Algorithms for the Construction of Constrained and Unconstrained Optimal Designs

Chapter 1 provides an introduction to the area of optimum experimental design for the linear regression design problem with parameter vector theta. This problem seeks to obtain a best inference for all or some of the components of theta by making the dispersion matrix of their estimates small in some sense. In this chapter we summarise the main criteria used for this purpose. Chapter 2 studies a class of multiplicative algorithms of the form Pi(r+1)=pi(r)(di,delta) / SigmaJipj(r)f(dj,delta) indexed by a function f(d,delta) which depends on the derivatives of the criterion &phis;(p) and a free parameter delta for solving problem (a). The performance of the algorithm is investigated in constructing D-optimal designs under optimal choices of the parameter delta, and in constructing c-optimal designs starting from difficult initial designs, using an optimal and fixed value of the parameter delta. The work for this chapter has appeared in Torsney and Alahmadi (1992). Chapter 3 considers the covariance and correlation criterion of problem(b). The only property we know of these criteria is homogeneity in the weights p of degree -2 and zero respectively. This type of criterion differs from the standard optimality criteria such as c-, D- and A-optimality criteria. It may have negative first partial derivatives. An explicit solution has been found for the optimal weights and the optimal value for the covariance criterion when the number of design points equals the number of parameters i.e J = k, while in the case when J > k we have explored a new version of the above algorithm for dealing with this type of problem. Chapters 4 and 5 are concerned with the solution of problem (c). In Chapter 4 we consider the case when the number of design points equals the number of the unknown parameters theta. In this case we find a class of designs which guarantees zero covariance. Zero covariance is guaranteed under a transformation of the design weights p to two or three sets of variables each of which forms a probability vector. We wish to maximise standard design criteria with respect to these weights. This yields an extension of problem (a) of Chapter 2 to that of maximising a criterion with respect to two probability vectors and we use a natural extension of the algorithm used for that problem. For the above mentioned results the efficiencies of the restricted optimal design under the zero covariance constraint relevant to the unrestricted optimal design has been calculated. Chapter 5 considers the case when the number of design points exceeds the number of the parameters. Using a Lagrangian approach, the problem is transformed to one of simultaneous maximisation of two functions of the same probability vector each of which is maximised at the same value of this vector and have a common maximum of zero. This yields another extension of problem (a). Chapter 6 summarises the results obtained in the preceding chapters as well as giving an indication of future work that could be done