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.     

D- and V-optimal population designs for the quadratic regression model with a random intercept term

In this paper D- and V-optimal population designs for the quadratic regression model with a random intercept term and with values of the explanatory variable taken from a set of equally spaced, non-repeated time points are considered. D-optimal population designs based on single-point individual designs were readily found but the derivation of explicit expressions for designs based on two-point individual designs was not straightforward and was complicated by the fact that the designs now depend on ratio of the variance components. Further algebraic results pertaining to d-point D-optimal population designs where d≥3 and to V-optimal population designs proved elusive. The requisite designs can be calculated by careful programming and this is illustrated by means of a simple example.     

Locally optimal designs for some dose–response models with continuous endpoints

We consider the problem of constructing static (or non sequential), approximate optimal designs for a class of dose–response models with continuous outcomes. We obtain conditions for a design being D-optimal or c-optimal. The designs are locally optimal in that they depend on the model parameters. The efficiency studies show that these designs have high efficiency when the mis-specification of the initial values of model parameters is not severe. A case study indicates that using an optimal design may result in a significant saving of resources.     

An algorithm for calculating D-optimal designs for polynomial regression through a fixed point

Optimal designs are required to make efficient statistical experiments. By using canonical moments, in 1980, Studden found D_s-optimal designs for polynomial regression models. On the other hand, integrable systems are dynamical systems whose solutions can be written down concretely. In this paper, polynomial regression models through a fixed point are discussed. In order to calculate D-optimal designs for these models, a useful relationship between canonical moments and discrete integrable systems is introduced. By using canonical moments and discrete integrable systems, a new algorithm for calculating D-optimal designs for these models is proposed.     

Optimal designs for discriminating between some extensions of the Michaelis–Menten model

In this paper some results on the computation of optimal designs for discriminating between nonlinear models are provided. In particular, some typical deviations of the Michaelis–Menten model are considered. A common deviation of this pharmacokinetic model consists on adding a linear term. If two linear models differ in one parameter the T-optimal design for discriminating between them is c-optimal for estimating the added linear term. This is not the case for nonlinear models.     

On D-optimal robust second order slope-rotatable designs

Das and Park (2006) introduced slope-rotatable designs overall directions for correlated observations which is known as A-optimal robust slope-rotatable designs. This article focuses D-optimal slope-rotatable designs for second-order response surface model with correlated observations. It has been established that robust second-order rotatable designs are also D-optimal robust slope-rotatable designs. A class of D-optimal robust second-order slope-rotatable designs has been derived for special correlation structures of errors.     

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.     

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.     

On D-optimal robust designs for lifetime improvement experiments

Locating the optimal operating conditions of the process parameters is critical in a lifetime improvement experiment. For log-normal lifetime distribution with compound error structure (i.e., symmetry, inter-class and intra-class correlation error structures), we have developed methods for construction of D-optimal robust first order designs. It is shown that D-optimal robust first order designs are always robust first order rotatable but the converse is not always true.     

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.     

Approximate E-optimal designs for the model of spring balance weighing with a constant bias

The present paper analyzes the linear regression model with a nonzero intercept term on the vertices of a d-dimensional unit cube. This setting may be interpreted as a model of weighing d objects on a spring balance with a constant bias. We give analytic formulas for E-optimal designs, as well as their minimal efficiencies under the class of all orthogonally invariant optimality criteria, proving the criterion-robustness of the E-optimal designs. We also discuss the D- and A-optimal designs for this model.     

Rank-order choice-based conjoint experiments: Efficiency and design

In a rank-order choice-based conjoint experiment, the respondent is asked to rank a number of alternatives of a number of choice sets. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered conditional logit model is derived which clearly shows how much additional information is provided by each extra ranking step. A simulation study shows that, besides the Bayesian D-optimal ranking design, the Bayesian D-optimal choice design is also an appropriate design for this type of experiments. Finally, it is shown that considerable improvements in estimation and prediction accuracy are obtained by including extra ranking steps in an experiment.     

Design optimality in multi-factor generalized linear models in the presence of an unrestricted quantitative factor

In this paper we show that product type designs are optimal in partially heteroscedastic multi-factor linear models. This result is applied to obtain locally D-optimal designs in multi-factor generalized linear models by means of a canonical transformation. As a consequence we can construct optimal designs for direct logistic response as well as for Bradley–Terry type paired comparison experiments.     

Optimal designs for response functions with a downturn

In many toxicological assays, interactions between primary and secondary effects may cause a downturn in mean responses at high doses. In this situation, the typical monotonicity assumption is invalid and may be quite misleading. Prior literature addresses the analysis of response functions with a downturn, but so far as we know, this paper initiates the study of experimental design for this situation. A growth model is combined with a death model to allow for the downturn in mean doses. Several different objective functions are studied. When the number of treatments equals the number of parameters, Fisher information is found to be independent of the model of the treatment means and on the magnitudes of the treatments. In general, A- and D_A-optimal weights for estimating adjacent mean differences are found analytically for a simple model and numerically for a biologically motivated model. Results on c-optimality are also obtained for estimating the peak dose and the EC₅₀ (the treatment with response half way between the control and the peak response on the increasing portion of the response function). Finally, when interest lies only in the increasing portion of the response function, we propose composite D-optimal designs.     

The analytic construction of D-optimal designs for the two-variable binary logistic regression model without interaction

Candidate locally D-optimal designs for the binary two-variable logistic model with no interaction, which comprise 3 and 4 support points lying in the first quadrant of the two-dimensional Euclidean space, were introduced by Haines et al. (D-optimal designs for logistic regression in two variables. In: Lopez-Fidalgo J, Rodrigez-Diaz JM, Torsney B, editors. MODA8 – advances in model-oriented designs and analysis. Heidelberg: Physica-Verlag; 2007. p. 91–98). The authors proved algebraically the global D-optimality of the 3-point design for the special case in which the intercept parameter is equal to?1.5434. However for other selected values of the intercept parameter, the global D-optimality of the proposed 3- and 4-point designs was only demonstrated numerically. In this paper, we provide analytical proofs of the D-optimality of these 3- and 4-point designs for all negative and zero intercept parameters of the binary two-variable logistic model with no interaction. The results are extended to the construction of D-optimal designs on a rectangular design space and illustrated by means of two examples of which one is a real example taken from the literature.     

Optimal designs for additive mixture model with heteroscedastic errors

This paper presents a study of D- and A-optimality of direct sum designs for additive mixture models when the errors are heteroscedastic. Sufficient conditions are given so that D- and A-optimal designs for additive mixture models can be constructed from the D- and A-optimal designs for homogeneous models in sub-mixture systems.     

Constructing D-optimal symmetric stated preference discrete choice experiments

We give new constructions for DCEs in which all attributes have the same number of levels. These constructions use several combinatorial structures, such as orthogonal arrays, balanced incomplete block designs and Hadamard matrices. If we assume that only the main effects of the attributes are to be used to explain the results and that all attribute level combinations are equally attractive, we show that the constructed DCEs are D-optimal.     

D s -optimal designs for Kozak's tree taper model

In this work, we study D _s -optimal design for Kozak's tree taper model. The approximate D _s -optimal designs are found invariant to tree size and hence create a ground to construct a general replication-free D _s -optimal design. Even though the designs are found not to be dependent on the parameter value p of the Kozak's model, they are sensitive to the s×1 subset parameter vector values of the model. The 12 points replication-free design (with 91% efficiency) suggested in this study is believed to reduce cost and time for data collection and more importantly to precisely estimate the subset parameters of interest.     

D-optimal designs for combined polynomial and trigonometric regression on a partial circle

Consider the D-optimal designs for a combined polynomial and trigonometric regression on a partial circle. It is shown that the optimal design is equally supported and the structure of the optimal design depends only on the length of the design interval and the support points are analytic functions of this parameter. Moreover, the Taylor expansion of the optimal support points can be determined efficiently by a recursive procedure. Examples are presented to illustrate the procedures for computing the optimal designs.     

A new lower bound to A2-optimality measure for multi-level and mixed-level column balanced designs and its applications

In this paper, a new lower bound to A₂-optimality measure is derived and is applied to multi-level and mixed-level column balanced designs. A₂-optimal multi-level and mixed-level designs are obtained by the application of the new lower bound.