Design of Experiments

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Design of Experiments

 

the branch of mathematical statistics dealing with the efficient organization of measurements that are subject to random errors. Experimental design usually conforms to the following scheme. A function f(θ, x) is to be measured with random errors; f(θ, x) depends on unknown parameters (the vector θ) and on variables x that, at the experimenter’s choice, can assume values from some admissible set X. The goal of the experiment is usually (1) to estimate some or all of the parameters or functions of the parameters or (2) to test some hypotheses about the parameters. Given the goal of the experiment, we formulate an optimality criterion for its design. By the design of an experiment we mean a selection of a set of values of the variables x in the experiment. As a rule, the parameters are estimated by the method of least squares and the hypotheses about them are tested by means of Fisher’s F-test, because of the optimal properties of these methods. With respect to the optimality criterion for a design with a given number of experiments, in both cases it turns out to be natural to select a certain function of the variances and the correlation coefficients of the estimates by the method of least squares. When f(θ, x) depends linearly on θ, the optimal design can often be constructed before the experiment is conducted; in other cases, the design is refined during the course of the experiment.

As an illustration, let us consider the determination of three weights θ1, θ2, and θ3 on a two-pan balance if the result of the mth experiment is the difference in weight in the contents of the two pans plus a random error m with mean 0 and variance σ2; that is,

where

Xim = (–1)kim

if the ith weight is in the kimth pan in the mth experiment, and xim = 0 if the ith weight is not weighed in the mth experiment. Having weighed each object separately n times (3n experiments), we can estimate the object’s weight by the method of least squares as the quantity

with variance σ2/n. For n = 8, we can achieve the same accuracy after weighing once each the eight different combinations of weights in which each weight lies either in one pan or the other; here, the estimate by the method of least squares is given by the formula

where i = 1, 2, 3.

The foundations for the design of experiments were laid by the English statistician R. Fisher in a work published in 1935. Fisher emphasized that the efficient design of experiments gives no less important a gain in accuracy than does the optimal processing of the results of measurements. The following branches of the design of experiments can be distinguished.

Factor analysis was historically the first branch. It arose in connection with agrobiological applications of the analysis of variance, a fact reflected in the terminology that has been preserved. Here the function f(θ, x) depends on a vector x of variables, or factors, with a finite number of possible values; the function characterizes the comparative effect of the values of each factor and of combinations of different factors. By algebraic and combinatorial methods there were constructed intuitively attractive designs that simultaneously and in a balanced fashion studied the effect of as great a number of factors as possible. It was subsequently shown that the constructed designs optimized certain natural characteristics of the least squares method of estimation.

The next area of experimental design—the seeking of optimal conditions for the occurrence of various processes—developed under the influence of applications in chemistry and engineering. The methods in this area are essentially modifications of ordinary methods for finding extrema while taking random errors of measurement into account.

There are methods specific to the design of screening experiments. In these experiments it is necessary to isolate the components of the vector x that most strongly affect f(θ, x). This is an important task in the initial stage of an investigation, when the number of components of x is large.

The present-day theory of the design of experiments was developed in the 1960’s. Its methods are closely associated with the theory of approximation of functions and mathematical programming. Optimal designs have been constructed and their properties studied for a broad range of models. Iterative algorithms have been developed that in many cases give satisfactory numerical solutions for problems of experimental design.

REFERENCES

Hicks, C. R. Osnovnye printsipy planirovaniia eksperimenta. Moscow, 1967. (Translated from English.)
Fedorov, V. V. Teoriia optimal’nogo eksperimenta. Moscow, 1971.

M. B. MALIUTOV

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