An Annealing Algorithm for Searching Optimal Block Designs

This paper is concerned with a computer algorithm for searching optimal block designs. The algorithm uses a technique called simulated annealing. Exchange and interchange steps are defined in a way similar to JONES and ECCLESTON (1980). Two examples are used to show the performance of the algorithm.     

Optimal spending functions for asymmetric group sequential designs

We present optimized group sequential designs where testing of a single parameter theta is of interest. We require specification of a loss function and of a prior distribution for theta. For the examples presented, we pre-specify Type I and II error rates and minimize the expected sample size over the prior distribution for theta. Minimizing the square of sample size rather than the sample size is found to produce designs with slightly less aggressive interim stopping rules and smaller maximum sample sizes with essentially identical expected sample size. We compare optimal designs using Hwang-Shih-DeCani and Kim-DeMets spending functions to fully optimized designs not restricted by a spending function family. In the examples selected, we also examine when there might be substantial benefit gained by adding an interim analysis. Finally, we provide specific optimal asymmetric spending function designs that should be generally useful and simply applied when a design with minimal expected sample size is desired.     

On the Role of Baseline Measurements for Crossover Designs under the Self and Mixed Carryover Effects Model

Summary .  It is well known that optimal designs are strongly model dependent. In this article, we apply the Lagrange multiplier approach to the optimal design problem, using a recently proposed model for carryover effects. Generally, crossover designs are not recommended when carryover effects are present and when the primary goal is to obtain an unbiased estimate of the treatment effect. In some cases, baseline measurements are believed to improve design efficiency. This article examines the impact of baselines on optimal designs using two different assumptions about carryover effects during baseline periods and employing a nontraditional crossover design model. As anticipated, baseline observations improve design efficiency considerably for two-period designs, which use the data in the first period only to obtain unbiased estimates of treatment effects, while the improvement is rather modest for three- or four-period designs. Further, we find little additional benefits for measuring baselines at each treatment period as compared to measuring baselines only in the first period. Although our study of baselines did not change the results on optimal designs that are reported in the literature, the problem of strong model dependency problem is generally recognized. The advantage of using multiperiod designs is rather evident, as we found that extending two-period designs to three- or four-period designs significantly reduced variability in estimating the direct treatment effect contrast.     

Optimal observation times in experimental epidemic processes

Summary .   This article describes a method for choosing observation times for stochastic processes to maximise the expected information about their parameters. Two commonly used models for epidemiological processes are considered: a simple death process and a susceptible-infected (SI) epidemic process with dual sources for infection spreading within and from outwith the population. The search for the optimal design uses Bayesian computational methods to explore the joint parameter-data-design space, combined with a method known as moment closure to approximate the likelihood to make the acceptance step efficient. For the processes considered, a small number of optimally chosen observations are shown to yield almost as much information as much more intensively observed schemes that are commonly used in epidemiological experiments. Analysis of the simple death process allows a comparison between the full Bayesian approach and locally optimal designs around a point estimate from the prior based on asymptotic results. The robustness of the approach to misspecified priors is demonstrated for the SI epidemic process, for which the computational intractability of the likelihood precludes locally optimal designs. We show that optimal designs derived by the Bayesian approach are similar for observational studies of a single epidemic and for studies involving replicated epidemics in independent subpopulations. Different optima result, however, when the objective is to maximise the gain in information based on informative and non-informative priors: this has implications when an experiment is designed to convince a naïve or sceptical observer rather than consolidate the belief of an informed observer. Some extensions to the methods, including the selection of information criteria and extension to other epidemic processes with transition probabilities, are briefly addressed.     

Optimal design of longitudinal data analysis using generalized estimating equation models

Longitudinal studies are often applied in biomedical research and clinical trials to evaluate the treatment effect. The association pattern within the subject must be considered in both sample size calculation and the analysis. One of the most important approaches to analyze such a study is the generalized estimating equation (GEE) proposed by Liang and Zeger, in which “working correlation structure” is introduced and the association pattern within the subject depends on a vector of association parameters denoted by ρ. The explicit sample size formulas for two‐group comparison in linear and logistic regression models are obtained based on the GEE method by Liu and Liang. For cluster randomized trials (CRTs), researchers proposed the optimal sample sizes at both the cluster and individual level as a function of sampling costs and the intracluster correlation coefficient (ICC). In these approaches, the optimal sample sizes depend strongly on the ICC. However, the ICC is usually unknown for CRTs and multicenter trials. To overcome this shortcoming, Van Breukelen et al. consider a range of possible ICC values identified from literature reviews and present Maximin designs (MMDs) based on relative efficiency (RE) and efficiency under budget and cost constraints. In this paper, the optimal sample size and number of repeated measurements using GEE models with an exchangeable working correlation matrix is proposed under the considerations of fixed budget, where “optimal” refers to maximum power for a given sampling budget. The equations of sample size and number of repeated measurements for a known parameter value ρ are derived and a straightforward algorithm for unknown ρ is developed. Applications in practice are discussed. We also discuss the existence of the optimal design when an AR(1) working correlation matrix is assumed. Our proposed method can be extended under the scenarios when the true and working correlation matrix are different.     

Planning incomplete block experiments when treatments are genetically related

Selection trials in plant and animal breeding, in incomplete blocks, are described by linear models with random effect parameters associated with treatments with known genetic covariance structure. It is now well known that the information on relatives can improve the analysis and many extensions of this model have been proposed, but no studies have been done on the consequences of this genetical relatedness among treatments for the optimality of block designs. Using a suitable optimality criterion, we show that the knowledge on relatedness may imply that the optimal design is not in the class of designs which are optimal for unrelated treatments. Implications for practical applications are discussed.     

Factorial and time course designs for cDNA microarray experiments

Microarrays are powerful tools for surveying the expression levels of many thousands of genes simultaneously. They belong to the new genomics technologies which have important applications in the biological, agricultural and pharmaceutical sciences. There are myriad sources of uncertainty in microarray experiments, and rigorous experimental design is essential for fully realizing the potential of these valuable resources. Two questions frequently asked by biologists on the brink of conducting cDNA or two-colour, spotted microarray experiments are 'Which mRNA samples should be competitively hybridized together on the same slide?' and 'How many times should each slide be replicated?' Early experience has shown that whilst the field of classical experimental design has much to offer this emerging multi-disciplinary area, new approaches which accommodate features specific to the microarray context are needed. In this paper, we propose optimal designs for factorial and time course experiments, which are special designs arising quite frequently in microarray experimentation. Our criterion for optimality is statistical efficiency based on a new notion of admissible designs; our approach enables efficient designs to be selected subject to the information available on the effects of most interest to biologists, the number of arrays available for the experiment, and other resource or practical constraints, including limitations on the amount of mRNA probe. We show that our designs are superior to both the popular reference designs, which are highly inefficient, and to designs incorporating all possible direct pairwise comparisons. Moreover, our proposed designs represent a substantial practical improvement over classical experimental designs which work in terms of standard interactions and main effects. The latter do not provide a basis for meaningful inference on the effects of most interest to biologists, nor make the most efficient use of valuable and limited resources.     

Optimal adaptive two‐stage designs for phase II cancer clinical trials

In oncology, single‐arm two‐stage designs with binary endpoint are widely applied in phase II for the development of cytotoxic cancer therapies. Simon's optimal design with prefixed sample sizes in both stages minimizes the expected sample size under the null hypothesis and is one of the most popular designs. The search algorithms that are currently used to identify phase II designs showing prespecified characteristics are computationally intensive. For this reason, most authors impose restrictions on their search procedure. However, it remains unclear to what extent this approach influences the optimality of the resulting designs. This article describes an extension to fixed sample size phase II designs by allowing the sample size of stage two to depend on the number of responses observed in the first stage. Furthermore, we present a more efficient numerical algorithm that allows for an exhaustive search of designs. Comparisons between designs presented in the literature and the proposed optimal adaptive designs show that while the improvements are generally moderate, notable reductions in the average sample size can be achieved for specific parameter constellations when applying the new method and search strategy.     

Searching for Optimal Block Designs when Block Effects are Random

An algorithm that searches for optimal block designs when block effects are random is described. Some situations in which such an algorithm will prove useful are considered. An illustrative example and some discussion are given.     

A divide-and-conquer approach to determine the Pareto frontier for optimization of protein engineering experiments

In developing improved protein variants by site-directed mutagenesis or recombination, there are often competing objectives that must be considered in designing an experiment (selecting mutations or breakpoints): stability versus novelty, affinity versus specificity, activity versus immunogenicity, and so forth. Pareto optimal experimental designs make the best trade-offs between competing objectives. Such designs are not "dominated"; that is, no other design is better than a Pareto optimal design for one objective without being worse for another objective. Our goal is to produce all the Pareto optimal designs (the Pareto frontier), to characterize the trade-offs and suggest designs most worth considering, but to avoid explicitly considering the large number of dominated designs. To do so, we develop a divide-and-conquer algorithm, Protein Engineering Pareto FRontier (PEPFR), that hierarchically subdivides the objective space, using appropriate dynamic programming or integer programming methods to optimize designs in different regions. This divide-and-conquer approach is efficient in that the number of divisions (and thus calls to the optimizer) is directly proportional to the number of Pareto optimal designs. We demonstrate PEPFR with three protein engineering case studies: site-directed recombination for stability and diversity via dynamic programming, site-directed mutagenesis of interacting proteins for affinity and specificity via integer programming, and site-directed mutagenesis of a therapeutic protein for activity and immunogenicity via integer programming. We show that PEPFR is able to effectively produce all the Pareto optimal designs, discovering many more designs than previous methods. The characterization of the Pareto frontier provides additional insights into the local stability of design choices as well as global trends leading to trade-offs between competing criteria.     

An optimal shape problem related to the realistic design of river fishways

A river fishway is a hydraulic structure that facilitates fish in overcoming obstacles (dams, waterfalls, etc.) to their spawning and other migrations in rivers. In this work we present a mathematical formulation of an optimal design problem for a vertical slot fishway, where the state system is given by the 2D shallow water equations fixing the height and velocity of water, the design variables are the geometry of the slots, and the objective function is determined by the existence of rest areas for fish and of a water velocity suitable for fish swimming capability. We also derive an expression for the gradient of the objective function via the adjoint system. From the numerical point of view, we present a characteristic-Galerkin method for solving the shallow water equations, and a direct search algorithm for the computation of the optimal design variables. Finally, we give numerical results obtained for a standard ten pools channel.     

Dual Designs and Their Application in Genetical Experiments

The dual of incomplete block designs has been studied with th́eir applications in genetical experiments. Partial diallel crosses (PDC) of type I have been constructed using balanced incomplete block (BIB) designs, partially balanced incomplete block (PBIB) designs and their dual designs. Simplified analysis of PDC has been presented using the dual property of these designs. List of optimal PDC having simple analysis has been given.     

Flexible Designs for Genomewide Association Studies

Summary .  Genomewide association studies attempting to unravel the genetic etiology of complex traits have recently gained attention. Frequently, these studies employ a sequential genotyping strategy: A large panel of markers is examined in a subsample of subjects, and the most promising markers are genotyped in the remaining subjects. In this article, we introduce a novel method for such designs enabling investigators to, for example, modify marker densities and sample proportions while strongly controlling the family-wise type I error rate. Loss of efficiency is avoided by redistributing conditional type I error rates of discarded markers. Our approach can be combined with cost optimal designs and entails a greater flexibility than all previously suggested designs. Among other features, it allows for marker selections based upon biological criteria instead of statistical criteria alone, or the option to modify the sample size at any time during the course of the project. For practical applicability, we develop a new algorithm, subsequently evaluate it by simulations, and illustrate it using a real data set.     

Efficient algorithms for optimal designs with correlated observations in pharmacokinetics and dose-finding studies

Random effects models are widely used in population pharmacokinetics and dose-finding studies. However, when more than one observation is taken per patient, the presence of correlated observations (due to shared random effects and possibly residual serial correlation) usually makes the explicit determination of optimal designs difficult. In this article, we introduce a class of multiplicative algorithms to be able to handle correlated data and thus allow numerical calculation of optimal experimental designs in such situations. In particular, we demonstrate its application in a concrete example of a crossover dose-finding trial, as well as in a typical population pharmacokinetics example. Additionally, we derive a lower bound for the efficiency of any given design in this context, which allows us on the one hand to monitor the progress of the algorithm, and on the other hand to investigate the efficiency of a given design without knowing the optimal one. Finally, we extend the methodology such that it can be used to determine optimal designs if there exist some requirements regarding the minimal number of treatments for several (in some cases all) experimental conditions.     

Decision-theoretic designs for phase II clinical trials allowing for competing studies

This article describes an approach to optimal design of phase II clinical trials using Bayesian decision theory. The method proposed extends that suggested by Stallard (1998, Biometrics 54, 279-294) in which designs were obtained to maximize a gain function including the cost of drug development and the benefit from a successful therapy. Here, the approach is extended by the consideration of other potential therapies, the development of which is competing for the same limited resources. The resulting optimal designs are shown to have frequentist properties much more similar to those traditionally used in phase II trials.     

Four Factor Balanced Optimal Designs of Resolution V in Terms of a Modified Trace Criterion

Balanced optimal two level fractional factorial designs of resolution V have been reported in series of papers by SRIVASTAVA and CHOPRA (1971a, 1971b, 1974) and CHOPRA and SRIVASTAVA (1973a, 1973b). The optimality criterion considered by them is the trace optimality which implies minimisation of the average variance of the estimated effects which include the general mean μ, main effects, A_i, and first order interactions, A_ij. In the present work some new optimal designs have been obtained on the basis of a modified criterion of trace optimality. The criterion requires minimisation of the average variance of each of the main effects, two factor interactions, etc. separately instead of minimising average variance of all the estimated effects.     

应用试验设计提高蝮蛇抗栓酶产量的研究

试验设计是研究使试验过程科学化、合理化的一门学科。本文介绍了应用试验设计方法寻找生产蝮蛇抗栓酶的最佳工艺,从而有效地提高了产品产量,并取得明显经济效益     

Bayesian optimal designs for Phase I clinical trials

A broad approach to the design of Phase I clinical trials for the efficient estimation of the maximum tolerated dose is presented. The method is rooted in formal optimal design theory and involves the construction of constrained Bayesian c- and D-optimal designs. The imposed constraint incorporates the optimal design points and their weights and ensures that the probability that an administered dose exceeds the maximum acceptable dose is low. Results relating to these constrained designs for log doses on the real line are described and the associated equivalence theorem is given. The ideas are extended to more practical situations, specifically to those involving discrete doses. In particular, a Bayesian sequential optimal design scheme comprising a pilot study on a small number of patients followed by the allocation of patients to doses one at a time is developed and its properties explored by simulation.     

Optimal designs for a multiresponse Emax model and efficient parameter estimation

The aim of dose finding studies is sometimes to estimate parameters in a fitted model. The precision of the parameter estimates should be as high as possible. This can be obtained by increasing the number of subjects in the study, N, choosing a good and efficient estimation approach, and by designing the dose finding study in an optimal way. Increasing the number of subjects is not always feasible because of increasing cost, time limitations, etc. In this paper, we assume fixed N and consider estimation approaches and study designs for multiresponse dose finding studies. We work with diabetes dose–response data and compare a system estimation approach that fits a multiresponse Emax model to the data to equation‐by‐equation estimation that fits uniresponse Emax models to the data. We then derive some optimal designs for estimating the parameters in the multi‐ and uniresponse Emax model and study the efficiency of these designs.     

Optimal complete block designs to adjust for interplot competition with a covariance analysis

Optimal complete block designs are given for variety trials with interplot competition. In the paper, competition is modeled using height differences between neighboring varieties. When variety heights are known at the design stage, a simulated annealing algorithm is proposed. The optimal designs tend to reduce height differences between neighboring varieties. When variety heights are not known in advance, neighbor-balanced designs at distances 1 and 2 are shown to have optimality properties.