A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

Design of experiments for locally weighted regression

We consider the design of experiments when estimation is to be performed using locally weighted regression methods. We adopt criteria that consider both estimation error (variance) and error resulting from model misspecification (bias). Working with continuous designs, we use the ideas developed in convex design theory to analyze properties of the corresponding optimal designs. Numerical procedures for constructing optimal designs are developed and applied to a variety of design scenarios in one and two dimensions. Among the interesting properties of the constructed designs are the following: (1) Design points tend to be more spread throughout the design space than in the classical case. (2) The optimal designs appear to be less model and criterion dependent than their classical counterparts.(3) While the optimal designs are relatively insensitive to the specification of the design space boundaries, the allocation of supporting points is strongly governed by the points of interest and the selected weight function, if the latter is concentrated in areas significantly smaller than the design region. Some singular and unstable situations occur in the case of saturated designs. The corresponding phenomenon is discussed using a univariate linear regression example.     

Using the Canonical Design Space to Obtain c-optimal Designs for the Quadratic Logistic Model

c-optimal designs for estimating the model parameters of the quadratic logistic regression model are considered. The designs are constructed via the canonical design space. It is shown that the number of design points varies between 1 and 4 depending on the parameter being estimated. Furthermore, formulae for finding the design points along with the corresponding design weights are derived.     

Equivalence of factorial designs with qualitative and quantitative factors

Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed.     

A multi-objective coordinate-exchange two-phase local search algorithm for multi-stratum experiments

A multi-stratum design is a useful tool for industrial experimentation, where factors that have levels which are harder to set than others, due to time or cost constraints, are frequently included. The number of different levels of hardness to set defines the number of strata that should be used. The simplest case is the split-plot design, which includes two strata and two sets of factors defined by their level of hardness-to-set. In this paper, we propose a novel computational algorithm which can be used to construct optimal multi-stratum designs for any number of strata and up to six optimality criteria simultaneously. Our algorithm allows the study of the entire Pareto front of the optimization problem and the selection of the designs representing the desired trade-off between the competing objectives. We apply our algorithm to several real case scenarios and we show that the efficiencies of the designs obtained present experimenters with several good options according to their objectives.     

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.     

Experimental designs for detecting synergy and antagonism between two drugs in a pre‐clinical study

The identification of synergistic interactions between combinations of drugs is an important area within drug discovery and development. Pre‐clinically, large numbers of screening studies to identify synergistic pairs of compounds can often be ran, necessitating efficient and robust experimental designs. We consider experimental designs for detecting interaction between two drugs in a pre‐clinical in vitro assay in the presence of uncertainty of the monotherapy response. The monotherapies are assumed to follow the Hill equation with common lower and upper asymptotes, and a common variance. The optimality criterion used is the variance of the interaction parameter. We focus on ray designs and investigate two algorithms for selecting the optimum set of dose combinations. The first is a forward algorithm in which design points are added sequentially. This is found to give useful solutions in simple cases but can lack robustness when knowledge about the monotherapy parameters is insufficient. The second algorithm is a more pragmatic approach where the design points are constrained to be distributed log‐normally along the rays and monotherapy doses. We find that the pragmatic algorithm is more stable than the forward algorithm, and even when the forward algorithm has converged, the pragmatic algorithm can still out‐perform it. Practically, we find that good designs for detecting an interaction have equal numbers of points on monotherapies and combination therapies, with those points typically placed in positions where a 50% response is expected. More uncertainty in monotherapy parameters leads to an optimal design with design points that are more spread out. Copyright © 2015 John Wiley & Sons, Ltd.     

CONSTRUCTION OF RESOLVABLE ROW-COLUMN DESIGNS BASED ON CONTRACTIONS

This paper defines the contraction of a resolvable row‐column design for more than two replicates. It shows that the (M,S)‐optimality criterion for the row‐column designs can be expressed simply in terms of the elements of the row and column incidence matrices of the contraction. This allows the development of a very fast algorithm to construct optimal or near‐optimal resolvable row‐column designs. The performance of such an algorithm is compared with an existing algorithm.     

Comparing and generating Latin Hypercube designs in Kriging models

In Computer Experiments (CE), a careful selection of the design points is essential for predicting the system response at untried points, based on the values observed at tried points. In physical experiments, the protocol is based on Design of Experiments, a methodology whose basic principles are questioned in CE. When the responses of a CE are modeled as jointly Gaussian random variables with their covariance depending on the distance between points, the use of the so called space-filling designs (random designs, stratified designs and Latin Hypercube designs) is a common choice, because it is expected that the nearer the untried point is to the design points, the better is the prediction. In this paper we focus on the class of Latin Hypercube (LH) designs. The behavior of various LH designs is examined according to the Gaussian assumption with exponential correlation, in order to minimize the total prediction error at the points of a regular lattice. In such a special case, the problem is reduced to an algebraic statistical model, which is solved using both symbolic algebraic software and statistical software. We provide closed-form computation of the variance of the Gaussian linear predictor as a function of the design, in order to make a comparison between LH designs. In principle, the method applies to any number of factors and any number of levels, and also to classes of designs other than LHs. In our current implementation, the applicability is limited by the high computational complexity of the algorithms involved.     

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.     

Cost-cautious designs for confirmatory bioassay

Confirmatory bioassay experiments take place in late stages of the drug discovery process when a small number of compounds have to be compared with respect to their properties. As the cost of the observations may differ considerably, the design problem is well specified by the cost of compound used rather than by the number of observations. We show that cost-efficient designs can be constructed using useful properties of the minimum support designs. These designs are particularly suited for studies where the parameters of the model to be estimated are known with high accuracy prior to the experiment, although they prove to be robust against typical inaccuracies of these values. When the parameters of the model can only be specified with ranges of values or by a probability distribution, we use a Bayesian criterion of optimality to construct the required designs. Typically, the number of their support points depends on the prior knowledge for the model parameters. In all cases we recommend identifying a set of designs with good statistical properties but different potential costs to choose from.     

Adaptive grid semidefinite programming for finding optimal designs

We find optimal designs for linear models using a novel algorithm that iteratively combines a semidefinite programming (SDP) approach with adaptive grid techniques. The proposed algorithm is also adapted to find locally optimal designs for nonlinear models. The search space is first discretized, and SDP is applied to find the optimal design based on the initial grid. The points in the next grid set are points that maximize the dispersion function of the SDP-generated optimal design using nonlinear programming. The procedure is repeated until a user-specified stopping rule is reached. The proposed algorithm is broadly applicable, and we demonstrate its flexibility using (i) models with one or more variables and (ii) differentiable design criteria, such as A-, D-optimality, and non-differentiable criterion like E-optimality, including the mathematically more challenging case when the minimum eigenvalue of the information matrix of the optimal design has geometric multiplicity larger than 1. Our algorithm is computationally efficient because it is based on mathematical programming tools and so optimality is assured at each stage; it also exploits the convexity of the problems whenever possible. Using several linear and nonlinear models with one or more factors, we show the proposed algorithm can efficiently find optimal designs.     

Intermediate Monitoring,Sample Size Reassessment,and Multi-Treatment Optimal Response-Adaptive Designs for Phase III Clinical Trials with More Than One Constraint

Optimal response-adaptive designs in Phase III clinical trial set up are becoming more and more current interest. In the present article, an optimal response-adaptive design is introduced for more than two treatments at hand. We minimize an objective function subject to more than one inequality constraints. For this purpose, we propose an extensive computer search algorithm. The proposed procedure is illustrated with extensive numerical computation and simulations. Some real data set is used to illustrate the proposed methodology.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

Lee discrepancy on mixed two- and three-level uniform augmented designs

Follow-up experiment is widely applied to various fields such as science and engineering, since it is an indispensable strategy, especially when some additional resources or information become available after the initial design of experiment is carried out. Moreover, some extra factors may be added in the follow-up experiment. One may augment the number of runs and/or factors for the purpose of application. In this paper, the issue of the uniform row augmented designs and column augmented designs with mixed two- and three-level is investigated. The uniformity of augmented designs is discussed under the Lee discrepancy, some lower bounds of Lee discrepancy for the augmented designs are obtained. The construction algorithm of the uniform augmented designs is given. Some numerical examples indicate that uniform augmented designs can be constructed with high efficiency.     

Saturated main-effect designs for factorial experiments

The results of a computer search for saturated designs for 2ⁿ factorial experiments with n runs is reported, (where n = 2 mod 4). A complete search of the design space is avoided by focussing on designs constructed from cyclic generators. A method of searching quickly for the best generators is given. The resulting designs are as good as, and sometimes better than, designs obtained via search algorithms reported in the literature. The addition of a further factor having three levels is also considered. Here, too, a complete search is avoided by restricting attention to the most efficient part of the design space under p-efficiency.     

A SEQUENTIAL METHOD OF CONSTRUCTING OPTIMAL BLOCK DESIGNS

An algorithm for the construction of a wide class of block designs including Balanced Incomplete Blocks (BIB) is described. The algorithm which allows the experimenter to give weights for a set of treatment contrasts uses an initial starting design to generate an optimal block design sequentially. The performance of the algorithm is illustrated by examples, and designs constructed by the algorithm compare favourably with designs generated by other methods.     

A class of saturated row–column designs

A new class of row–column designs is proposed. These designs are saturated in terms of eliminating two-way heterogeneity with an additive model. The (m,s)-criterion is used to select optimal designs. It turns out that all (m,s)-optimal designs are binary. Square (m,s)-optimal designs are constructed and they are treatment-connected. Thus, all treatment contrasts are estimable regardless of the row and column effects.     

Discrete M-robust designs for regression models

M-robust designs are defined and constructed for misspecified linear regression models with possibly autocorrelated errors on a discrete design space. These designs minimize the mean-squared errors if linear regression models are correct with uncorrelated errors, subject to two robust constraints which control the change of the bias and the change of variance under model departures. Simulated annealing algorithm is applied to construct M-robust designs. Examples are given to show M-robust designs and compare them with minimax robust designs.     

Optimal parametric design with applications to pharmacokinetic and pharmacodynamic trials

This paper considers optimal parametric designs, i.e. designs represented by probability measures determined by a set of parameters, for nonlinear models and illustrates their use in designs for pharmacokinetic (PK) and pharmacokinetic/pharmacodynamic (PK/PD) trials. For some practical problems, such as designs for modelling PK/PD relationship, this is often the only feasible type of design, as the design points follow a PK model and cannot be directly controlled. Even for ordinary design problems the parametric designs have some advantages over the traditional designs, which often have too few design points for model checking and may not be robust to model and parameter misspecifications. We first describe methods and algorithms to construct the parametric design for ordinary nonlinear design problems and show that the parametric designs are robust to parameter misspecification and have good power for model discrimination. Then we extend this design method to construct optimal repeated measurement designs for nonlinear mixed models. We also use this parametric design for modelling a PK/PD relationship and propose a simulation based algorithm. The application of parametric designs is illustrated with a three-parameter open one-compartment PK model for the ordinary design and repeated measurement design, and an Emax model for the phamacokinetic/pharmacodynamic trial design.