Equally spaced design points in polynomial regression: A comparison of systematic sampling methods with the optimal design of experiments

In some situations an experimenter may desire to have equally spaced design points. Three methods of obtaining such points on the interval [—1,1]—namely systematic random sampling, centrally located systematic sampling, and a purposive systematic sampling method which includes the endpoints - 1 and 1 as two of the design points-are evaluated under the D-optimal and G-optimal criteria. These methods are also compared to the optimal designs in polynomial regression and to the limiting designs of Kiefer and Studden (1976).     

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.     

Optimum designs for estimating EDp based on raw optical density data

For raw optical density (ROD) data, such as those generated in biological assays employing an ELISA plate reader, ED_p-optimal designs are identified for a family of homogeneous non-linear models with two parameters. In every case, the theoretical ED_p-optimal design is a design with one or two support points. These theoretical optimal designs might not be suitable for many practical applications. To overcome this shortcoming, we have specified ED_p-optimal designs within the class of k-point equally spaced and uniform designs. The efficiency robustness of these designs with respect to initial nominal values of the parameters have been investigated.     

Robustness properties of minimally‐supported Bayesian D‐optimal designs for heteroscedastic models

Bayesian D‐optimal designs supported on a fixed number of points were found by Dette & Wong (1998) for estimating parameters in a polynomial model when the error variance depends exponentially on the explanatory variable. The present authors provide optimal designs under a broader class of error variance structures and investigate the robustness properties of these designs to model and prior distribution assumptions. A comparison of the performance of the optimal designs relative to the popular uniform designs is also given. The authors' results suggest that Bayesian D‐optimal designs suported on a fixed number of points are more likely to be globaly optimal among all designs if the prior distribution is symmetric and is concentrated around its mean.     

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.     

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.     

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.     

Computing A-optimal Designs for Weighted Polynomial Regression by Taylor Expansion

This article is concerned with the problem of constructing A-optimal design for polynomial regression with analytic weight function on the interval [m ? a, m + a], m, a > 0. It is shown that the structure of the optimal design depends on a and weight function only, as a close to 0. Moreover, if the weight function is an analytic function a, then a scaled version of optimal support points, and weights are analytic functions of a at a = 0. We make use of a Taylor expansion to provide a recursive procedure for calculating the A-optimal designs. Examples are presented to illustrate the procedures for computing the optimal designs.     

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.     

E-optimal designs for polynomial regression without intercept

We give all E-optimal designs for the mean parameter vector in polynomial regression of degree d without intercept in one real variable. The derivation is based on interplays between E-optimal design problems in the present and in certain heteroscedastic polynomial setups with intercept. Thereby the optimal supports are found to be related to the alternation points of the Chebyshev polynomials of the first kind, but the structure of optimal designs essentially depends on the regression degree being odd or even. In the former case the E-optimal designs are precisely the (infinitely many) scalar optimal designs, where the scalar parameter system refers to the Chebyshev coefficients, while for even d there is exactly one optimal design. In both cases explicit formulae for the corresponding optimal weights are obtained. Remarks on extending the results to E-optimality for subparameters of the mean vector (in heteroscdastic setups) are also given.     

OPTIMAL DESIGNS FOR FITTING A PROPORTIONAL HAZARDS REGRESSION MODEL TO DATA SUBJECT TO CENSORING

Suppose the probability model for failure time data, subject to censoring, is specified by the hazard function λ(t)exp(β_T x), where x is a vector of covariates. Analytical difficulties involved in finding the optimal design are avoided by assuming that λ is completely specified and by using D-optimality based on the information matrix for β Optimal designs are found to depend on β, but some results of practical consequence are obtained. It is found that censoring does not affect the choice of design appreciably when β^T x ≥ 0 for all points of the feasible region, but may have an appreciable effect when β_ix_i 0, for all i and all points in the feasible experimental region. The nature of the effect is discussed in detail for the cases of one and two parameters. It is argued that in practical biomedical situations the optimal design is almost always the same as for uncensored data.     

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.     

Some robust designs for polynomial regression models

The author identifies static optimal designs for polynomial regression models with or without intercept. His optimality criterion is an average between the D‐optimality criterion for the estimation of low‐degree terms and the D₈‐optimality criterion for testing the significance of higher degree terms. His work relies on classical results concerning canonical moments and the theory of continued fractions.     

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.     

Optimal blocked main effects plans

In this paper, we consider the construction of optimal blocked main effects designs where m two-level factors are to be studied in N runs which are partitioned into b blocks of equal size. For N ≡ 2±od4 sufficient conditions are derived for a design to be Φ _f optimal among all designs having main effects occurring equally often at their high and low levels within blocks and then this result is extended to the class of all designs for the case when the block size is two. Methods of constructing designs satisfying the sufficient conditions derived are also given.     

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     

Designs for weighted least squares regression, with estimated weights

We study designs, optimal up to and including terms that are O(n ^?1), for weighted least squares regression, when the weights are intended to be inversely proportional to the variances but are estimated with random error. We take a finite, but arbitrarily large, design space from which the support points are to be chosen, and obtain the optimal proportions of observations to be assigned to each point. Specific examples of D- and I-optimal design for polynomial responses are studied. In some cases the same designs that are optimal under homoscedasticity remain so for a range of variance functions; in others there tend to be more support points than are required in the homoscedastic case. We also exhibit minimax designs, that minimize the maximum, over finite classes of variance functions, value of the loss. These also tend to have more support points, often resulting from the breaking down of replicates into clusters.     

Optimal design for classification of functional data

We study the design problem for the optimal classification of functional data. The goal is to select sampling time points so that functional data observed at these time points can be classified accurately. We propose optimal designs that are applicable to either dense or sparse functional data. Using linear discriminant analysis, we formulate our design objectives as explicit functions of the sampling points. We study the theoretical properties of the proposed design objectives and provide a practical implementation. The performance of the proposed design is evaluated through simulations and real data applications. The Canadian Journal of Statistics 48: 285–307; 2020 © 2019 Statistical Society of Canada     

Locally optimal designs for binary dose‐response models

In this article, we consider the problem of seeking locally optimal designs for nonlinear dose‐response models with binary outcomes. Applying the theory of Tchebycheff Systems and other algebraic tools, we show that the locally D‐, A‐, and c‐optimal designs for three binary dose‐response models are minimally supported in finite, closed design intervals. The methods to obtain such designs are presented along with examples. The efficiencies of these designs are also discussed. The Canadian Journal of Statistics 46: 336–354; 2018 © 2018 Statistical Society of Canada     

Admissible two-stage designs for phase II cancer clinical trials that incorporate the expected sample size under the alternative hypothesis

two‐stage studies may be chosen optimally by minimising a single characteristic like the maximum sample size. However, given that an investigator will initially select a null treatment e?ect and the clinically relevant di?erence, it is better to choose a design that also considers the expected sample size for each of these values. The maximum sample size and the two expected sample sizes are here combined to produce an expected loss function to ?nd designs that are admissible. Given the prior odds of success and the importance of the total sample size, minimising the expected loss gives the optimal design for this situation. A novel triangular graph to represent the admissible designs helps guide the decision‐making process. The H ₀‐optimal, H ₁‐optimal, H ₀‐minimax and H ₁‐minimax designs are all particular cases of admissible designs. The commonly used H ₀‐optimal design is rarely good when allowing stopping for e?cacy. Additionally, the δ‐minimax design, which minimises the maximum expected sample size, is sometimes admissible under the loss function. However, the results can be varied and each situation will require the evaluation of all the admissible designs. Software to do this is provided. Copyright © 2012 John Wiley & Sons, Ltd.