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In this paper we mainly investigate the problem of optimal designs for multi-factor regression models with partially known heteroscedastic structure. The Bayesian Φ q -optimality criterion proposed by Dette and Wong (Ann Stat 24:2108–2127, 1996), which closely resembles Kiefer's Φ k -class of criteria, and the standardized maximin D-optimal criterion are considered. More precisely, for heteroscedastic Kronecker product models, it is shown that the product designs formed from optimal designs for sub-models with a single factor are optimal under the two robust criteria. For additive models with intercept, however, sufficient conditions are given in order to search for Bayesian Φ q -optimal and standardized maximin D-optimal product designs. Finally, several examples are presented to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Cluster randomized trial design may raise financial concerns because the cost to recruit an additional cluster is much higher than to enroll an additional subject in subject-level randomized trials. Therefore, it is desirable to develop an optimal design. For local optimal designs, optimization means the minimum variance of the estimated treatment effect under the total budget. The local optimal design derived from the variance needs the input of an association parameter ρ in terms of a "working" correlation structure R (ρ) in the generalized estimating equation models. When the range of ρ instead of an exact value is available, the parameter space is defined as the range of ρ and the design space is defined as enrollment feasibility, for example, the number of clusters or cluster size. For any value ρ within the range, the optimal design and relative efficiency for each design in the design space is obtained. Then, for each design in the design space, the minimum relative efficiency within the parameter space is calculated. MaxiMin design is the optimal design that maximizes the minimum relative efficiency among all designs in the design space. Our contributions are threefold. First, for three common measures (risk difference, risk ratio, and odds ratio), we summarize all available local optimal designs and MaxiMin designs utilizing generalized estimating equation models when the group allocation proportion is predetermined for two-level and three-level parallel cluster randomized trials. We then propose the local optimal designs and MaxiMin designs using the same models when the group allocation proportion is undecided. Second, for partially nested designs, we develop the optimal designs for three common measures under the setting of equal number of subjects per cluster and exchangeable working correlation structure in the intervention group. Third, we create three new Statistical Analysis System (SAS) macros and update two existing SAS macros for all the optimal designs. We provide two examples to illustrate our methods. [ABSTRACT FROM AUTHOR]

Model-based optimal designs for regression problems with multiple objectives are common in practice. The traditional approach is to construct an optimal design for the most important objective and hope that the design performs well for the other objectives. Analytical approaches are challenging because the objectives are often competitive and their relative importance has to be incorporated at the onset of the design construction. There are also no general and efficient algorithms for searching such designs for user-specified nonlinear models and criteria. We propose a new and effective approach for finding multiple-objective optimal designs via the CVX software and demonstrate it can efficiently find different types of multiple-objective optimal designs after the optimization problems are carefully formulated as convex optimization problems appropriate for CVX use. We provide three biomedical applications and show our MATLAB code producing the same few multiple-objective optimal designs reported in the statistical literature. MATLAB code files are available online in the of this article. [ABSTRACT FROM AUTHOR]

Cluster randomized trials evaluate the effect of a treatment on persons nested within clusters, with clusters being randomly assigned to treatment. The optimal sample size at the cluster and person level depends on the study cost per cluster and per person, and the outcome variance at the cluster and the person level. The variances are unknown in the design stage and can differ between treatment arms. As a solution, this paper presents a Maximin design that maximizes the minimum relative efficiency (relative to the optimal design) over the variance parameter space, for trials with two treatment arms and a quantitative outcome. This maximin relative efficiency design (MMRED) is compared with a published Maximin design which maximizes the minimum efficiency (MMED). Both designs are also compared with the optimal designs for homogeneous costs and variances (balanced design) and heterogeneous costs and homogeneous variances (cost‐conscious design), for a range of variances based upon three published trials. Whereas the MMED is balanced under high uncertainty about the treatment‐to‐control variance ratio, the MMRED then tends towards a balanced budget allocation between arms, leading to an unbalanced sample size allocation if costs are heterogeneous, similar to the cost‐conscious design. Further, the MMRED corresponds to an optimal design for an intraclass correlation (ICC) in the lower half of the assumed ICC range (optimistic), whereas the MMED is the optimal design for the maximum ICC within the ICC range (pessimistic). Attention is given to the effect of the Welch–Satterthwaite degrees of freedom for treatment effect testing on the design efficiencies. [ABSTRACT FROM AUTHOR]

Fully-sequential (i.e., with design points added one-at-a-time) space-filling designs are useful for global surrogate modeling of expensive computer experiments when the number of design points required to achieve a suitable accuracy is unknown in advance. We develop and investigate three fully-sequential space-filling (FSSF) design algorithms that are conceptually simple and computationally efficient and that achieve much better space-filling properties than alternative methods such as Sobol sequences and more complex batch-sequential methods based on sliced or nested optimal Latin hypercube designs (LHDs). Remarkably, at each design size in the sequence, our FSSF algorithms even achieve much better space-filling properties than a one-shot LHD optimized for that specific size. The algorithms we propose also scale well to very large design sizes. We provide an R package to implement the approaches. [ABSTRACT FROM AUTHOR]

To estimate the mean of a quantitative variable in a hierarchical population, it is logistically convenient to sample in two stages (two-stage sampling), i.e. selecting first clusters, and then individuals from the sampled clusters. Allowing cluster size to vary in the population and to be related to the mean of the outcome variable of interest (informative cluster size), the following competing sampling designs are considered: sampling clusters with probability proportional to cluster size, and then the same number of individuals per cluster; drawing clusters with equal probability, and then the same percentage of individuals per cluster; and selecting clusters with equal probability, and then the same number of individuals per cluster. For each design, optimal sample sizes are derived under a budget constraint. The three optimal two-stage sampling designs are compared, in terms of efficiency, with each other and with simple random sampling of individuals. Sampling clusters with probability proportional to size is recommended. To overcome the dependency of the optimal design on unknown nuisance parameters, maximin designs are derived. The results are illustrated, assuming probability proportional to size sampling of clusters, with the planning of a hypothetical survey to compare adolescent alcohol consumption between France and Italy. [ABSTRACT FROM AUTHOR]

The cost-effectiveness of interventions (e.g. new medical therapies or health care technologies) is often evaluated in randomized clinical trials, where individuals are nested within clusters, for instance patients within general practices. In such two-level cost-effectiveness trials, one can randomly assign treatments to individuals within clusters (multicentre trial) or to entire clusters (cluster randomized trial). Such trials need careful planning to evaluate the cost-effectiveness of interventions within the available research resources. The optimal number of clusters and the optimal number of subjects per cluster for both types of cost-effectiveness trials can be determined by using optimal design theory. However, the construction of the optimal design requires information on model parameters, which may be unknown at the planning stage of a trial. To overcome this problem, a maximin strategy is employed. We have developed a computer program SamP2CeT in R to perform these sample size calculations. SamP2CeT provides a graphical user interface which enables the researchers to optimize the numbers of clusters and subjects per cluster in their cost-effectiveness trial as a function of study costs and outcome variances. In case of insufficient knowledge on model parameters, SamP2CeT also provides safe numbers of clusters and subjects per cluster, based on a maximin strategy. SamP2CeT can be used to calculate the smallest budget needed for a user-specified power level, the largest power attainable with a user-specified budget, and also has the facility to calculate the power for a user-specified design. Recent methodological developments on sample size and power calculation for two-level cost-effectiveness trials have been implemented in SamP2CeT. This program is user-friendly, as illustrated for two published cost-effectiveness trials. [ABSTRACT FROM AUTHOR]

Cluster randomized trials evaluate the effect of a treatment on persons nested within clusters, where treatment is randomly assigned to clusters. Current equations for the optimal sample size at the cluster and person level assume that the outcome variances and/or the study costs are known and homogeneous between treatment arms. This paper presents efficient yet robust designs for cluster randomized trials with treatment-dependent costs and treatment-dependent unknown variances, and compares these with 2 practical designs. First, the maximin design (MMD) is derived, which maximizes the minimum efficiency (minimizes the maximum sampling variance) of the treatment effect estimator over a range of treatment-to-control variance ratios. The MMD is then compared with the optimal design for homogeneous variances and costs (balanced design), and with that for homogeneous variances and treatment-dependent costs (cost-considered design). The results show that the balanced design is the MMD if the treatment-to control cost ratio is the same at both design levels (cluster, person) and within the range for the treatment-to-control variance ratio. It still is highly efficient and better than the cost-considered design if the cost ratio is within the range for the squared variance ratio. Outside that range, the cost-considered design is better and highly efficient, but it is not the MMD. An example shows sample size calculation for the MMD, and the computer code (SPSS and R) is provided as supplementary material. The MMD is recommended for trial planning if the study costs are treatment-dependent and homogeneity of variances cannot be assumed. [ABSTRACT FROM AUTHOR]

Latin Hypercube Designs (LHDs) are crucial in the meta-modeling field. An LHD with maximal minimum distance between any pair of points guarantees a high quality of the model. We introduce a generalization of the Maximin LHD construction, the Maximin LHD completion, in which some points of the design are already fixed. We study the complexity of this problem and prove that it is NP-complete and it does not admit any approximation algorithm in most practical cases. [ABSTRACT FROM AUTHOR]

In metamodeling, the choice of sampling points is crucial for the quality of the model. In this context, the maximin Latin hypercube designs (LHD), with their spacefilling and noncollapsing properties, are particularly efficient. To this day, there is no polynomial time algorithm that produces optimal maximin LHDs, i.e., in which the minimum distance between two points (the separation distance) is maximal.We are interested in LHDs with a separation distance as large as possible. The algorithm we propose, IES, gives an approximate solution to the LHD problem regardless of its dimension and size with a theoretical performance guarantee. We introduce two upper bounds for the separation distance to find its approximation ratio. Its performance is compared with the best metaheuristic algorithm known for this problem, an appropriate simulated annealing scheme. Our algorithm defeats the metaheuristic algorithm for large instances of the problem while having a very short running time. [ABSTRACT FROM AUTHOR]

A framework for designing and analyzing computer experiments is presented, which is constructed for dealing with functional and scalar inputs and scalar outputs. For designing experiments with both functional and scalar inputs, a two-stage approach is suggested. The first stage consists of constructing a candidate set for each functional input. During the second stage, an optimal combination of the found candidate sets and a Latin hypercube for the scalar inputs is sought. The resulting designs can be considered to be generalizations of Latin hypercubes. Gaussian process models are explored as metamodel. The functional inputs are incorporated into the Kriging model by applying norms in order to define distances between two functional inputs. We propose the use of B-splines to make the calculation of these norms computationally feasible. [ABSTRACT FROM AUTHOR]

Designing studies such that they have a high level of power to detect an effect or association of interest is an important tool to improve the quality and reproducibility of findings from such studies. Since resources (research subjects, time, and money) are scarce, it is important to obtain sufficient power with minimum use of such resources. For commonly used randomized trials of the treatment effect on a continuous outcome, designs are presented that minimize the number of subjects or the amount of research budget when aiming for a desired power level. This concerns the optimal allocation of subjects to treatments and, in case of nested designs such as cluster-randomized trials and multicenter trials, also the optimal number of centers versus the number of persons per center. Since such optimal designs require knowledge of parameters of the analysis model that are not known in the design stage, in particular outcome variances, maximin designs are presented. These designs guarantee a prespecified power level for plausible ranges of the unknown parameters and minimize research costs for the worst-case values of these parameters. The focus is on a 2-group parallel design, the AB/BA crossover design, and cluster-randomized and multicenter trials with a continuous outcome. How to calculate sample sizes for maximin designs is illustrated for examples from nutrition. Several computer programs that are helpful in calculating sample sizes for optimal and maximin designs are discussed as well as some results on optimal designs for other types of outcomes., (Copyright © 2023 American Society for Nutrition. Published by Elsevier Inc. All rights reserved.)

This paper deals with the optimal sample sizes for a multicentre trial in which the cost-effectiveness of two treatments in terms of net monetary benefit is studied. A bivariate random-effects model, with the treatment-by-centre interaction effect being random and the main effect of centres fixed or random, is assumed to describe both costs and effects. The optimal sample sizes concern the number of centres and the number of individuals per centre in each of the treatment conditions. These numbers maximize the efficiency or power for given research costs or minimize the research costs at a desired level of efficiency or power. Information on model parameters and sampling costs are required to calculate these optimal sample sizes. In case of limited information on relevant model parameters, sample size formulas are derived for so-called maximin sample sizes which guarantee a power level at the lowest study costs. Four different maximin sample sizes are derived based on the signs of the lower bounds of two model parameters, with one case being worst compared to others. We numerically evaluate the efficiency of the worst case instead of using others. Finally, an expression is derived for calculating optimal and maximin sample sizes that yield sufficient power to test the cost-effectiveness of two treatments. [ABSTRACT FROM AUTHOR]

When comparing two different kinds of group therapy or two individual treatments where patients within each arm are nested within care providers, clustering of observations may occur in both arms. The arms may differ in terms of (a) the intraclass correlation, (b) the outcome variance, (c) the cluster size, and (d) the number of clusters, and there may be some ideal group size or ideal caseload in case of care providers, fixing the cluster size. For this case, optimal cluster numbers are derived for a linear mixed model analysis of the treatment effect under cost constraints as well as under power constraints. To account for uncertain prior knowledge on relevant model parameters, also maximin sample sizes are given. Formulas for sample size calculation are derived, based on the standard normal as the asymptotic distribution of the test statistic. For small sample sizes, an extensive numerical evaluation shows that in a two-tailed test employing restricted maximum likelihood estimation, a safe correction for both 80% and 90% power, is to add three clusters to each arm for a 5% type I error rate and four clusters to each arm for a 1% type I error rate. [ABSTRACT FROM AUTHOR]

For cluster randomized and multicentre trials evaluating the effect of a treatment on persons nested within clusters, equations have been published to compute the optimal sample sizes at the cluster and person level as a function of sampling costs and intraclass correlation (ICC). Here, optimal means maximum power and precision for a given sampling budget, or minimum sampling costs for a given power and precision. However, the ICC is usually unknown, and the optimal sample sizes depend strongly on this ICC. To overcome this local optimality problem, this study presents Maximin designs (MMDs) based on relative efficiency (RE) and efficiency. These designs perform well over a range of possible ICC values either in terms of RE compared with the locally optimal designs, or in terms of minimum efficiency (maximum variance) of the treatment effect estimator. The use of MMDs is illustrated using information from many cluster randomized trials in primary care. It is concluded that MMDs and the optimal design for an ICC halfway its assumed range are efficient for a range of ICC values and recommendable for practical use. This requires that trial reports mention the study cost per cluster and person. [ABSTRACT FROM AUTHOR]

In this paper, the optimal sample sizes at the cluster and person levels for each of two treatment arms are obtained for cluster randomized trials where the cost-effectiveness of treatments on a continuous scale is studied. The optimal sample sizes maximize the efficiency or power for a given budget or minimize the budget for a given efficiency or power. Optimal sample sizes require information on the intra-cluster correlations (ICCs) for effects and costs, the correlations between costs and effects at individual and cluster levels, the ratio of the variance of effects translated into costs to the variance of the costs (the variance ratio), sampling and measuring costs, and the budget. When planning, a study information on the model parameters usually is not available. To overcome this local optimality problem, the current paper also presents maximin sample sizes. The maximin sample sizes turn out to be rather robust against misspecifying the correlation between costs and effects at the cluster and individual levels but may lose much efficiency when misspecifying the variance ratio. The robustness of the maximin sample sizes against misspecifying the ICCs depends on the variance ratio. The maximin sample sizes are robust under misspecification of the ICC for costs for realistic values of the variance ratio greater than one but not robust under misspecification of the ICC for effects. Finally, we show how to calculate optimal or maximin sample sizes that yield sufficient power for a test on the cost-effectiveness of an intervention. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In longitudinal trials, the number of accrual groups and their sizes should carefully be chosen to ensure a desired power to detect a specified treatment effect. Methods are proposed to obtain a cost-effective combination of the number and size of accrual groups that provides high efficiency at minimal cost. We focus on trials where an event occurs at any point in time, but it is recorded on a discrete scale. The Weibull survival function is considered for modeling the underlying time to event. By using a cost function, it is shown that the ratio of the cost of recruiting and treating subjects to the cost of measuring them and also the survival pattern highly influence the optimal combination of the number and size of accrual groups. A maximin approach is further presented to obtain robust designs with respect to poor specification of these modeling parameters. We also show the application of the proposed optimal design methodology using real examples. [ABSTRACT FROM AUTHOR]

This paper studies efficient designs for simultaneous model discrimination among polynomial regression models up to degree k. Based on the $${\Phi_{\boldsymbol{\beta}}}$$ -optimality criterion proposed by Dette (Ann Stat 22:890-903, ), a maximin $${\Phi_{\boldsymbol{\beta}^{*}}}$$ -optimal discriminating design is derived in terms of canonical moments for $${k\in\mathbb{N}}$$ . Theoretical and numerical results show that the proposed design performs well for model discrimination in most of the considered models. [ABSTRACT FROM AUTHOR]

When setting up a computer experiment, it has become a standard practice to select the inputs spread out uniformly across the available space. These so-called space-filling designs are now ubiquitous in corresponding publications and conferences. The statistical folklore is that such designs have superior properties when it comes to prediction and estimation of emulator functions. In this paper we want to review the circumstances under which this superiority holds, provide some new arguments and clarify the motives to go beyond space-filling. An overview over the state of the art of space-filling is introducing and complementing these results. [ABSTRACT FROM AUTHOR]

Hormesis is a widely observed phenomenon in many branches of life sciences, ranging from toxicology studies to agronomy, with obvious public health and risk assessment implications. We address optimal experimental design strategies for determining the presence of hormesis in a controlled environment using the recently proposed Hunt-Bowman model. We propose alternative models that have an implicit hormetic threshold, discuss their advantages over current models, and construct and study properties of optimal designs for (i) estimating model parameters, (ii) estimating the threshold dose, and (iii) testing for the presence of hormesis. We also determine maximin optimal designs that maximize the minimum of the design efficiencies when we have multiple design criteria or there is model uncertainty where we have a few plausible models of interest. We apply these optimal design strategies to a teratology study and show that the proposed designs outperform the implemented design by a wide margin for many situations. [ABSTRACT FROM AUTHOR]

Abstract: Space filling designs are important for deterministic computer experiments. Even a single experiment can be very time consuming and can have many input parameters. Furthermore the underlying function generating the output is often nonlinear. Thus, the computer experiment has to be designed carefully. There exist many design criteria, which can be numerically optimized. Here, a method is developed, which does not need algorithmic optimization. A mesh of nearly regular simplices is constructed and the vertices of the simplices are used as potential design points. The extraction of a design from these meshes is very fast and easy to implement once the underlying mesh has been constructed. The extracted designs are highly competitive regarding the maximin design criterion and it is easy to extract designs for nonstandard design spaces. [Copyright &y& Elsevier]

The design of computer experiments is an important step in black-box evaluation and optimization processes. When dealing with multiple black-box functions the need often arises to construct designs for all black boxes jointly, instead of individually. These so-called nested designs are particularly useful as training and test sets for fitting and validating metamodels, respectively. Furthermore, nested designs can be used to deal with linking parameters and sequential evaluations. In this paper, we introduce one-dimensional nested maximin designs. We show how to nest two designs optimally and develop a heuristic to nest three and four designs. These nested maximin designs can be downloaded from the website . Furthermore, it is proven that the loss in space-fillingness, with respect to traditional maximin designs, is at most 14.64 and 19.21%, when nesting two and three designs, respectively. [ABSTRACT FROM AUTHOR]

Abstract: The Arrhenius equation is widely used to describe the relationship between the rate of a chemical reaction and the temperature. However, in some cases more precision is needed and a Modified Arrhenius (MA) model, allowing the linear parameter to be temperature-dependent, appears as the correct alternative to the plain model. Optimal designs for the Arrhenius equation have been already computed, for instance in Rodríguez-Aragón and López Fidalgo [L.J. Rodríguez-Aragón and J. López-Fidalgo (2005). Optimal designs for the Arrhenius equation. Chemometr Intell Lab Syst 77 131–138.] for independent and normally-distributed errors with constant variance and in Rodríguez-Torreblanca and Rodríguez-Díaz [C. Rodríguez-Torreblanca and J.M. Rodríguez Díaz (2007). Locally D- and c-optimal designs for Poisson and Negative Binomial regression models. Metrika 66 161–172.] for different variance structures. However, the MA model has not been studied at the same level. In this work, optimal designs for this last equation will be computed for a general design space and different optimality criteria, and their performance will be shown through convenient examples. A robustness analysis when a wrong choice of the initial values for the parameters is made or some of the hypothesis on the model are not fulfilled will be performed, in order to be able to choose the best design for each situation. [Copyright &y& Elsevier]

Abstract: In this paper robust and efficient designs are derived for several exponential decay models. These models are widely used in chemistry, pharmacokinetics or microbiology. We propose a maximin approach, which determines the optimal design such that a minimum of the D-efficiencies (taken over a certain range) becomes maximal. Analytic solutions are derived if optimization is performed in the class of minimal supported designs. In general the optimal designs with respect to the maximin criterion have to be determined numerically and some properties of these designs are also studied. We also illustrate the benefits of our approach by reanalysing a data example from the literature. [Copyright &y& Elsevier]

Abstract: A maximin criterion is used to find optimal designs for the logistic random intercept model with dichotomous independent variables. The dichotomous independent variables can be subdivided into variables for which the distribution is specified prior to data sampling, called variates, and into variables for which the distribution is not specified prior to data sampling, but is obtained from data sampling, called covariates. The proposed maximin criterion maximizes the smallest possible relative efficiency not only with respect to all possible values of the model parameters, but also with respect to the joint distribution of the covariates. We have shown that, under certain conditions, the maximin design is balanced with respect to the joint distribution of the variates. The proposed method will be used to plan a (stratified) clinical trial where variates and covariates are involved. [Copyright &y& Elsevier]

The problem of finding optimal calibration designs for dichotomous item response theory (IRT) models has been extensively studied in the literature. In this study, this problem will be extended to polytomous IRT models. Focus is given to items described by the nominal response model (NRM). The optimization's objective is to minimize the generalized variance of item parameter estimates. For nonlinear models, Fisher's information depends on the unknown parameters to be estimated. By assuming knowledge of the latter, D-optimal designs can be found, however, with the inconvenience that these are only locally optimal. A modified maximin approach, based on the relative efficiency of calibration designs with respect to the D-optimal design, is introduced as a viable method to circumvent the problem of local optimality. The performance of maximin designs is compared to that of alternative calibration designs, those with normally and uniformly distributed latent trait values. [ABSTRACT FROM AUTHOR]

There are a lot of papers discussing the testing and estimation problems on the parameters under order restrictions. They are, however, mostly for the balanced case and there are only a few papers dealing with the design problems. The purpose of the present paper is to note that the favorite tests for the balanced case are not necessarily good for the extremely unbalanced case where a simple minimax test or a linear trend test behaves better. [Copyright &y& Elsevier]