Bayesian D-optimal Designs Research Articles

Penalized Bayesian D-Optimal Designs for Regression Models of Continuous Response

We propose extending the use of desirability functions in Bayesian optimal designs for regression models. This technique generates experimental designs with good statistical inference properties according to Bayesian optimal design theory and practical features, as defined by an investigator. These practical features are defined by a penalty function, using an overall desirability function, which is added to a Bayesian $D$-optimal design criterion to penalize impractical experimental designs. This methodology is illustrated by two examples of regression models: quadratic and exponential. Then, we compare designs obtained for different prior distributions of unknown parameters by efficiency calculations and simulation study. Results show that the $D$-efficiencies of the penalized designs relative to the non-penalized Bayesian D-optimal designs are competitive.

Bayesian D-Optimal Design of Experiments with Quantitative and Qualitative Responses

Systems with both quantitative and qualitative responses are widely encountered in many applications. Design of experiment methods are needed when experiments are conducted to study such systems. Classic experimental design methods are unsuitable here because they often focus on one type of response. In this paper, we develop a Bayesian D-optimal design method for experiments with one continuous and one binary response. Both noninformative and conjugate informative prior distributions on the unknown parameters are considered. The proposed design criterion has meaningful interpretations regarding the D-optimality for the models for both types of responses. An efficient point-exchange search algorithm is developed to construct the local D-optimal designs for given parameter values. Global D-optimal designs are obtained by accumulating the frequencies of the design points in local D-optimal designs, where the parameters are sampled from the prior distributions. The performances of the proposed methods are evaluated through two examples.

A Method for Augmenting Supersaturated Designs with Newly Added Factors

Follow-up experimental designs are popularly used in industry. In practice, some important factors may be neglected for various reasons in the first-stage experiment and they need to be added in the next stage. In this paper, we propose a method for augmenting supersaturated designs with newly added factors and augmented levels using the Bayesian D-optimality criterion. In addition, we suggest using the integrated Bayesian D-optimal augmented design to plan the follow-up experiment when the newly added factors have been allowed to vary in an appropriate region. Examples and simulation results show that the augmented designs perform well in improving identified rates of latent factor effects.

Bayesian sequential design for sensitivity experiments with hybrid responses

In experimental design, a common problem seen in practice is when the result includes one binary response and multiple continuous responses. However, this problem receives scant attention. Most studies pertaining to this problem usually consider the situation in which the continuous responses are independent of the stimulus level condition on the binary response. However, in many practical applications, real data show that this conditional independent assumption is not always appropriate. This article considers a new model for the dependent situation and a corresponding sequential design is proposed under the decision-theoretic framework. To deal with the problem of complex computation involved in searching for optimal designs, fast algorithms are presented using two strategies to approximate the optimal criterion, denoted as SI-optimal design and Bayesian D-optimal design, respectively. Simulation studies based on data from a Chinese chemical material factory show that the proposed methods perform well in estimating the interesting quantiles.

Saturated and supersaturated order-of-addition designs

Van Nostrand’s pairwise order model has been the most popular model for analyzing order-of-addition experiments. Here we construct designs of minimal size for the pairwise order model, considering both D-optimality, as well as the prediction-based I- and G-optimality criteria. Even these saturated designs grow large when the number of components exceeds 10. Thus, we also propose Bayesian D-optimal designs, which may have fewer runs than the number of parameters for the pairwise order model. These supersaturated designs may be used as screening designs for applications where the number of components is large.

DOPE: D-Optimal Pooling Experimental design with application for SARS-CoV-2 screening.

ObjectiveTesting individuals for the presence of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the pathogen causing the coronavirus disease 2019 (COVID-19), is crucial for curtailing transmission chains. Moreover, rapidly testing many potentially infected individuals is often a limiting factor in controlling COVID-19 outbreaks. Hence, pooling strategies, wherein individuals are grouped and tested simultaneously, are employed. Here, we present a novel pooling strategy that builds on the Bayesian D-optimal experimental design criterion.Materials and MethodsOur strategy, called DOPE (D-Optimal Pooling Experimental design), is built on a novel Bayesian formulation of pooling. DOPE defines optimal pooled tests as those maximizing the mutual information between data and infection states. We estimate said mutual information via Monte-Carlo sampling and employ a discrete optimization heuristic to maximize it.ResultsWe compare DOPE to other, commonly used pooling strategies, as well as to individual testing. DOPE dominates the other strategies as it yields lower error rates while utilizing fewer tests. We show that DOPE maintains this dominance for a variety of infection prevalence values.DiscussionDOPE has several additional advantages over common pooling strategies: it provides posterior distributions of the probability of infection rather than only binary classification outcomes; it naturally incorporates prior information of infection probabilities and test error rates; and finally, it can be easily extended to include other, newly discovered information regarding COVID-19.ConclusionDOPE can substantially improve accuracy and throughput over current pooling strategies. Hence, DOPE can facilitate rapid testing and aid the efforts of combating COVID-19 and other future pandemics.

Optimal designs for homoscedastic functional polynomial measurement error models

This paper considers the construction of optimal designs for homoscedastic functional polynomial measurement error models. The general equivalence theorems are given to check the optimality of a given design, based on the locally and Bayesian D-optimality criteria. The explicit characterizations of the locally and Bayesian D-optimal designs are provided. The results are illustrated by numerical analysis for a quadratic polynomial measurement error model. Numerical results show that the error-variances ratio and the model parameter are the important factors for the both optimal designs. Moreover, it is shown that the Bayesian D-optimal design is more robust and effective compared with the locally D-optimal design, if the error-variances ratio or the model parameter is misspecified.

Analysis of consumer preferences for prepaid mobile internet packages in Iran: A Discrete Choice Experiment

As a first discrete choice experiment in Iran emerging telecommunication market, this paper studies consumer preferences for prepaid mobile internet packages with a combined software and paper-based interview. A two-stage Bayesian D-optimal design procedure is deployed to design choice sets of mobile internet packages from four main attributes. The utility structure and customers’ willingness-to-pay for mobile internet packages are analyzed. Findings/originality: The results indicate that even with a considerable price reduction, consumers avoid prepaying for data plans with commitment periods longer than six months and high traffic volume. Traffic volume and brand attributes are recognized as the two most influential factors on consumers’ behavior. Simulating the market demonstrates the competition between mobile internet operators in Iran market. The statistics express a significant effect of consumers’ current mobile operator on their preferences for the brand attribute.

Generalized Bayesian D-optimal supersaturated multistratum designs

Supersaturated designs are useful in the initial stage of experiments to identify important factors from many of interest with a small number of runs. Traditional supersaturated designs were mainly constructed for completely randomized experiments, which have single-stratum structures. They cannot be used for experiments that have multistratum structures, such as the split-plot, strip-plot, and staggered-level experiments. How to construct supersaturated multistratum designs for complex experiments has gained much attention recently. In this paper, we consider the situation in which the experimenters have prior knowledge of which factors are more likely to be important (called the primary factors) than the others (called the potential factors). By taking primary and potential factors into account, we propose an approach using the generalized Bayesian D (GBD) criterion to construct a new class of supersaturated multistratum designs. The GBD-optimal supersaturated multistratum designs provide guidelines on how to assign factors to the designs, which enhances efficiency on identifying active factors. A case study shows that the proposed supersaturated design (32 runs with 19 factors) is as effective as the full 26 factorial design (64 runs with 6 factors) to identify important factors in a battery cell experiment.

BAYESIAN APPROACH TO THE D-OPTIMAL FOR MIXTURE EXPERIMENTAL DESIGN

A mixture experiment is a special case of response surface methodology in which the value of the components are proportions. In case there are constraints on the proportions, the experimental region can be not a simplex. The classical designs such as a simplex-lattice design or a simplex-centroid design, in some cases, cannot fit to the problem. In this case, optimal design come up as a solution. A D-optimal design is seeking a design in which minimizing the covariance of the model parameter. Some model parameters are important and some of them are less important. As the priority of the parameters, the prior information of parameters is needed in advance. This brings to a Bayesian D-optimal design. This research was focus on a baking experiment in which consisted of three ingredients with lower bounds on the proportion of the ingredients. The assumption model was a quadratic model. Due to the priority of the model parameters, the Bayesian D-optimal design was used to solve the problem. A point-exchange algorithm was developed to find the optimal design. Nineteen candidates is used to choose twelve design points. It found that the potential term is feasible to the actual model and design points represent overall points in the design area.

Data-driven multistratum designs with the generalized Bayesian [formula omitted]-[formula omitted] criterion for highly uncertain models

Multistratum designs have gained much attention recently. Most criteria, such as the D criterion, select multistratum designs based on a given model that is assumed to be true by the experimenters. However, when the true model is highly uncertain, the model used for selecting the optimal design can be seriously misspecified. If this is the case, then the selected multistratum design will be not efficient for fitting the true model. To deal with the problem of high uncertain models, we propose the generalized Bayesian D-D (GBDD) criterion, which selects multistratum designs based on the experimental data. Under the framework of multistratum structures, we develop theorems and formula that are used for conducting Bayesian analysis and extracting information about the true model from the data to reduce model uncertainty. The GBDD criterion is easy and flexible in use. We provide several examples to demonstrate how to construct the GBDD-optimal split-plot, strip-plot, and staggered-level designs. By comparing with the D-optimal designs and one-stage generalized Bayesian D-optimal designs, we show that the GBDD-optimal designs have higher efficiency on fitting the true models. The extensions of the GBDD criterion for more complicated cases, such as more than two stages of experiments and more than one class of potential terms, are also developed.

Particle swarm based algorithms for finding locally and Bayesian D-optimal designs

When a model-based approach is appropriate, an optimal design can guide how to collect data judiciously for making reliable inference at minimal cost. However, finding optimal designs for a statistical model with several possibly interacting factors can be both theoretically and computationally challenging, and this issue is rarely discussed in the literature. We propose nature-inspired metaheuristic algorithms, like particle swarm optimization (PSO) and its variants, to solve such optimization problems. We demonstrate that such techniques, which are easy to implement, can find different types of optimal designs for models with several factors efficiently. To facilitate use of such algorithms, we provide computer codes to generate tailor made optimal designs and evaluate efficiencies of competing designs. As applications, we apply PSO and find Bayesian optimal designs for Exponential models useful in HIV studies and re-design a car-refuelling study for a Logistic model with ten factors and some interacting factors.

Non-sequential augmentation strategies to address separation in logistic regression

Previous research on small sample multi-factor D-optimal designs for the logistic regression model has demonstrated that these designs are prone to encountering separation, a phenomenon where the responses are completely or quasi-completely separable by a hyperplane in the design space. Separation causes the non-existence of maximum likelihood parameter estimates and represents a serious problem for model fitting purposes. In this paper, several non-sequential design augmentation strategies, where additional experimental trials are performed following an initial experiment that has encountered separation, are investigated. Small local and Bayesian D-optimal initial designs are generated for several representative logistic regression models, and a Monte Carlo simulation methodology is then used to investigate the effectiveness of each augmentation strategy in eliminating separation. Results of the simulation study show that augmenting design runs (trials) in regions of maximum prediction variance (MPV) is the most effective strategy for eliminating separation. However, MPV augmentation tends to produce designs with lower D-efficiencies. The paper illustrates that MPV augmentation reliably eliminates separation and can be used in practice to obtain usable parameter estimates for the logistic regression model.

Assessing the relative importance of health and conformation traits in the cavalier king Charles spaniel

BackgroundThe selection of a future breeding dog is a complicated task, in which disease characteristics and different traits have to be combined and weighed against one another. Truncation selection, that is the exclusion of affected animals, may be very inefficient when selecting on a large number of traits, and may result in a reduction of the genetic diversity in a population or breed. Selection could be facilitated by the use of a selection index that combines multiple traits or breeding values into one score. This however requires a consideration of their relative value according to their economic weight, which is difficult to express in monetary units for health traits. The use of a choice experiment to derive non-market values might be a solution to this problem. This is a pilot study to assess the potential use of choice experiments to ascertain the public preference and relative importance attached to health- and conformation traits in the selection of a Cavalier King Charles spaniel. The focus was on two prevalent disorders, mitral valve disease and syringomyelia, and on several important conformation traits such as muzzle length and eye shape. Based on available prior information, a Bayesian D-optimal design approach was used to develop a choice experiment and the resulting choice sets.ResultsEvery participant (breeder or owner) in the choice experiment was presented with a total of 17 choice sets, in which at most four traits could vary to reduce the cognitive burden. A total of 114 respondents participated in the choice experiment and results showed that respondents (breeders/owners) current attitudes were directed towards health (syringomyelia and mitral valve disease), followed by eye shape and level of inbreeding.ConclusionsThis approach identifies the value breeders and owners attach to certain traits in the breeding objective. The resulting relative weights, represented as the logworths obtained from the choice experiment, could be an alternative to economic weights. They could be implemented as a weight when breeding values are available, but more study on this topic will be necessary. A challenge in this approach is to scale up the experiment with additional traits. Moreover, for other traits, the genetic parameters and correlations should be known first, in order to include them in the health selection index as well.

Singular prior distributions in Bayesian D-optimal design for nonlinear models

For Bayesian D-optimal design, we define a singular prior distribution for the model parameters as a prior distribution such that the determinant of the Fisher information matrix has a prior geometric mean of zero for all designs. For such a prior distribution, the Bayesian D-optimality criterion fails to select a design. For the exponential decay model, we characterize singularity of the prior distribution in terms of the expectations of a few elementary transformations of the parameter. For a compartmental model and several multi-parameter generalized linear models, we establish sufficient conditions for singularity of a prior distribution. For the generalized linear models we also obtain sufficient conditions for non-singularity. In the existing literature, weakly informative prior distributions are commonly recommended as a default choice for inference in logistic regression. Here it is shown that some of the recommended prior distributions are singular, and hence should not be used for Bayesian D-optimal design. Additionally, methods are developed to derive and assess Bayesian D-efficient designs when numerical evaluation of the objective function fails due to ill-conditioning, as often occurs for heavy-tailed prior distributions. These numerical methods are illustrated for logistic regression.

D-optimal Designs with Ordered Categorical Data

Cumulative link models have been widely used for ordered categorical responses. Uniform allocation of experimental units is commonly used in practice, but often suffers from a lack of efficiency. We consider D-optimal designs with ordered categorical responses and cumulative link models. For a predetermined set of design points, we derive the necessary and sufficient conditions for an allocation to be locally D-optimal and develop efficient algorithms for obtaining approximate and exact designs. We prove that the number of support points in a minimally supported design only depends on the number of predictors, which can be much less than the number of parameters in the model. We show that a D-optimal minimally supported allocation in this case is usually not uniform on its support points. In addition, we provide EW D-optimal designs as a highly efficient surrogate to Bayesian D-optimal designs. Both of them can be much more robust than uniform designs.

JMASM45: A computer program for Bayesian D-optimal binary repeated measurements designs (Matlab)

Planners of longitudinal studies of binary responses in applied sciences have not yet benefitted from optimal designs, which have been shown to improve precision of model parameter estimates, due to absence of a computer program. An interactive computer program for Bayesian optimal binary repeated measurements designs is presented for this purpose.

Special issue in honor of Kathryn Chaloner

We are honored to be guest editors for this special issue of Applied Stochastic Models in Business and Industry, dedicated to Kathryn Chaloner's life and achievements. Contributions from her colleagues and friends in this volume are devoted to Kathryn's early research interest in Bayesian optimal design. She was a pioneer in this field and had a broad impact on the subsequent literature on experimental design. Kathryn was internationally known for her research in Bayesian Statistics, which included design of experiments, outlier detection, prior elicitation, and clinical trials, and for her advancements in the study of HIV/AIDS. She was an accomplished researcher, teacher, and mentor. She actively encouraged women and under-represented minority students to join the field of biostatistics and provided mentorship, support, and encouragement. The special issue contains seven papers. The first three papers present some of the recent work in Bayesian optimal designs for nonlinear models, binary responses, and experiments with adversarial components. This is followed by two articles on Bayesian designs in clinical trials. The last two articles of the special issue involve engineering design problems. The paper by Giovagnoli considers the binary response models of Chaloner and Larntz (1989, Journal of Statistical Planning and Inference, 21:191–208) and develops an adaptive version of A-optimal Bayesian designs that are revised as additional data become available during the experiment. Adaptive Bayesian compound designs is also discussed as another extension. Konstantinou and Dette develop approximate Bayesian D-optimal designs for nonlinear regression models that involve covariates that cannot be observed directly. Applications to specific nonlinear models including the exponential regression model are considered and characterizations of the D-optimal saturated designs are presented. Optimal sample size selection for experiments is considered by De Santis and Gubbiotti in an adversarial setting where the decision makers have different prior opinions while having common utility functions and data. Using a Bayesian decision approach with a quadratic loss function, the authors present results for the one-parameter exponential family and investigate effect of the priors on the optimal sample size. Müller, Xu, and Thall discuss advantages and limitations of Bayesian decision theoretic approach in design of clinical trials. In doing so, the authors present a specific case study and point out the limitations such as the computational difficulties involved in solving sequential problems as well as selection of utility functions capturing different stakeholder interests. The article by Ventz, Parmigiani, and Trippa also consider limitations of the Bayesian framework in design of clinical trials and propose a strategy to alleviate these. The proposed strategy involves combining Bayesian designs with frequentist metrics such as error rates, confidence intervals, etc. that are commonly used by medical community and regulatory agencies. The authors show that this can be achieved by an inclusion of frequentist constraints into the Bayesian formulation. The constrained decision theoretic framework is illustrated via several applications and computational algorithms are presented for implementing the proposed approach. Nakamura, Seepaul, Kadane, and Reeja-Jayan discuss design of experiments in materials science. The authors consider use of a batch-sequential experiment and a regression model to determine optimal levels of input variables for synthesis of ceramic materials. It is illustrated that the optimal settings produce more reliable results than other experiments. The final article by Polson and Soyer considers Bayesian design of accelerated life tests for reliability assessment. An augmented probability simulation (APS) approach using Lindley's (1976, Annals of Statistics, 4:1–10) conjugate utility functions is proposed to compute optimal designs in an efficient manner. The proposed approach is illustrated using single and multiple point designs. Use of particle based methods for APS is also discussed as an alternative to Markov chain Monte Carlo methods.

Selecting an Orthogonal or Nonorthogonal Two-Level Design for Screening

ABSTRACTThis article presents a comparison of criteria used to characterize two-level designs for screening purposes. To articulate the relationships among criteria, we focus on 7-factor designs with 16–32 runs and 11-factor designs with 20–48 runs. Screening based on selected designs for each of the run sizes considered is studied with simulation using a forward selection procedure and the Dantzig selector. This article compares Bayesian D-optimal designs, designs created algorithmically to optimize estimation capacity over various model spaces, and orthogonal designs by estimation-based criteria and simulation. In this way, we furnish both general insights regarding various design approaches, as well as a guide to make a choice among a few final candidate designs. Supplementary materials for this article are available online.

The Bayesian D-optimal design for simple beta regression model in two cases (both known and unknown nuisance parameter)

ABSTRACTAccording to investigated topic in the context of optimal designs, various methods can be used to obtain optimal design, of which Bayesian method is one. In this paper, considering the model and the features of the information matrix, this method (Bayesian optimality criterion) has been used for obtaining optimal designs which due to the variation range of the model parameters, prior distributions such as Uniform, Normal and Exponential have been used and the results analysed.