An algorithm for finding efficient test-control block designs with correlated observations

Prior to the 1970's a substantial literature had accumulated on the theory of optimal design, particularly of optimal linear regression design. To a certain extent the study of the subject had been piecemeal, different criteria of optimality having been studied separately. Also to a certain extent the topic was regarded as being largely of theoretical interest and as having little value for the practising statistician. However during this decade two significant developments occurred. It was observed that the various different optimality criteria had several mathematical properties in common; and general algorithms for constructing optimal design measures were developed. From the first of these there emerged a general theory of remarkable simplicity and the second at least raised the possibility that the theory would have more practical value. With respect to the second point there does remain a limiting factor as far as designs that are optimal for parameter estimation are concerned, and this is that the theory assumes that the model be collected is known a priori. This of course underlying data to is seldom the case in practice and it often happens that designs which are optimal for parameter estimation allow no possibility of model validation. For this reason the theory of design for parameter estimation may well have to be combined with a theory of model validation before its practical potential is fully realized. Nevertheless discussion in this monograph is limited to the theory of design optimal for parameter estimation.

Heyman’s theorem and the Prager-Marcal-Shield theory of optimal plastic design are generalized to include the cost of joints (connections) in the optimization procedure. The significance of the cost of joints was emphasized by Witteveen and Tolman at a International union of Theoretical and Applied Mechanics Symposium and by Shanley in the context of aircraft structures.

The theory of optimal experimental designs is concerned with the construction of designs that are optimum with respect to some statistical criteria. Some of these criteria include the alphabetic optimality criteria such as; D-, A-, E-, T-, G- and C- criterion. Compound optimality criteria are those that combine two or more alphabetic optimality criteria. Design that require optimality criteria have specific desired properties that do very well in one design and at the same time perform poorly in another design. Thus, a compound optimality criterion gives a balance to the desirability of any two or more alphabetic optimality criteria. The present paper aims to introduce CD- and DT- criteria which are compound optimality criteria for second order rotatable designs constructed using Balanced Incomplete Block Designs (BIBDs) in four dimensions.

29 Review of optimal bayes designs

Since most problems treated in Part II concern plastic design, it will be useful to review a general theory of optimal plastic design, which was initiated by Marcal and Prager30, and Prager and Shield31, and extended by Charrett and Rozvany32, and Save33.

We consider a dose-finding trial in phase IIB of drug development. For choosing an appropriate design for this trial the specification of two points is critical: an appropriate model for describing the dose-effect relationship, and the specification of the aims of the trial (objectives), which will be the focus in the present paper. For many situations it is essential to have a robust trial objective that has little risk of changing during the complete trial due to external information. An important and realistic objective of a dose-finding trial is to obtain precise information about key parts of the dose-effect curve. We reflect this goal in a statistical optimality criterion and derive efficient designs using optimal design theory. In particular, we determine nonadaptive Bayesian optimal designs, i.e., designs which are not changed by information obtained from an interim analysis. Compared with a traditional balanced design for this trial, it is shown that the optimal design is substantially more efficient. This implies either a gain in information, or essential savings in sample size. Further, we investigate an adaptive Bayesian optimal design that uses different optimal designs before and after an interim analysis, and we compare the adaptive with the nonadaptive Bayesian optimal design. The basic concept is illustrated using a modification of a recent AstraZeneca trial.

For the class of repeated measurements designs based on $t$ treatments, $n$ experimental units and two periods, the following results are obtained. 1. The equivalence of the information matrices of such repeated measurements designs and of certain block designs is established. The implication of this equivalence on the optimality of both repeated measurements designs and block designs is explored. 2. A family of universally optimal designs or $A$-optimal designs is constructed depending whether or not $n$ divides $t$. 3. Families of optimal designs for residual effects and for comparing test treatments with a control are constructed.

The selection of items from a calibrated item bank for fixed-form tests is an optimal test design problem; this problem has been handled in the literature by mathematical programming models. A similar problem, however, arises when items are selected for an adaptive test or for testlets. This paper focuses on the similarities of optimal design of fixed-form tests, adaptive tests, and testlets within the framework of the general theory of optimal designs. A sequential design procedure is proposed that uses these similarities. This procedure not only enables optimal design of fixed-form tests, adaptive tests, and testlets, but is also very flexible. The procedure is easy to apply, and consistent estimates for the trait level distribution are obtained. Index terms: adaptive tests, consistency, efficiency, optimal test design, sequential procedure, test design, testlets.

In experiments with mixtures that involve process variables, if the response function is expressed as the sum of a function of mixture components and a function of process variables, then the parameters in the mixture part and in the process part can be estimated independently using orthogonal block designs. This paper is concerned with such a block design for parameter estimation in the mixture part of a quadratic mixture model for three mixture components. The behaviour of the eigenvalues of the moment matrix of the design is investigated in detail, the design is optimized according to E- and Aoptimality criteria, and the results are compared together with a known result on Doptimality. It is found that this block design is robust with respect to these diff erent optimality criteria against the shifting of experimental points. As a result, we recommend experimental points of the form (a, b, c) in the simplex S2, where c=0, b=1-a, and a can be any value in the range 0.17+/-0.02.

Motivated by a question of C.-S. Cheng on optimal block designs, this paper describes the symmetric matrices with entries 0, +1 and −1, zero diagonal, least eigenvalue strictly greater than −2, and constant row sum. I also describe briefly the motivation for the question. Let A be a symmetric n× n matrix with entries 0, +1 and −1, with zero diagonal and constant row sums, having least eigenvalue greater than −2. The aim of this paper is to describe such matrices. Of course, we may assume that the matrix is “connected”, that is, not permutation-equivalent to one of the form ( B O O C ) . We also note that the constant row sum c is an eigenvalue, and so c ≥ −1. Such a matrix is represented by a set of vectors in a spherical root system, and hence (apart from finitely many examples represented in the exceptional root systems E6, E7 and E8) by either a tree with oriented edges, or a unicyclic graph with edges either signed or oriented. We give a test for recognising when such a graph represents a matrix satisying the conditions of the question. There are many examples. All matrices occurring in the exceptional root systems are determined. 1 Least eigenvalue −1 As a warmup, I consider the case where the least eigenvalue is −1.

Optimal block designs for comparing test treatments with a control when k> υ

The electrical behaviour of a system, such as an electrode–tissue interface (ETI) or a biological tissue, can be used for its characterization. One way of accomplishing this goal consists of measuring the electrical impedance, that is, the opposition that a system exhibits to an alternating current flow as a function of frequency. Subsequently, experimental impedance data are fitted to an electrical equivalent circuit (EEC model) whose parameters can be correlated with the electrode processes occurring in the ETI or with the physiological state of a tissue. The EEC used in this paper is a reasonable approach for simple bio-electrodes or cell membranes, assuming ideal capacitances. We use the theory of optimal experimental design to identify the frequencies in which the impedance is measured, as well as the number of measurement repetitions, in such a way that the EEC parameters can be optimally estimated. Specifically, we calculate approximate and exact D-optimal designs by optimizing the determinant of the information matrix by adapting two of the most algorithms that are routinely used nowadays (REX random exchange algorithm and KL exchange algorithm). The D-efficiency of the optimal designs provided by the algorithms was compared with the design commonly used by experimenters and it is shown that the precision of the parameter estimates can be increased.

The theory of optimal design has proved its importance in many applications in engineering and applied mathematics. The redesign of old problems such as a capacitor in an ionized medium has motivated new studies and new plasma diagnostic devices. Here we present some examples of optimal design of electrodes in Electrostatics and Electromagnetism with applications in plasma physics. In our first example, we present a study of the interaction between a uniform plasma and a planar substrate. In order to increase the uniformity of etching and deposition processes, we suggest an optimal design for substrate-holders by using a guard-ring in the shape of a ‘doughnut’. The possible consequences on plasma processes are analyzed and discussed. We would also like to illustrate the techniques of optimal design in space science for modifying the design of electrodes for a planar plasma capacitor (a plasma capacitor is a device for measuring the electron density in the Ionosphere). This reduces edges effects. Other techniques of optimal design are presented, such as the measurement of the conductivity in the Stratosphere by Gerdien condensers. Finally, although our last example does not involve numerical simulations, the interaction plasma-object is of great importance in space science from telecommunications satellites to scientific instruments. A great danger for scientific instruments comes from the charging of insulators. Many devices are supported by a boom, which is often an insulator. We suggest different techniques for hiding an insulator.

This paper deals with the optimal design of monolithic piezoelectric microactuators with integrated proprioceptive sensors. Dedicated to the microrobotic and micromechatronic fields, this work details the modelling and the characterization of compliant structures with integrated actuating and sensing elements. The proposed optimal design procedure addresses not only static criteria but also dynamic ones. This leads to microdevices which are better performing with regards to mechanical (displacement, force, etc) and control (dynamics, stability, precision) characteristics. The efficient design of such devices is achieved using a flexible building block method. A topological optimization method combined with an evolutionary algorithm is used to optimize the design of a truss-like planar structure. This method chooses the best location among the different piezoelectric elements. Different mechanical, actuation or sensing elements are accordingly chosen from a data bank. From the control point of view, optimization criteria are considered, to enforce the observability of the vibrational dominant modes of the structure. Therefore, control and observation Gramians are exploited in the optimal design to shape the open-loop frequency response of both actuation and sensing functions of the integrated device. In the last part of the paper, based on these results, the optimal design and manufacture of an innovative piezoelectric flexible microgripper is proposed. The prototype is manufactured from a monolithic piezoelectric material (PIC 151). Its reduced size (15 mm × 18 mm) fits the requirements of both microrobotic and micromechatronic applications, and it is suitable for micromanipulation tasks. Closing the paper, the characterization and the performance of this integrated microactuator and the efficiency of the optimal design procedure for micromechatronic applications are shown.

Improving parameter estimation with an efficient sequential probabilistic collocation-based optimal design method