The optimality of a design depends on the statistical model and is assessed with respect to a statistical criterion, which is related to the variance-matrix of the estimator. Specifying an appropriate model and specifying a suitable criterion function both require understanding of statistical theory and practical knowledge with designing experiments.
One criterion is A-optimality, which seeks to minimize the trace of the inverse of the information matrix. This criterion results in minimizing the average variance of the estimates of the regression coefficients.
This criterion maximizes a quantity measuring the mutual column orthogonality of X and the determinant of the information matrix.
This criterion maximizes the discrepancy between two proposed models at the design locations.
Other optimality-criteria are concerned with the variance of predictions:
A popular criterion is G-optimality, which seeks to minimize the maximum entry in the diagonal of the hat matrix X(X'X)−1X'. This has the effect of minimizing the maximum variance of the predicted values.
A second criterion on prediction variance is I-optimality, which seeks to minimize the average prediction variance over the design space.
A third criterion on prediction variance is V-optimality, which seeks to minimize the average prediction variance over a set of m specific points.
Catalogs of optimal designs occur in books and in software libraries.
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.
Some advanced topics in optimal design require more statistical theory and practical knowledge in designing experiments.
Model dependence and robustness
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. Therefore, it is important to benchmark the performance of designs under alternative models.
Choosing an optimality criterion and robustness
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. Cornell writes that
since the [traditional optimality] criteria . . . are variance-minimizing criteria, . . . a design that is optimal for a given model using one of the . . . criteria is usually near-optimal for the same model with respect to the other 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. The experience of practitioners like Cornell and the "universal optimality" theory of Kiefer suggest that robustness with respect to changes in the optimality-criterion is much greater than is robustness with respect to changes in the model.
Flexible optimality criteria and convex analysis
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. Users may use a standard optimality-criterion or may program a custom-made criterion.
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.
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. Alternative terminology for "Bayesian" optimality includes "on-average" optimality or "population" optimality.
Scientific experimentation is an iterative process, and statisticians have developed several approaches to the optimal design of sequential experiments.
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. D. Gergonne in 1815 (Stigler). In English, two early contributions were made by Charles S. Peirce and Kirstine Smith.
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.
Discretizing probability-measure designs
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. Such a result was proved by Kôno and Kiefer in their works on response-surface designs for quadratic models. The Kôno–Kiefer analysis explains why optimal designs for response-surfaces can have discrete supports, which are very similar as do the less efficient designs that have been traditional in response surface methodology.
Charles S. Peirce proposed an economic theory of scientific experimentation in 1876, 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 1882 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.
[....] Unfortunately practice generally precedes theory, and it is the usual fate of mankind to get things done in some boggling way first, and find out afterward how they could have been done much more easily and perfectly.
^The adjective "optimum" (and not "optimal") "is the slightly older form in English and avoids the construction 'optim(um) + al´—there is no 'optimalis' in Latin" (page x in Optimum Experimental Designs, with SAS, by Atkinson, Donev, and Tobias).
^Traditionally, statisticians have evaluated estimators and designs by considering some summary statistic of the covariance matrix (of a mean-unbiased estimator), usually with positive real values (like the determinant or matrix trace). Working with positive real-numbers brings several advantages: If the estimator of a single parameter has a positive variance, then the variance and the Fisher information are both positive real numbers; hence they are members of the convex cone of nonnegative real numbers (whose nonzero members have reciprocals in this same cone).
For several parameters, the covariance-matrices and information-matrices are elements of the convex cone of nonnegative-definite symmetric matrices in a partiallyordered vector space, under the Loewner (Löwner) order. This cone is closed under matrix-matrix addition, under matrix-inversion, and under the multiplication of positive real-numbers and matrices.
An exposition of matrix theory and the Loewner-order appears in Pukelsheim.
^The above optimality-criteria are convex functions on domains of symmetric positive-semidefinite matrices: See an on-line textbook for practitioners, which has many illustrations and statistical applications:
^Iterative methods and approximation algorithms are surveyed in the textbook by Atkinson, Donev and Tobias and in the monographs of Fedorov (historical) and Pukelsheim, and in the survey article by Gaffke and Heiligers.
^Bayesian designs are discussed in Chapter 18 of the textbook by Atkinson, Donev, and Tobias. More advanced discussions occur in the monograph by Fedorov and Hackl, and the articles by Chaloner and Verdinelli and by DasGupta. Bayesian designs and other aspects of "model-robust" designs are discussed by Chang and Notz.
^As an alternative to "Bayesian optimality", "on-average optimality" is advocated in Fedorov and Hackl.
^Chernoff, H. (1972) Sequential Analysis and Optimal Design, SIAM Monograph.
^Zacks, S. (1996) "Adaptive Designs for Parametric Models". In: Ghosh, S. and Rao, C. R., (Eds) (1996). Design and Analysis of Experiments, Handbook of Statistics, Volume 13. North-Holland. ISBN0-444-82061-2. (pages 151–180)
^Henry P. Wynn wrote, "the modern theory of optimum design has its roots in the decision theory school of U.S. statistics founded by Abraham Wald" in his introduction "Jack Kiefer's Contributions to Experimental Design", which is pages xvii–xxiv in the following volume:
^The discretization of optimal probability-measure designs to provide approximately optimal designs is discussed by Atkinson, Donev, and Tobias and by Pukelsheim (especially Chapter 12).
^Regarding designs for quadratic response-surfaces, the results of Kôno and Kiefer are discussed in Atkinson, Donev, and Tobias.
Mathematically, such results are associated with Chebyshev polynomials, "Markov systems", and "moment spaces": See
^Peirce, C. S. (1882), "Introductory Lecture on the Study of Logic" delivered September 1882, published in Johns Hopkins University Circulars, v. 2, n. 19, pp. 11–12, November 1882, see p. 11, Google BooksEprint. Reprinted in Collected Papers v. 7, paragraphs 59–76, see 59, 63, Writings of Charles S. Peirce v. 4, pp. 378–82, see 378, 379, and The Essential Peirce v. 1, pp. 210–14, see 210–1, also lower down on 211.
Logothetis, N.; Wynn, H. P. (1989). Quality through design: Experimental design, off-line quality control, and Taguchi's contributions. Oxford U. P. pp. 464+xi. ISBN978-0-19-851993-5.
Textbooks emphasizing block designs
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). Bailey's exercises and discussion of randomization both emphasize statistical concepts (rather than algebraic computations).