhile in the literature (e.g., Hampel, Ronchetti, Rousseeuw, and Stahel 1986; Huber 1981; Hettmansperger 1984), this book presents a philosophy and a variety of techniques that will be entirely new to many readers. As a result, readers must be cautioned to persevere through the somewhat unfamiliar terminology (e.g., bioptimal, biefficient, bieffective, biconditional, bioptimal-like, biefficient-like, bieffective-like). It is an im- portant book for one who wishes to pursue the study of robust statistics and, perhaps more importantly, for one in statistical research to see how problems in classical statistics (optimal equivariant location and scale estimators in

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356 Journal of the American Statistical Association, March 1994

large samples from single situations) can lead to a whole new theory (finite sample robustness) and new tools for solving them (configural polysampling).

Karen KAFADAR National Cancer Instituite

REFERENCES Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A. (1986), Robutst

Staltistic.s. The Approach BCIsed on Influten e Funct-ions, New York: John Wiley. Hettmansperger, T. P. (1984), Statistical Ink(rence Based on Ranks, New York: John

Wiley. Huber, P. J. (1981), Robutst Statislics. New York: John Wiley. Lehmann, E. L. (1983). Theorl' of Point Estination, New York: John Wiley.

Statistical Models Based on Counting Processes. Per Kragh ANDERSEN, Ornulf BORGAN, Richard D. GILL and Niels KEIDING. New York: Springer-Verlag, 1993. xi + 767 pp. $69.

This book is a comprehensive treatment of the counting process approach to event history analysis. The authors characterize this field as the study of individuals moving through a small number of states, including as an im- portant subset the field of survival analysis, where the two states are alive and dead. The data from such studies are often subject to varying kinds of incompleteness, including right-censoring, left-censoring, and left-truncation. Counting processes and their transformation to martingales provide con- venient models for such data and powerful tools for the investigation of the statistical properties of natural descriptive and test statistics in these settings.

Counting processes, martingales, filtrations, and product integrals can be formidable concepts for the uninitiated, but they are fundamentally quite intuitive. The authors do an excellent job of providing the requisite math- ematical background and notation while keeping the intuition in the fore- ground. Detailed road maps are provided in the introductory chapter for readers with varying backgrounds and interests to get the most out of the book. Other strong points include the use of examples, introduced early and used throughout; introductory material for each chapter telling the reader what lies ahead; and extensive bibliographical notes at the end of each chapter. The breadth of the material covered is illustrated by the fact that the examples come from medicine, biology, social science, anthropology, and software reliability.

An obvious comparison is with the recent book Counting Processes and Survival Anali'sis (Fleming and Harrington 1991), which has more math- ematical detail (some in appendixes) and is more focused on the proportional hazards model for right-censored data. In contrast, Statistical Models Based on Cotnining Processes includes material on parametric models, nonpara- metric additive hazards models, frailty models, and multiple time scales. It also includes a greater variety of methods for model checking and diagnostics, whereas Fleming and Harrington's book concentrates on martingale residuals. Both books could be used as a text for second- or third-year graduate students of statistics or biostatistics with a good background in real analysis and measure theory, although Fleming and Harrington's book is perhaps a bit more natural in this regard.

Our own research and experience is primarily in the field of survival analysis, and our graduate training preceeded many of the developments outlined here. One of us (JC) was privileged to learn from Anderson, Borgan, Gill, and Keiding as this book was being developed, and the other (PYL) came to this material through a course using Fleming and Harrington's text as it was being written. We have both benefited greatly from reviewing this impressive reference, which contains a wealth of powerful mathematics, practical examples, and analytic insights, as well as a complete integration of historical developments and recent advances in event history analysis.

John CROWLEY Ping-Yu Llu

Fred Hulichinson Cancer Research Center

REFERENCE

Fleming. T. R.. and Harrington, D. P. (1991), Counting Procvses and Survival Analvsis, New York: John Wiley.

Cross-Over Experiments: Design, Analysis and Application. David A. RATKOWSKY, Marc A. EVANS, and J. Richard ALLDREDGE. New York: Marcel Dekker, 1993. ix + 446 pp. $1 10.

In crossover experiments, each subject receives several treatments, one after another. Difficult problems arise in the analysis and interpretation of

such experiments. The effect of a treatment may linger, changing the response to the next treatment. Observations within a subject are correlated, com- plicating the analysis. Model formulation is difficult, and in some designs the parameters of interest are not identifiable. On the other hand, crossover designs control variability by applying several treatments to the same subject. These designs may also be desirable if it is difficult to recruit subjects but inexpensive to collect measurements on each subject at several points in time.

To design a crossover experiment, the investigator must decide how many observations per subject will be collected, which treatment sequences will be applied, and how many subjects will be assigned to each treatment se- quence. For instance, one might decide to observe each subject at three time periods, applying treatments A and B in the sequences ABB and BAA, and assigning 10 subjects to each sequence. This is a two-treatment, three-period, two-sequence design.

This book focuses on the following model for a t treatment, p period, s sequence design with continuous response. The response of the kth subject assigned to sequence i observed at time period j is

Y,lk = A + Y, + Irx + Sik + Td(i,j) + I{ j > 1 } Xd(i,J-1) + e,,k (*)

where yi is a sequence effect, ir, is a period effect, Sk, is a random subject effect, d iS the "direct effect" of treatment d, Xd iS the "first-order carryover effect" of treatment d, and d( i, j) is the treatment assigned to the jth period of sequence i. The -y's, 7r'S, T's, and X's are assumed to sum to 0. The random variables Si, and ei,k are assumed to be normally distributed with mean 0.

The first-order carryover effect is the residual effect of the treatment applied in the previous period. More elaborate models can be formulated by including higher-order carryover effects, interactions between treatments and periods, and so on. One might assume that the random variables Sik and e#k are mutually independent with variances o-a and Se, or one might assume a more complicated covariance structure.

The two sources of random variability, between subjects and period-to- period within a subject, require a split-plot analysis. Ideally one would like to regard treatment effects as within-subject effects, but this is not valid in all crossover designs. This is one of the difficulties in analyzing crossover data.

In this book commonly used crossover designs and their analysis for both continuous and categorical responses are examined. Continuous data are analyzed using least squares only; nonparametric and robust approaches are not considered. Data examples are analyzed using SAS. The book uses no mathematics beyond matrix algebra, so it is accessible to researchers in the biomedical, agricultural, and social sciences as well as to statisticians.

The difficulties and complexities of crossover data are most clearly illus- trated in the widely used two-treatment, two-period, two-sequence design. This is the major topic that the authors treat theoretically. Unfortunately, they commit a major theoretical error, so that their analysis of this design is flawed.

In this design, treated in Chapter 3, n, subjects receive treatments in the sequence AB and n2 subjects receive treatments in the sequence BA. The authors stress that the parameters in (*) are not identifiable. This is easily seen, because there are only four sequence-period means to be estimated, but (* ) contains five parameters. Any valid treatment of this design requires an assumption that at least one function of the parameters is 0. Of course such assumptions cannot be tested from available data. The authors consider four possible assumptions (parameterizations 1, 2, 3, and 4) and their in- terpretations. Unfortunately, they go astray when they consider the corre- sponding data analyses.

The authors claim that the random subject effect is eliminated if one assumes that the fixed sequence effect is absent. This is obviously untrue if subjects have been selected at random from a population. Hence two of their four models are incorrectly analyzed.

The authors go on to criticize the standard analysis of (*) for the two- treatment, two-period, two-sequence design, which appears in the work of Jones and Kenward (1989). Evidently the authors believe that Jones and Kenward are analyzing "parameterization 3," even though Jones and Ken- ward explicitly formulate and correctly analyze "parameterization 1." The authors' discussion of Milliken and Johnson's ( 1984) treatment of this design also suffers from a misunderstanding of the correct interpretation of the analysis.

One source of difficulty is the inherent aliasing of the parameters in this design. The carryover effects of treatments, which one would wish to regard as within-subject parameters, are actually confounded with sequence effects. In addition, as the authors note elsewhere, direct and carryover effects do not separate into orthogonal contrasts in many crossover designs. This prob- lem is particularly acute in the two-treatment, two-period, two-sequence design.

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