Optimal experimental design
Specifying an appropriate model and specifying a suitable criterion function both require understanding of statistical theory and practical knowledge with designing experiments.
In addition, major statistical systems like SAS and R have procedures for optimizing a design according to a user's specification.
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.
Such "discrimination experiments" are especially important in the biostatistics supporting pharmacokinetics and pharmacodynamics, following the work of Cox and Atkinson.
Scientific experimentation is an iterative process, and statisticians have developed several approaches to the optimal design of sequential experiments.
[27] 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).
[30] In computational optimal control, D. Judin & A. Nemirovskii and Boris Polyak has described methods that are more efficient than the (Armijo-style) step-size rules introduced by G. E. P. Box in response-surface methodology.
There are several methods of finding an optimal design, given an a priori restriction on the number of experimental runs or replications.
Prudent statisticians examine the other optimal designs, whose number of experimental runs differ.
Such optimal probability-measure designs solve a mathematical problem that neglected to specify the cost of observations and experimental runs.
[33] In 1815, an article on optimal designs for polynomial regression was published by Joseph Diaz Gergonne, according to Stigler.
Charles S. Peirce proposed an economic theory of scientific experimentation in 1876, which sought to maximize the precision of the estimates.
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.
(Kirstine Smith had been a student of the Danish statistician Thorvald N. Thiele and was working with Karl Pearson in London.)
Optimal block designs are discussed in the advanced monograph by Shah and Sinha and in the survey-articles by Cheng and by Majumdar.
