A comparison of confidence regions and designs in estimation of a ratio

A computer simulation is performed to compare confidence regions arising from Fie1ler's theorem and approximate large sample intervals in estimating the point of extremum in quadratic regression. In addition, two designs, one of which is motivated by optimal design theory, are compared. These comparisons are made by examining the confidence level and accuracy of the regions, as well as the concentration of the standard point estimator, in a variety of settings. The results have implications for the more general problem of estimating a ratio of linear combinations in the general linear model.     

On some DS-optimal designs in spherical regions

For polynomial regression over spherical regions the d-th order D_s-optimal designs for the λ-th order models are derived for 1 ≤ λ ≤ d ≤ 4. Efficiencies of these designs with respect to the λ-th order D-optimal designs are obtained. The effects of estimating addtional parameters due to an m-th order model (d ≥ m >>λ) on the efficiencies are investigated.     

A new characterization of Elfving's method for high dimensional computation

We give a new characterization of Elfving's (1952) method for computing c-optimal designs in k dimensions which gives explicit formulae for the k unknown optimal weights and k unknown signs in Elfving's characterization. This eliminates the need to search over these parameters to compute c-optimal designs, and thus reduces the computational burden from solving a family of optimization problems to solving a single optimization problem for the optimal finite support set. We give two illustrative examples: a high dimensional polynomial regression model and a logistic regression model, the latter showing that the method can be used for locally optimal designs in nonlinear models as well.     

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.     

Designs for large degree polynomial regression

Limiting designs are considered for estimating the individual coefficients in a large degree polynomial regression.     

Optimal design for quasi-likelihood estimation in Poisson regression with random coefficients

Count data may be described by a Poisson regression model. If random coefficients are involved, maximum likelihood is not feasible and alternative estimation methods have to be employed. For the approach based on quasi-likelihood estimation a characterization of design optimality is derived and optimal designs are determined numerically for an example with random slope parameters.     

Optimal Designs for Random Coefficient Regression Models with Heteroscedastic Errors

The purpose of this article is to present the optimal designs based on D-, G-, A-, I-, and D _β-optimality criteria for random coefficient regression (RCR) models with heteroscedastic errors. A sufficient condition for the heteroscedastic structure is given to make sure that the search of optimal designs can be confined at extreme settings of the design region when the criteria satisfy the assumption of the real valued monotone design criteria. Analytical solutions of D-, G-, A-, I-, and D _β-optimal designs for the RCR models are derived. Two examples are presented for random slope models with specific heteroscedastic errors.     

D-optimal designs for beta regression models with single predictor

The problem of construction of D-optimal designs for beta regression models involving one predictor is considered for the mean-precision parameterization suggested by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions. J Appl Stat. 2004;31:799–815]. Here we use the logit link function for the mean sub-model. These designs are presented and discussed for unrestricted as well as restricted design regions by considering the precision parameter as (1) a known constant and (2) an unknown constant. Efficiency comparison of obtained designs with commonly used equi-weighted, equi-spaced designs is made to recommend designs for practical use. Real-life applications are given to show the usefulness of these designs.     

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.     

Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

On Exact K-optimal Designs Minimizing the Condition Number

A new design criterion based on the condition number of an information matrix is proposed to construct optimal designs for linear models, and the resulting designs are called K-optimal designs. The relationship between exact and asymptotic K-optimal designs is derived. Since it is usually hard to find exact optimal designs analytically, we apply a simulated annealing algorithm to compute K-optimal design points on continuous design spaces. Specific issues are addressed to make the algorithm effective. Through exact designs, we can examine some properties of the K-optimal designs such as symmetry and the number of support points. Examples and results are given for polynomial regression models and linear models for fractional factorial experiments. In addition, K-optimal designs are compared with A-optimal and D-optimal designs for polynomial regression models, showing that K-optimal designs are quite similar to A-optimal designs.     

Efficient design of experiments in the Monod model

Summary. Estimation and experimental design in a non-linear regression model that is used in microbiology are studied. The Monod model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, water research, pharmacokinetics and plant physiology. It is proved that least squares estimates are asymptotically unbiased and normally distributed. The asymptotic covariance matrix of the estimator is the basis for the construction of efficient designs of experiments. In particular locally D -, E - and c -optimal designs are determined and their properties are studied theoretically and by simulation. If certain intervals for the non-linear parameters can be specified, locally optimal designs can be constructed which are robust with respect to a misspecification of the initial parameters and which allow efficient parameter estimation. Parameter variances can be decreased by a factor of 2 by simply sampling at optimal times during the experiment.     

Bayesian D-Optimal Designs for Poisson Regression Models

By incorporating informative and/or historical knowledge of the unknown parameters, Bayesian experimental design under the decision-theory framework can combine all the information available to the experimenter so that a better design may be achieved. Bayesian optimal designs for generalized linear regression models, especially for the Poisson regression model, is of interest in this article. In addition, lack of an efficient computational method in dealing with the Bayesian design leads to development of a hybrid computational method that consists of the combination of a rough global optima search and a more precise local optima search. This approach can efficiently search for the optimal design for multi-variable generalized linear models. Furthermore, the equivalence theorem is used to verify whether the design is optimal or not.     

A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs

Fractional factorial (FF) designs are no doubt the most widely used designs in experimental investigations due to their efficient use of experimental runs. One price we pay for using FF designs is, clearly, our inability to obtain estimates of some important effects (main effects or second order interactions) that are separate from estimates of other effects (usually higher order interactions). When the estimate of an effect also includes the influence of one or more other effects the effects are said to be aliased. Folding over an FF design is a method for breaking the links between aliased effects in a design. The question is, how do we define the foldover structure for asymmetric FF designs, whether regular or nonregular? How do we choose the optimal foldover plan? How do we use optimal foldover plans to construct combined designs which have better capability of estimating lower order effects? The main objective of the present paper is to provide answers to these questions. Using the new results in this paper as benchmarks, we can implement a powerful and efficient algorithm for finding optimal foldover plans which can be used to break links between aliased effects.     

Bayesian A- and D-optimal designs for gamma regression model with inverse link function

Bayesian optimal designs have received increasing attention in recent years, especially in biomedical and clinical trials. Bayesian design procedures can utilize the available prior information of the unknown parameters so that a better design can be achieved. With this in mind, this article considers the Bayesian A- and D-optimal designs of the two- and three-parameter Gamma regression model. In this regard, we first obtain the Fisher information matrix of the proposed model and then calculate the Bayesian A- and D-optimal designs assuming various prior distributions such as normal, half-normal, gamma, and uniform distribution for the unknown parameters. All of the numerical calculations are handled in R software. The results of this article are useful in medical and industrial researches.     

Minimax A-, c-, and I-optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A-, c-, and I-optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.     

A closer look at de-aliasing effects using an efficient foldover technique

Foldover techniques are used to reduce the confounding when some important effects (usually lower order effects) cannot be estimated independently. This article develops an efficient foldover mechanism for symmetric or asymmetric designs, whether regular or nonregular. In this paper, we take the uniformity criteria (UC) as the optimality measures to construct the optimal combined designs (initial design plus its corresponding foldover design) which have better capability of estimating lower order effects. The relationship between any initial design and its combined design is studied. A comparison study between the combined designs via different UC is provided. Equivalence between any combined design and its complementary combined design is investigated, which is a very useful constraint that reduce the search space. Using our results as benchmarks, we can implement a powerful algorithm for constructing optimal combined designs. Our work covers as well as gives results better than recent works of about 20 articles in the last few years as special cases. So this article is a good reference for constructing effective designs.     

A candidate-set-free algorithm for generating D-optimal split-plot designs

Summary.  We introduce a new method for generating optimal split-plot designs. These designs are optimal in the sense that they are efficient for estimating the fixed effects of the statistical model that is appropriate given the split-plot design structure. One advantage of the method is that it does not require the prior specification of a candidate set. This makes the production of split-plot designs computationally feasible in situations where the candidate set is too large to be tractable. The method allows for flexible choice of the sample size and supports inclusion of both continuous and categorical factors. The model can be any linear regression model and may include arbitrary polynomial terms in the continuous factors and interaction terms of any order. We demonstrate the usefulness of this flexibility with a 100-run polypropylene experiment involving 11 factors where we found a design that is substantially more efficient than designs that are produced by using other approaches.     

Marginally restricted sequential D‐optimal designs

In many experiments, not all explanatory variables can be controlled. When the units arise sequentially, different approaches may be used. The authors study a natural sequential procedure for “marginally restricted” D‐optimal designs. They assume that one set of explanatory variables (x₁) is observed sequentially, and that the experimenter responds by choosing an appropriate value of the explanatory variable x₂. In order to solve the sequential problem a priori, the authors consider the problem of constructing optimal designs with a prior marginal distribution for x₁. This eliminates the influence of units already observed on the next unit to be designed. They give explicit designs for various cases in which the mean response follows a linear regression model; they also consider a case study with a nonlinear logistic response. They find that the optimal strategy often consists of randomizing the assignment of the values of x₂.     

Applications and Implementations of Continuous Robust Designs

In the literature concerning the construction of robust optimal designs, many resulting designs turn out to have densities. In practice, an exact design should tell the experimenter what the support points are and how many subjects should be allocated to each of these points. In particular, we consider a practical situation in which the number of support points allowed is constrained. We discuss an intuitive approach, which motivates a new implementation scheme that minimizes the loss function based on the Kolmogorov and Smirnov distance between an exact design and the optimal design having a density. We present three examples to illustrate the application and implementation of a robust design constructed: one for a nonlinear dose-response experiment and the other two for general linear regression. Additionally, we perform some simulation studies to compare the efficiencies of the exact designs obtained by our optimal implementation with those by other commonly used implementation methods.