The last twenty years have witnessed a significant growth of interest in optimal factorial designs, under possible model uncertainty, via the minimum aberration and related criteria. The present book gives, for the first time in book form, a comprehensive and up-to-date account of this modern theory. Many major classes of designs are covered in the book. While maintaining a high level of mathematical rigor, it also provides extensive design tables for research and practical purposes. In order to equip the readers with the necessary background, some foundational concepts and results are developed in Chapter 2.

Apart from being useful to researchers and practitioners, the book can form the core of a graduate level course in experimental design. In the estimation theory for statistical models with one real parameter , the reciprocal of the variance of an "efficient" estimator is called the " Fisher information " for that estimator. When the statistical model has several parameters , however, the mean of the parameter-estimator is a vector and its variance is a matrix. The inverse matrix of the variance-matrix is called the "information matrix". Because the variance of the estimator of a parameter vector is a matrix, the problem of "minimizing the variance" is complicated.

Using statistical theory , statisticians compress the information-matrix using real-valued summary statistics ; being real-valued functions, these "information criteria" can be maximized. Other optimality-criteria are concerned with the variance of predictions:. In many applications, the statistician is most concerned with a "parameter of interest" rather than with "nuisance parameters".

## Optimal design

More generally, statisticians consider linear combinations of parameters, which are estimated via linear combinations of treatment-means in the design of experiments and in the analysis of variance ; such linear combinations are called contrasts. Statisticians can use appropriate optimality-criteria for such parameters of interest and for more generally for contrasts. In addition, major statistical systems like SAS and R have procedures for optimizing a design according to a user's specification. The experimenter must specify a model for the design and an optimality-criterion before the method can compute an optimal design.

## A Modern Theory of Factorial Design - Rahul Mukerjee, C.F. J. Wu - Google Книги

Some advanced topics in optimal design require more statistical theory and practical knowledge in designing experiments. Since the optimality criterion of most optimal designs is based on some function of the information matrix, the 'optimality' of a given design is model dependent: While an optimal design is best for that model , its performance may deteriorate on other models. On other models , an optimal design can be either better or worse than a non-optimal design. The choice of an appropriate optimality criterion requires some thought, and it is useful to benchmark the performance of designs with respect to several optimality criteria.

Indeed, there are several classes of designs for which all the traditional optimality-criteria agree, according to the theory of "universal optimality" of Kiefer. High-quality statistical software provide a combination of libraries of optimal designs or iterative methods for constructing approximately optimal designs, depending on the model specified and the optimality criterion.

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Users may use a standard optimality-criterion or may program a custom-made criterion. All of the traditional optimality-criteria are convex or concave functions , and therefore optimal-designs are amenable to the mathematical theory of convex analysis and their computation can use specialized methods of convex minimization. In particular, the practitioner can specify a convex criterion using the maxima of convex optimality-criteria and nonnegative combinations of optimality criteria since these operations preserve convex functions.

For convex optimality criteria, the Kiefer - Wolfowitz equivalence theorem allows the practitioner to verify that a given design is globally optimal. If an optimality-criterion lacks convexity , then finding a global optimum and verifying its optimality often are difficult. When scientists wish to test several theories, then a statistician can design an experiment that allows optimal tests between specified models. Such "discrimination experiments" are especially important in the biostatistics supporting pharmacokinetics and pharmacodynamics , following the work of Cox and Atkinson.

When practitioners need to consider multiple models , they can specify a probability-measure on the models and then select any design maximizing the expected value of such an experiment.

### Associated Data

Such probability-based optimal-designs are called optimal Bayesian designs. Such Bayesian designs are used especially for generalized linear models where the response follows an exponential-family distribution. The use of a Bayesian design does not force statisticians to use Bayesian methods to analyze the data, however. Indeed, the "Bayesian" label for probability-based experimental-designs is disliked by some researchers.

Scientific experimentation is an iterative process, and statisticians have developed several approaches to the optimal design of sequential experiments. Sequential analysis was pioneered by Abraham Wald. Optimal designs for response-surface models are discussed in the textbook by Atkinson, Donev and Tobias, and in the survey of Gaffke and Heiligers and in the mathematical text of Pukelsheim.

The blocking of optimal designs is discussed in the textbook of Atkinson, Donev and Tobias and also in the monograph by Goos. The earliest optimal designs were developed to estimate the parameters of regression models with continuous variables, for example, by J. Gergonne in Stigler. In English, two early contributions were made by Charles S.

Peirce and Kirstine Smith. Pioneering designs for multivariate response-surfaces were proposed by George E. However, Box's designs have few optimality properties. Indeed, the Box—Behnken design requires excessive experimental runs when the number of variables exceeds three. The optimization of sequential experimentation is studied also in stochastic programming and in systems and control. Popular methods include stochastic approximation and other methods of stochastic optimization.

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Much of this research has been associated with the subdiscipline of system identification. Nemirovskii and Boris Polyak has described methods that are more efficient than the Armijo-style step-size rules introduced by G. Box in response-surface methodology. Adaptive designs are used in clinical trials , and optimal adaptive designs are surveyed in the Handbook of Experimental Designs chapter by Shelemyahu Zacks. There are several methods of finding an optimal design, given an a priori restriction on the number of experimental runs or replications.

Some of these methods are discussed by Atkinson, Donev and Tobias and in the paper by Hardin and Sloane. Of course, fixing the number of experimental runs a priori would be impractical. Prudent statisticians examine the other optimal designs, whose number of experimental runs differ. In the mathematical theory on optimal experiments, an optimal design can be a probability measure that is supported on an infinite set of observation-locations.

Such optimal probability-measure designs solve a mathematical problem that neglected to specify the cost of observations and experimental runs. Nonetheless, such optimal probability-measure designs can be discretized to furnish approximately optimal designs. In some cases, a finite set of observation-locations suffices to support an optimal design.

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In , an article on optimal designs for polynomial regression was published by Joseph Diaz Gergonne , according to Stigler. Peirce proposed an economic theory of scientific experimentation in , which sought to maximize the precision of the estimates. Peirce's optimal allocation immediately improved the accuracy of gravitational experiments and was used for decades by Peirce and his colleagues. In his published lecture at Johns Hopkins University , Peirce introduced experimental design with these words:.

Logic will not undertake to inform you what kind of experiments you ought to make in order best to determine the acceleration of gravity, or the value of the Ohm; but it will tell you how to proceed to form a plan of experimentation. Kirstine Smith proposed optimal designs for polynomial models in Kirstine Smith had been a student of the Danish statistician Thorvald N. Thiele and was working with Karl Pearson in London.

The textbook by Atkinson, Donev and Tobias has been used for short courses for industrial practitioners as well as university courses. Optimal block designs are discussed by Bailey and by Bapat. The first chapter of Bapat's book reviews the linear algebra used by Bailey or the advanced books below.