The construction of optimal designs for dose-escalation studies

Methods for the construction of A-, MV-, D- and E-optimal designs for dose-escalation studies are presented. Algebraic results proved elusive and explicit expressions for the requisite optimal designs are only given for a restricted class of traditional designs. Recourse to numerical procedures and heuristics is therefore made. Complete enumeration of all possible designs is discussed but is, as expected, highly computer intensive. Two exchange algorithms, one based on block exchanges and termed the Block Exchange Algorithm and the other a candidate-set-free algorithm based on individual exchanges and termed the Best Move Algorithm, are therefore introduced. Of these the latter is the most computationally effective. The methodology is illustrated by means of a range of carefully selected examples.     

An evolutionary algorithm for A-optimal incomplete block designs

Evolutionary algorithms are heuristic stochastic search and optimization techniques with principles taken from natural genetics. They are procedures mimicking the evolution process of an initial population through genetic transformations. This paper is concerned with the problem of finding A-optimal incomplete block designs for multiple treatment comparisons represented by a matrix of contrasts. An evolutionary algorithm for searching optimal, or nearly optimal, incomplete block designs is described in detail. Various examples regarding the application of the algorithm to some well-known problems illustrate the good performance of the algorithm     

COMPUTER-AIDED CONSTRUCTION OF SMALL (M, S)-OPTIMAL INCOMPLETE BLOCK DESIGNS

A new exchange algorithm for the construction of (M, S)-optimal incomplete block designs (IBDS) is developed. This exchange algorithm is used to construct 973 (M, S)-optimal IBDs (v, k, b) for v= 4,…,12 (varieties) with arbitrary v, k (block size) and b (number of blocks). The efficiencies of the “best” (M, S)-optimal IBDs constructed by this algorithm are compared with the efficiencies of the corresponding nearly balanced incomplete block designs (NBIBDs) of Cheng(1979), Cheng & Wu (1981) and Mitchell & John(1976).     

E- and R-optimality of block designs for treatment-control comparisons

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple linear constraints. Based on this characterization, we obtain a class of E-optimal exact designs for unequal block sizes. In the studied model, we provide a statistical interpretation for wide classes of E-optimal designs. Moreover, we show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.     

Optimal Designs for 2 k Choice Experiments

Abstract

In this article we establish the choice sets in the D-optimal design for a choice experiment for testing main effects and for testing main effects and two-factor interactions, when there are k attributes, each with two levels, for choice set size m. We also give a method to construct optimal and near-optimal designs with small numbers of choice sets.     

Marginally restricted D-optimal designs for correlated observations

Two practical degrees of complexity may arise when designing an experiment for a model of a real life case. First, some explanatory variables may not be under the control of the practitioner. Secondly, the responses may be correlated. In this paper three real life cases in this situation are considered. Different covariance structures are studied and some designs are computed adapting the theory of marginally restricted designs for correlated observations. An exchange algorithm given by Brimkulov's algorithm is also adapted to marginally restricted D–optimality and it is applied to a complex situation.     

Block designs for incomplete factorial treatment structures with two factors

Abstract

In this paper, the problem of obtaining efficient block designs for incomplete factorial treatment structure with two factors excluding one treatment combination for estimation of dual versus single treatment contrasts is considered. The designs have been obtained using the A-optimal completely randomized designs and modified strongest treatment interchange algorithm. A catalog of efficient block designs has been prepared for m₁?=?3, 4 and m₂?=?2, b?≤?10 and k?≤?9 and for m₁?=?3,4 and m₂?=?3, 4, b?≤?10 and k?≤?10.     

An algorithm for constructing optimal resolvable incomplete block designs

This paper describes an efficient algorithm for the construction of optimal or near-optimal resolvable incomplete block designs (IBDs) for any number of treatments v < 100. The performance of this algorithm is evaluated against known lattice designs and the 414 or-designs of Patterson & Williams [36]. For the designs under study, it appears that our algorithm is about equally effective as the simulated annealing algorithm of Venables & Eccleston [42]. An example of the use of our algorithm to construct the row (or column) components of resolvable row-column designs is given.     

AN ALGORITHM FOR THE CONSTRUCTION OF OPTIMAL OR NEAR-OPTIMAL CHANGE-OVER DESIGNS

This paper presents an algorithm for the construction of optimal or near optimal change-over designs for arbitrary numbers of treatments, periods and units. Previous research on optimality has been either theoretical or has resulted in limited tabulations of small optimal designs. The algorithm consists of a number of steps:first find an optimal direct treatment effects design, ignoring residual effects, and then optimise this class of designs with respect to residual effects. Poor designs are avoided by judicious application of the (M, S)-optimality criterion, and modifications of it, to appropriate matrices. The performance of the algorithm is illustrated by examples.     

A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

Food shelf life: estimation and optimal design

Shelf life is a specified percentile of the time-until-spoilage distribution of a food product. This paper investigates statistical properties of various estimators of shelf life and develops a genetic algorithm for finding near-optimal staggered designs for estimation of shelf life. MLEs and their associated confidence intervals for shelf life have smaller bias, better performance, and better coverage than the corresponding ad hoc regression-based estimates. However, performance of MLEs for common sample sizes must be evaluated by simulation. The genetic algorithm, coded as an SAS macro, searched the design space well and generated near-optimal designs as measured by improvement to a simulation-based performance measure.     

A comparison study of effectiveness and robustness of robust economic designs of T2 chart using genetic algorithm

ABSTRACT

Recently, researchers have tried to design the T² chart economically to achieve the minimum possible quality cost; however, when T² chart is designed, it is important to consider multiple scenarios. This research presents the robust economic designs of the T² chart where there is more than one scenario. An illustrative example is used to demonstrate the effect of the model parameters on the optimal designs. The genetic algorithm optimization method is employed to obtain the optimal designs. Simulation studies show that the robust economic designs of T² chart are more effective than traditional economic design in practice.     

Some optimal block designs with nested rows and columns for research on alternative methods of limiting slug damage

Some criteria of optimality of a block design with nested rows and columns are considered. The criteria are based on the eigenvalues of the information matrix C or on the eigenvalues of the matrix C with respect to a diagonal matrix R of treatment replications. New constructions of some optimal block designs with nested rows and columns are presented for application to special plant protection experiments.     

AN ALGORITHM FOR CONSTRUCTING OPTIMAL RESOLVABLE ROW-COLUMN DESIGNS

This paper describes an effective algorithm for constructing optimal or near-optimal resolvable row-column designs (RCDs) with up to 100 treatments. The performance of this algorithm is assessed against 20 2-replicate resolvable RCDs of Patterson & Robinson (1989) and 17 resolvable RCDs based on generalized cyclic designs (GCDs) of Ipinyomi & John (1985). The use of the algorithm to construct RCDs with contiguous replicates is discussed.     

Optimal balanced block designs for correlated observations

The construction of universally optimal designs, if such exist, is difficult to obtain, especially when there are some nuisance effects or correlated errors. The hub correlation is a special correlation structure with applications to experiments in genetics, networks and other areas in industry and agriculture. There may be restrictions on the correlation values of the hub structure depending on the experiment. Optimality of block designs under hub correlation has been studied for the case of a constant correlation value. In this article, we consider the hub structure when one of the correlation values is different from the others, and the universally optimal block designs, binary or non-binary, are theoretically obtained. Also, we introduce an algorithm to construct the optimal designs. The Canadian Journal of Statistics 48: 596–604; 2020 © 2020 Statistical Society of Canada     

An Optimal Reliability and Maintainability Design of a Searching System

In this article, we deal with an optimal reliability and maintainability design problem of a searching system with complex structures. The system availability and life cycle cost are used as optimization criteria and estimated by simulation. We want to determine MTBF (Mean Time between Failures) and MTTR (Mean Time to Repair) for all components and ALDT (Administrative and Logistics Delay Times) of the searching system in order to minimize the life cycle cost and to satisfy the target system availability. A hybrid genetic algorithm with a heuristic method is proposed to find near-optimal solutions and compared with a general genetic algorithm.     

COMPUTER AIDED-CONSTRUCTION OF D-OPTIMAL 2m FRACTIONAL DESIGNS OF RESOLUTION V

A new exchange algorithm for construction of 2^mD-optimal fractional factorial design (FFD) is devised. This exchange algorithm is a modification of the one due to Fedorov (1969, 1972) and is an improvement over similar algorithm due to Mitchell (1974) and Galil & Kiefer (1980). This exchange algorithm is then used to construct 54 D-optimal 2^m-FFD's of resolution V for m = 4,5,6.     

A-Optimal and MV-Optimal Incomplete Block Designs for Comparing Successive Treatments

Consider an incomplete block experiment in which observations are taken from t treatments using an incomplete block design with b blocks of size k < t. Suppose the interest is in estimating the differences of effects of successive treatments. This may occur, for example, if the treatments are different dosages or concentrations of a compound. This article presents A-optimal and MV-optimal incomplete block designs for estimating the the differences of successive treatment effects. Tables of optimal designs are given for k < t ≤ 5 with b ≤ 40.