Optimal experimental design

In the design of experiments, optimal experimental designs are a class of experimental designs that are optimal with respect to some statistical criterion. The creation of this field of statistics has been credited to Danish statistician Kirstine Smith.
In the design of experiments for estimating statistical models, optimal designs allow parameters to be estimated without bias and with minimum variance. A non-optimal design requires a greater number of experimental runs to estimate the parameters with the same precision as an optimal design. In practical terms, optimal experiments can reduce the costs of experimentation.
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.

Advantages

Optimal designs offer three advantages over sub-optimal experimental designs:

Optimal designs reduce the costs of experimentation by allowing statistical models to be estimated with fewer experimental runs.
Optimal designs can accommodate multiple types of factors, such as process, mixture, and discrete factors.
Designs can be optimized when the design-space is constrained, for example, when the mathematical process-space contains factor-settings that are practically infeasible.
Minimizing the variance of estimators

Experimental designs are evaluated using statistical criteria.
It is known that the least squares estimator minimizes the variance of mean-unbiased estimators. In the estimation theory for statistical models with one real parameter, the reciprocal of the variance of an estimator is called the "Fisher information" for that estimator. Because of this reciprocity, minimizing the variance corresponds to maximizing the information.
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. The traditional optimality-criteria are invariants of the information matrix; algebraically, the traditional optimality-criteria are functionals of the eigenvalues of the information matrix.

A-optimality
*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.
C-optimality
*This criterion minimizes the variance of a best linear unbiased estimator of a predetermined linear combination of model parameters.
D-optimality
*A popular criterion is D-optimality, which seeks to minimize |⁻¹|, or equivalently maximize the determinant of the information matrix X'X of the design. This criterion results in maximizing the differential Shannon information content of the parameter estimates.
E-optimality
*Another design is E-optimality, which maximizes the minimum eigenvalue of the information matrix.
S-optimality
*This criterion maximizes a quantity measuring the mutual column orthogonality of X and the determinant of the information matrix.
T-optimality
*This criterion maximizes the discrepancy between two proposed models at the design locations.

Other optimality-criteria are concerned with the variance of predictions:

G-optimality
*A popular criterion is G-optimality, which seeks to minimize the maximum entry in the diagonal of the hat matrix X⁻¹X'. This has the effect of minimizing the maximum variance of the predicted values.
I-optimality
*A second criterion on prediction variance is I-optimality, which seeks to minimize the average prediction variance over the design space.
V-optimality
*A third criterion on prediction variance is V-optimality, which seeks to minimize the average prediction variance over a set of m specific points.
Contrasts

In many applications, the statistician is most concerned with a "parameter of interest" rather than with "nuisance parameters". 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 contrasts.

Implementation

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.

Practical considerations

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
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.
All of the traditional optimality-criteria are convex functions, and therefore optimal-designs are amenable to the mathematical theory of convex analysis and their computation can use specialized methods of convex minimization. The practitioner need not select exactly one traditional, optimality-criterion, but can specify a custom criterion. In particular, the practitioner can specify a convex criterion using the maxima of convex optimality-criteria and nonnegative combinations of optimality criteria. For convex optimality criteria, the Kiefer-Wolfowitz allows the practitioner to verify that a given design is globally optimal. The Kiefer-Wolfowitz is related with the Legendre-Fenchel conjugacy for convex functions.
If an optimality-criterion lacks convexity, then finding a global optimum and verifying its optimality often are difficult.

Model uncertainty and Bayesian approaches

Model selection

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.

Bayesian experimental design

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. Such probability-based optimal-designs are called optimal Bayesian designs. Such Bayesian designs are used especially for generalized linear models.
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.

Iterative experimentation

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. In 1972, Herman Chernoff wrote an overview of optimal sequential designs, while adaptive designs were surveyed later by S. Zacks. Of course, much work on the optimal design of experiments is related to the theory of optimal decisions, especially the statistical decision theory of Abraham Wald.

Response-surface methodology

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. In English, two early contributions were made by Charles S. Peirce and .
Pioneering designs for multivariate response-surfaces were proposed by George E. P. Box. However, Box's designs have few optimality properties. Indeed, the Box–Behnken design requires excessive experimental runs when the number of variables exceeds three.
Box's "central-composite" designs require more experimental runs than do the optimal designs of Kôno.