# Optimal test planning

The optimal experimental design , also called optimal experimental design , is a branch of the statistical experimental design .

The aim is to construct experimental plans so that z. B. With a given number of attempts, the identification of the unknown variables of interest in terms of a suitable optimality criterion succeeds as best as possible.

The optimal design of experiments for linear models , especially linear regression models , is particularly advanced and mathematically mature . For non-linear models there are also approaches to optimal test planning; in a special case see e.g. B. under model-based test planning .

## Example: simple linear regression

The variances of the estimated regression parameters (slope and absolute term of the regression line) depend on the values ​​of the independent variables at the observation points. If the values ​​of the independent variables can be adjusted within a certain test range, they should be selected so that the estimated regression parameters have the smallest possible variances. If you z. B. want to make 10 attempts and can set the values ​​in the interval , it is usually z. B. not optimal to distribute these 10 attempts evenly over . It is best to carry out 5 attempts at and 5 attempts at . Then you can, which is also clearly visible, determine the straight line “most securely”. ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle [a, b]}$ ${\ displaystyle [a, b]}$ ${\ displaystyle x = a}$ ${\ displaystyle x = b}$ ## Optimal design of experiments in the general linear regression model

The unknown parameters in a classical model of linear multiple regression are estimated true to expectation and optimally for variance using the least squares estimation (see Gauss-Markov theorem ). Optimal variance here means: with the smallest covariance matrix in the sense of the Loewner partial order . The inverse of this covariance matrix is ​​called the (Fisherman's) information matrix . Since these matrices depend on the independent variables at the observation points, all common optimality criteria are functionals of the information matrix or the covariance matrix of the least squares estimator .

## Examples of optimality criteria

D-optimality: maximizing the determinant of the information matrix. This criterion leads to a minimization of the volume of the confidence ellipsoid for the unknown parameters of the linear regression model.

A-optimality: minimizing the trace of the inverse information matrix, leads to the minimization of the mean variance of the estimated parameters.

E-optimality: Maximizing the minimum eigenvalue of the information matrix , leads to the minimization of the maximum possible variance of the components of the estimated parameter vector.

Some criteria relate to the variance of a prediction in the linear regression model, e.g. B. the

G-optimality: leads to the minimization of the maximum possible variance of the forecast in a specified forecast range.

## Theory of optimal design of experiments

Initially, the dependency on the number of tests is dispensed with and test plans are viewed as a standardized measure over the test range. For optimality criteria, which are functionals of the information matrix, one can restrict oneself to discrete test plan dimensions, see and find optimal plans with the means of convex optimization . So-called equivalence theorems play a major role here, see, which provide an equivalent optimality criterion for a given criterion, which often provides options for the iterative determination of approximately optimal plans. For the concrete application of an optimal test plan dimension for a given sample size , the weights of this (discrete) plan dimension have to be rounded to multiples of . ${\ displaystyle n}$ ${\ displaystyle n}$ ${\ displaystyle 1 / n}$ ## Historical development

Kirstine Smith (1918) already used the criterion of G-optimality, see. One can speak of a theory of optimal test planning since Gustav Elfving (1952), see, but especially since Jack Kiefer (1959).

In Germany, in particular, Hans Walter Bandemer (1973) (see), Friedrich Pukelsheim (1980) (see) and H. Dette contributed to the development of optimal test planning, see e.g. B.

## Standard works

• VV Fedorov: Theory of Optimal Experiments. Academic Press, New York 1972
• SD Silvey: Optimal design. Chapman and Hall, London 1980
• A. Pazman: Foundations of Optimum Experimental Design. Reichel, Dordrecht 1986
• AC Atkinson, AN Donev: Optimum Experimental Design. Clarendon Press, Oxford 1992
• F. Pukelsheim: Optimal Design of Experiments. Wiley, New York 1993

German language works

• H. Bandemer, (with a collective of authors): Theory and application of optimal test planning. Volume 1, Akademieverlag, Berlin 1977.
• O. Krafft: Linear statistical models and optimal test plans. Vandenhoeck & Ruprecht, Göttingen 1978.
• H. Bandemer, W. Näther: Theory and application of optimal test planning. Volume 2, Akademieverlag, Berlin 1980.

## Individual evidence

1. J. Kiefer: Optimum experimental design. In: Journal of Royal Statistical Society, Ser. B. 21, 1959, pp. 272-319.
2. J. Kiefer: General equivalence theory for optimum designs (approximate theory). In: Annals of Statistic. 2, 1974, pp. 849-879.
3. K. Smith: On the standard deviations of adjusted and interpolated values ​​of an observed polynomial function and its constants and the guidance they give towards a proper choice of the distribution of observations. In: Biometrika. 12, 1918, pp. 1-85.
4. G. Elfving: optimum allocation in linear regression theory. In: Annals of Mathematical Statistics. 23, 1952, pp. 255-262.
5. Hans Bandemer, Andreas Bellmann, Wolfhart Jung, Klaus Richter: Optimal experimental planning. Akademieverlag Berlin 1973.
6. ^ F. Pukelsheim: On linear regression designs which maximize information. In: Journal of Statistical Planning and Inference. 16, 1980, pp. 339-364.
7. H. Dette: A generalization of D- and D 1 -optimal designs in polynomial regression. In: Annals of Statistics. 18, 1990, pp. 1784-1804.