Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D‐optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian _q‐optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.     

Locally and maximin optimal designs for multi-factor nonlinear models

This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.     

Efficient maximin distance designs for experiments in mixtures

In this paper, different dissimilarity measures are investigated to construct maximin designs for compositional data. Specifically, the effect of different dissimilarity measures on the maximin design criterion for two case studies is presented. Design evaluation criteria are proposed to distinguish between the maximin designs generated. An optimization algorithm is also presented. Divergence is found to be the best dissimilarity measure to use in combination with the maximin design criterion for creating space-filling designs for mixture variables.     

Implementation of maximin efficient designs in dose‐finding studies

This paper considers the maximin approach for designing clinical studies. A maximin efficient design maximizes the smallest efficiency when compared with a standard design, as the parameters vary in a specified subset of the parameter space. To specify this subset of parameters in a real situation, a four‐step procedure using elicitation based on expert opinions is proposed. Further, we describe why and how we extend the initially chosen subset of parameters to a much larger set in our procedure. By this procedure, the maximin approach becomes feasible for dose‐finding studies. Maximin efficient designs have shown to be numerically difficult to construct. However, a new algorithm, the H‐algorithm, considerably simplifies the construction of these designs. We exemplify the maximin efficient approach by considering a sigmoid Emax model describing a dose–response relationship and compare inferential precision with that obtained when using a uniform design. The design obtained is shown to be at least 15% more efficient than the uniform design. © 2014 The Authors. Pharmaceutical Statistics Published by John Wiley & Sons Ltd.     

Some robust design strategies for percentile estimation in binary response models

For the problem of percentile estimation of a quantal response curve, the authors determine multiobjective designs which are robust with respect to misspecifications of the model assumptions. They propose a maximin approach based on efficiencies which leads to designs that are simultaneously efficient with respect to various choices of link functions and parameter regions. Furthermore, the authors deal with the problems of designing model and percentile robust experiments. They give various examples of such designs, which are calculated numerically.     

Simplex based space filling designs

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.     

Optimal design for the Bradley–Terry paired comparison model

Paired comparisons are a popular tool for questionnaires in psychological marketing research. The quality of the statistical analysis of the responses heavily depends on the design, i.e. the choice of the alternatives in the comparisons. In this paper we show that the structure of locally optimal designs changes substantially with the parameters in the underlying utility. This fact is illustrated by elementary examples, where the optimal designs can be completely characterized. As an alternative maximin efficient designs are proposed which perform well for all parameter settings. Research supported by grant Ho 1286 of the German Research Council (Deutsche Forschungsgemeinschaft).     

Optimal Latin hypercube designs for the Kullback–Leibler criterion

Space-filling designs are commonly used for selecting the input values of time-consuming computer codes. Computer experiment context implies two constraints on the design. First, the design points should be evenly spread throughout the experimental region. A space-filling criterion (for instance, the maximin distance) is used to build optimal designs. Second, the design should avoid replication when projecting the points onto a subset of input variables (non-collapsing). The Latin hypercube structure is often enforced to ensure good projective properties. In this paper, a space-filling criterion based on the Kullback–Leibler information is used to build a new class of Latin hypercube designs. The new designs are compared with several traditional optimal Latin hypercube designs and appear to perform well.     

On Robust and Efficient Designs for Risk Estimation in Epidemiological Studies

Abstract.  We consider the design problem for the estimation of several scalar measures suggested in the epidemiological literature for comparing the success rate in two samples. The designs considered so far in the literature are local in the sense that they depend on the unknown probabilities of success in the two groups and are not necessarily robust with respect to their misspecification. A maximin approach is proposed to obtain efficient and robust designs for the estimation of the relative risk, attributable risk and odds ratio, whenever a range for the success rates can be specified by the experimenter. It is demonstrated that the designs obtained by this method are usually more efficient than the commonly used uniform design, which allocates equal sample sizes to the two groups.     

Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Maximin efficient design of experiment for exponential regression models

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.     

Robust designs for linear mixed effects models

Summary.  In health sciences, medicine and social sciences linear mixed effects models are often used to analyse time-structured data. The search for optimal designs for these models is often hampered by two problems. The first problem is that these designs are only locally optimal. The second problem is that an optimal design for one model may not be optimal for other models. In this paper the maximin principle is adopted to handle both problems, simultaneously. The maximin criterion is formulated by means of a relative efficiency measure, which gives an indication of how much efficiency is lost when the uncertainty about the models over a prior domain of parameters is taken into account. The procedure is illustrated by means of three growth studies. Results are presented for a vocabulary growth study from education, a bone gain study from medical research and an epidemiological decline in height study. It is shown that, for the mixed effects polynomial models that are applied to these studies, the maximin designs remain highly efficient for different sets of models and combinations of parameter values.     

On the asymptotic efficiency of aligned-rank tests in randomized block designs

This paper is concerned with the ranking-after-alignment procedure, the alignment being made on the mean, in randomized block designs. The asymptotic efficiencies, as the number of blocks goes to infinity, of a class of aligned-rank tests, relative to the maximin most powerful test based on aligned observations, are established and studied. Some asymptotic efficiencies under doubleexponentiality are also obtained using Monte Carlo methods.     

A maximin criterion for the logistic random intercept model with covariates

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.     

Parallel, AA/BB, AB/BA and Balaam's design: efficient and maximin choices when testing the treatment effect in a mixed effects linear regression

When examining the effect of treatment A versus B, there may be a choice between a parallel group design, an AA/BB design, an AB/BA cross‐over and Balaam's design. In case of a linear mixed effects regression, it is examined, starting from a flexible function of the costs involved and allowing for subject dropout, which design is most efficient in estimating this effect. For no carry‐over, the AB/BA cross‐over design is most efficient as long as the dropout rate at the second measurement does not exceed 2ρ/(1 + ρ), ρ being the intraclass correlation. For steady‐state carry‐over, depending on the costs involved, the dropout rate and ρ, either a parallel design or an AA/BB design is most efficient. For types of carry‐over that allow for self carry‐over, interest is in the direct treatment effect plus the self carry‐over effect, with either an AA/BB or Balaam's design being most efficient. In case of insufficient knowledge on the dropout rate or ρ, a maximin strategy is devised: choose the design that minimizes the maximum variance of the treatment estimator. Such maximin designs are derived for each type of carry‐over. Copyright © 2012 John Wiley & Sons, Ltd.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

A semi-infinite programming based algorithm for finding minimax optimal designs for nonlinear models

Minimax optimal experimental designs are notoriously difficult to study largely because the optimality criterion is not differentiable and there is no effective algorithm for generating them. We apply semi-infinite programming (SIP) to solve minimax design problems for nonlinear models in a systematic way using a discretization based strategy and solvers from the General Algebraic Modeling System (GAMS). Using popular models from the biological sciences, we show our approach produces minimax optimal designs that coincide with the few theoretical and numerical optimal designs in the literature. We also show our method can be readily modified to find standardized maximin optimal designs and minimax optimal designs for more complicated problems, such as when the ranges of plausible values for the model parameters are dependent and we want to find a design to minimize the maximal inefficiency of estimates for the model parameters.     

The Construction of Efficient Two-replicate Row–Column Designs for Use in Field Trials

Two-replicate row–column designs are often used for field trials in multisite tree or plant breeding programmes. With only two replicates for each trial, it is important to use designs with optimal or near optimal efficiency factors. This paper presents an algorithm for generating such designs. The method extends the contraction approach of Bailey and Patterson to any set of parameters and uses the factorial design construction algorithm of Williams and John to generate designs. Our experience with the algorithm is that it produces designs that are at least as good as, and often much better and more quickly generated than, those obtained by other recent computer algorithms.     

CVX‐based algorithms for constructing various optimal regression designs

CVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada     

Combinatorial Equivalence of Fractional Factorial Designs

Two symmetrical fractional factorial designs are said to be combinatorially equivalent if one design can be obtained from another by reordering the runs, relabeling the factors and relabeling the levels of one or more factors. This article presents concepts of ordered distance frequency matrix, distance frequency vector, and reduced distance frequency vector for a design. Necessary conditions for two designs to be combinatorial equivalent based on these concepts are presented. A new algorithm based on the results obtained is proposed to check combinatorial non-equivalence of two factorial designs and several illustrating examples are provided.