Design of experiments for parameter estimation

The best known application of experiment design is to find the extremum of a quantity depending on further variables by observing its value at appropriately selected points (refs. 43-46). In this section, however, consideration is restricted to design methods, purported to increase the reliability of estimates when fitting a model to observations. 

A - point design is described by the design matrix Xk, consisting of rows. The i-th row of the matrix specify the values of the the independent variables to be selected in the i-th experiment. Depending on the linearity or nonlinearity of the model, the design matrix affects the covariance matrix Cp of the estimates according to the expressions (3.30) and (3.45), respectively. The covariance matrix, in turn, determines the joint confidence region (3.32) 

To obtain a meaningful extremum problem the number of experiments and the set of feasible vectors of the independent variables T are fixed. In most cases T is defined by inequalities x1- x x, i = l,2.k. Though introducing penalty functions such constrained extremum problems can be solved by the methods and modules described in Section 2.4, this direct approach is usually very inefficient. In fact, experiment design is not easy. The dimensionality of the extremum problem is high, the extrema are partly on the boundaries of the feasible region T, and since the objective functions are 

Example 3.10.2 Approximate D - optimal design for estimating Michaelis-Menten parameters 

Starting with the substrate values x = [S ] in Table 3.4, we construct a nearly D - optimal design to estimate the parameters of the response function 

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Draper, N. R., and W. G. Hunter, Design of experiments for parameter estimation in multiresponse situations, Biometrika, 53, 525-533 (1966). 

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