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.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Robust designs for experiments with blocks

ABSTRACT

For experiments running in field plots or over time, the observations are often correlated due to spatial or serial correlation, which leads to correlated errors in a linear model analyzing the treatment means. Without knowing the exact correlation matrix of the errors, it is not possible to compute the generalized least-squares estimator for the treatment means and use it to construct optimal designs for the experiments. In this paper, we propose to use neighborhoods to model the covariance matrix of the errors, and apply a modified generalized least-squares estimator to construct robust designs for experiments with blocks. A minimax design criterion is investigated, and a simulated annealing algorithm is developed to find robust designs. We have derived several theoretical results, and representative examples are presented.     

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.     

Robust prediction and extrapolation designs for censored data

In this paper we present the construction of robust designs for a possibly misspecified generalized linear regression model when the data are censored. The minimax designs and unbiased designs are found for maximum likelihood estimation in the context of both prediction and extrapolation problems. This paper extends preceding work of robust designs for complete data by incorporating censoring and maximum likelihood estimation. It also broadens former work of robust designs for censored data from others by considering both nonlinearity and much more arbitrary uncertainty in the fitted regression response and by dropping all restrictions on the structure of the regressors. Solutions are derived by a nonsmooth optimization technique analytically and given in full generality. A typical example in accelerated life testing is also demonstrated. We also investigate implementation schemes which are utilized to approximate a robust design having a density. Some exact designs are obtained using an optimal implementation scheme.     

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.     

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.     

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.     

Minimax designs for approximately linear models with AR(1) errors

We obtain designs for linear regression models under two main departures from the classical assumptions: (1) the response is taken to be only approximately linear, and (2) the errors are not assumed to be independent, but to instead follow a first-order autoregressive process. These designs have the property that they minimize (a modification of) the maximum integrated mean squared error of the estimated response, with the maximum taken over a class of departures from strict linearity and over all autoregression parameters ρ,|ρ,| < 1, of fixed sign. Specific methods of implementation are discussed. We find that an asymptotically optimal procedure for AR(1) models consists of choosing points from that design measure which is optimal for uncorrelated errors, and then implementing them in an appropriate order.     

Optimal discrimination designs for exponential regression models

We investigate optimal designs for discriminating between exponential regression models of different complexity, which are widely used in the biological sciences; see, e.g., Landaw [1995. Robust sampling designs for compartmental models under large prior eigenvalue uncertainties. Math. Comput. Biomed. Appl. 181–187] or Gibaldi and Perrier [1982. Pharmacokinetics. Marcel Dekker, New York]. We discuss different approaches for the construction of appropriate optimality criteria, and find sharper upper bounds on the number of support points of locally optimal discrimination designs than those given by Caratheodory's Theorem. These results greatly facilitate the numerical construction of optimal designs. Various examples of optimal designs are then presented and compared to different other designs. Moreover, to protect the experiment against misspecifications of the nonlinear model parameters, we adapt the design criteria such that the resulting designs are robust with respect to such misspecifications and, again, provide several examples, which demonstrate the advantages of our approach.     

On the Choice of a Prior for Bayesian D-Optimal Designs for the Logistic Regression Model with a Single Predictor

The Bayesian design approach accounts for uncertainty of the parameter values on which optimal design depends, but Bayesian designs themselves depend on the choice of a prior distribution for the parameter values. This article investigates Bayesian D-optimal designs for two-parameter logistic models, using numerical search. We show three things: (1) a prior with large variance leads to a design that remains highly efficient under other priors, (2) uniform and normal priors lead to equally efficient designs, and (3) designs with four or five equidistant equally weighted design points are highly efficient relative to the Bayesian D-optimal designs.     

A relaxation procedure for calculating (G–)minimax optimal designs

Summary: In nonlinear statistical models, standard optimality functions for experimental designs depend on the unknown parameters of the model. An appealing and robust concept for choosing a design is the minimax criterion. However, so far, minimax optimal designs have been calculated efficiently under various restrictive conditions only. We extend an iterative relaxation scheme originally proposed by Shimizu and Aiyoshi (1980) and prove its convergence under very general assumptions which cover a variety of situations considered in experimental design. Application to different specific design criteria is discussed and issues of practical implementation are addressed. First numerical results suggest that the method may be very efficient with respect to the number of iterations required.*Supported by a grant from the Deutsche Forschungsgemeinschaft. We are grateful to a referee for his constructive suggestions.     

On construction of exact experimental designs from discrete ones

In the paper we try to evaluate approximately optimum exact designs of required goodness from a known approximately optimum discrete design. In dependence on this discrete design we define some subsets of the set of all considered exact designs.

Then we look for sufficiency conditions, on which elements of these subsets are approximately optimum (exact) designs of the required goodness. Moreover we offer a method to find such elements, if they exist at all. Finally we prove some statements concerning the existence of such elements for two classes of (optimum) criteria.     

Multiplicative algorithms for computing optimum designs

We study a new approach to determine optimal designs, exact or approximate, both for the uncorrelated case and when the responses may be correlated. A simple version of this method is based on transforming design points on a finite interval to proportions of the interval. Methods for determining optimal design weights can therefore be used to determine optimal values of these proportions. We explore the potential of this method in a range of examples encompassing linear and non-linear models, some assuming a correlation structure and some with more than one design variable.     

Robust sequential designs for approximately linear models

We consider the problem of the sequential choice of design points in an approximately linear model. It is assumed that the fitted linear model is only approximately correct, in that the true response function contains a nonrandom, unknown term orthogonal to the fitted response. We also assume that the parameters are estimated by M-estimation. The goal is to choose the next design point in such a way as to minimize the resulting integrated squared bias of the estimated response, to order n^-1. Explicit applications to analysis of variance and regression are given. In a simulation study the sequential designs compare favourably with some fixed-sample-size designs which are optimal for the true response to which the sequential designs must adapt.     

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.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal designs.     

NESTING OPTIMAL MAIN EFFECTS PLANS AND OPTIMAL FOLDOVER DESIGNS

ABSTRACT

Optimal main effects plans (MEPs) and optimal foldover designs can often be performed as a series of nested optimal designs. Then, if the experiment cannot be completed due to time or budget constraints, the fraction already performed may still be an optimal design. We show that the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEP for 4t factors in 4t + 2 points nested within it. In general, the optimal MEP for 4t factors in 4t + 4 points does not contain the optimal MEPs for 4t factors in 4t + 1, 4t + 2, or 4t + 3 points and the optimal MEP for 4t + 1 factors in 4t + 4 points does not contain the optimal MEPs for 4t + 1 factors in 4t + 2 or 4t + 3 points. We also show that the runs in an orthogonal design for 4t factors in 4t + 4 points, and the optimal foldover designs obtained by folding, should be performed in a certain sequence in order to avoid the possibility of a singular X'X matrix.     

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.     

Bayes A-optimal designs for comparing test treatments with a control

The problem of comparing v test treatments simultaneously with a control treatment when k, v ⩾ 3 is considered. Following the work of Majumdar (1992), we use exact design theory to derive Bayes A-optimal block designs and optimal Г-minimax designs for a more general prior assumption for the one-way elimination of heterogeneity model. Examples of robust optimal designs, highly efficient designs, and the comparisons of the approximate optimal designs that are derived by our methods and by some other existing rounding-off schemes when using Owen's procedure are also provided.