Approximate D-optimal Designs Research Articles

Efficient computational algorithms for approximate optimal designs

In this paper, we propose two simple yet efficient computational algorithms to obtain approximate optimal designs for multi-dimensional linear regression on a large variety of design spaces. We focus on the two commonly used optimal criteria, D- and A-optimal criteria. For D-optimality, we provide an alternative proof for the monotonic convergence for D-optimal criterion and propose an efficient computational algorithm to obtain the approximate D-optimal design. We further show that the proposed algorithm converges to the D-optimal design and then proves that the approximate D-optimal design converges to the continuous D-optimal design under certain conditions. For A-optimality, we provide an efficient algorithm to obtain approximate A-optimal design and conjecture the monotonicity of the proposed algorithm. Numerical comparisons suggest that the proposed algorithms perform well and they are comparable or superior to some existing algorithms.

Barycentric algorithm for computing D-optimal size- and cost-constrained designs of experiments

In this paper, we study the problem of D-optimal experimental design under two linear constraints, which can be interpreted as simultaneous restrictions on the size and on the cost of the experiment. For computing a size- and cost-constrained approximate D-optimal design, we propose a specification of the “barycentric” multiplicative algorithm with sequential removal of redundant design points. We analytically prove convergence results for the proposed algorithm and numerically demonstrate its favorable properties compared to competing methods.

A Simulated Annealing Algorithm for D-Optimal Design for 2-Way and 3-Way Polynomial Regression with Correlated Observations

Much of the previous work in D-optimal design for regression models with correlated errors focused on polynomial models with a single predictor variable, in large part because of the intractability of an analytic solution. In this paper, we present a modified, improved simulated annealing algorithm, providing practical approaches to specifications of the annealing cooling parameters, thresholds, and search neighborhoods for the perturbation scheme, which finds approximate D-optimal designs for 2-way and 3-way polynomial regression for a variety of specific correlation structures with a given correlation coefficient. Results in each correlated-errors case are compared with traditional simulated annealing algorithm, that is, the SA algorithm without our improvement. Our improved simulated annealing results had generally higher D-efficiency than traditional simulated annealing algorithm, especially when the correlation parameter was well away from 0.

Exact D-optimal designs for a linear log contrast model with mixture experiment for three and four ingredients

This study investigates the exact D-optimal designs of the linear log contrast model using the mixture experiment suggested by Aitchison and Bacon-Shone (1984) and the design space restricted by Lim (1987) and Chan (1988). Results show that for three ingredients, there are six extreme points that can be divided into two non-intersect sets S1 and S2. An exact N-point D-optimal design for N=3p+q,p≥1,1≤q≤2 arranges equal weight n/N,0≤n≤p at the points of S1 (S2) and puts the remaining weight (N−3n)/N on the points of S2 (S1) as evenly as possible. For four ingredients and N=6p+q,p≥1,1≤q≤5, an exact N-point design that distributes the weights as evenly as possible among the six supports of the approximate D-optimal design is exact D-optimal.

Formulated Beta-Cyfluthrin Shows Wide Divergence in Toxicity among Bird Species

It is generally assumed that the toxicity of pyrethroid insecticides to birds is negligible, though few species have been tested. The oral acute toxicity of formulated beta-cyfluthrin was determined for canaries (Serinus sp.), shiny cowbirds (Molothrus bonariensis), and eared doves (Zenaida auriculata). Single doses were administered to adults by gavage. Approximate lethal doses 50 (LD50) and their confidence intervals were determined by approximate D-optimal design. Canaries were found to be substantially more sensitive to formulated beta-cyfluthrin (LD50 = (170 ± 41) mg/kg) than the other two species tested (LD50 = (2234 ± 544) mg/kg and LD50 = (2271 ± 433) mg/kg, resp.). The LD50 obtained for canaries was also considerably lower than typical toxicity values available in the literature for pyrethroids. This study emphasizes the need for testing a broader range of species with potentially toxic insecticides, using modern up and down test designs with minimal numbers of birds.

Approximate D-optimal designs of experiments on the convex hull of a finite set of information matrices

Abstract In the paper we solve the problem of D ℋ-optimal design on a discrete experimental domain, which is formally equivalent to maximizing determinant on the convex hull of a finite set of positive semidefinite matrices. The problem of D ℋ-optimality covers many special design settings, e.g., the D-optimal experimental design for multivariate regression models. For D ℋ-optimal designs we prove several theorems generalizing known properties of standard D-optimality. Moreover, we show that D ℋ-optimal designs can be numerically computed using a multiplicative algorithm, for which we give a proof of convergence. We illustrate the results on the problem of D-optimal augmentation of independent regression trials for the quadratic model on a rectangular grid of points in the plane.

D-Optimal designs for weighted polynomial regression—A functional approach

This paper is concerned with the problem of computing approximateD-optimal design for polynomial regression with analytic weight function on a interval [m 0-a,m 0+a]. It is shown that the structure of the optimal design depends ona and weight function. Moreover, the optimal support points and weights are analytic functions ofa ata=0. We make use of a Taylor expansion to provide a recursive procedure for calculating theD-optimal designs.

Optimal designs for a growth curve model

Seber and Wild (Nonlinear Regression, Wiley, New York (1989) 343) considered the nonlinear fitting problem for a growth curve model E( Y| x)= β 0+ β 1 x+ β 2 x α . In this paper, we will determine the D-, A- and E-optimal designs for this model, with α known, on the interval [a,b], a<b . If a⩾0 and α>1, then the D-, A- and E-optimal designs are supported on three points including the two end-points and one interior point of [ a, b]. Moreover, the D- and E-optimal designs share the same support points. If b=− a=1 and α=2,3,…, then the D-, A- and E-optimal designs are symmetrically supported on four points including the two end-points. For the interval of the form [ a,1] (−1⩽ a<0), and α=2,3,…, the optimal designs are supported on either three or four points. Moreover, the exact n-point D-optimal designs are shown numerically to be supported at the support of the approximate D-optimal designs if a⩾− 1 2 ,α⩾2 and n⩾3.

Exact D-optimal designs for polynomial regression without intercept

For the regression model fk(x)=(x,x2,…,xk)T on [a,1],−1⩽a<1, the exact n-point D-optimal designs are proved to be ones that put mass as equally as possible among the support points of the approximate D-optimal design for fk(x) if n⩾k,a=−1 and k=2,4, if n⩾k,a⩾(2−10)/6 and k=2, and if n⩾2k,−0.0876991⩽a⩽(5−1)/25 and k=3. For the other cases, the properties of exact D-optimal designs are discussed on the basis of an intensive numerical study.

Exact and approximate D-optimal designs in polynomial regression

If the sample sizen is large enough, then the exact polynomial regression designs obtained by rounding the weights of the approximate D-optimal design to integral multiples of 1/n are D-optimal. This was shown by Salaevskii (1966) and Gaffke (1987). In this note, an efficient algorithm to determine the minimum sample sizend for a polynomial model of degreed is derived from a condition given by Huang (1987). Under an additional assumption we show that the conditions of Gaffke and Huang are equivalent; we verify the additional assumption for polynomial degreed≤40.

On D-optimal designs for linear models under correlated observations with an application to a linear model with multiple response

Abstract In the general linear model we set conditions under which an exact D-optimal design for uncorrelated observations with common variance is also D-optimal for correlated observations. Further we determine conditions under which approximate D-optimal designs can be considered as approximate D-optimal designs for correlated observations. Then these results are applied to a regression model with multiple response generalizing Theorem 1 of Krafft and Schaefer (J. Multivariate Anal. 42, 1992). In the above context, however, a serious problem may arise if the covariance matrix is not known; for the Gauss-Markov estimator with respect to a D-optimal design does not need to be calculable for the correlated case. This leads to D-optimal-invariant designs introduced by Bischoff (Ann. Inst. Statist. Math., 44, 1992); such a design τ∗ remains D-optimal when the covariance matrix is changed, and additionally the Gauss-Markov estimator with respect to the design τ∗ stays fixed. For regression models with multiple response we determine classes of covariance matrices for which a D-optimal design for uncorrelated observations with common variance is D-optimal-invariant. As examples we consider linear models where each response belongs to a regression model with intercept term.

D-Optimal designs for a multivariate regression model

Considered is a linear regression model with a one-dimensional control variable and an m-dimensional response variable y. The components of y may be correlated with known covariance matrix. Let B be the covariance matrix of the Gauss-Markoff estimator for the unknown parameter vector of the model. Under rather mild assumptions on the set of regression functions a factorization lemma for det B is proved which implies that D-optimal designs do not depend on the covariance matrix of y. This allows the use of recent results of Dette to determine approximate D-optimal designs for polynomial regression. A partial result for exact D-optimal designs is given too.

Marginally RestrictedD-Optimal Designs

Abstract In experimental design it often happens that values of some of the relevant carriers cannot be specified by the experimenter. We consider the problem of obtaining approximate D-optimal designs when the design space is a product space and the carriers associated with one margin are not subject to control. An equivalence theorem for D-optimal designs is presented. The essential ingredients of iterative schemes for generating designs are discussed and an example is presented.