A note on robustness of D-optimal block designs for two-colour microarray experiments

Two-colour microarray experiments form an important tool in gene expression analysis. Due to the high risk of missing observations in microarray experiments, it is fundamental to concentrate not only on optimal designs but also on designs which are robust against missing observations. As an extension of Latif et al. (2009), we define the optimal breakdown number for a collection of designs to describe the robustness, and we calculate the breakdown number for various D-optimal block designs. We show that, for certain values of the numbers of treatments and arrays, the designs which are D-optimal have the highest breakdown number. Our calculations use methods from graph theory.     

Optimal regression designs in the presence of random block effects

A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.     

A- and D-optimal designs for a log contrast model for experiments with mixtures

A- and D-optimal designs are investigated for a log contrast model suggested by Aitchison & Bacon-Shone for experiments with mixtures. It is proved that when the number of mixture components q is an even integer, A- and D-optimal designs are identical; and when q is an odd integer, A- and D-optimal designs are different, but they share some common support points and are very close to each other in efficiency. Optimal designs with a minimum number of support points are also constructed for 3, 4, 5 and 6 mixture components.     

A note on A-efficiency of connected designs

Khatri (1982) justified a statement “Among the variance balanced designs, one should choose the design which has the minimum number of plots”, in terms of A-efficiency. Examples of connected block designs are here presented to disprove Khatri's statement. Some general results are also provided to stress our statements.     

Extensions of D-optimal minimal designs for symmetric mixture models

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide the desirable response characteristics in finished products. D-optimal minimal designs have been considered for a variety of mixture models, including Scheffé's linear, quadratic, and cubic models. Usually, these D-optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the lack-of-fit (LOF) tests. Also, the majority of the design points in D-optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. In this article, extensions of the D-optimal minimal designs are developed for a general mixture model to allow additional interior points in the design space to enable prediction of the entire response surface. Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. We compare the proposed designs with Cornell (1986 Cornell, J.A. (1986). A comparison between two ten-point designs for studying three-component mixture systems. J. Qual. Technol. 18(1):1–15.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) two 10-point designs for the LOF test by simulations.     

Bayesian D-optimal designs for exponential regression models

We consider the Bayesian D-optimal design problem for exponential growth models with one, two or three parameters. For the one-parameter model conditions on the shape of the density of the prior distribution and on the range of its support are given guaranteeing that a one-point design is also Bayesian D-optimal within the class of all designs. In the case of two parameters the best two-point designs are determined and for special prior distributions it is proved that these designs are Bayesian D-optimal. Finally, the exponential growth model with three parameters is investigated. The best three-point designs are characterized by a nonlinear equation. The global optimality of these designs cannot be proved analytically and it is demonstrated that these designs are also Bayesian D-optimal within the class of all designs if gamma-distributions are used as prior distributions.     

Some sufficient conditions for the type 1 optimality of block designs

In this paper, we investigate the problem of determining block designs which are optimal under type 1 optimality criteria within various classes of designs having υ treatments arranged in b blocks of size k. The solutions to two optimization problems are given which are related to a general result obtained by Cheng (1978) and which are useful in this investigation. As one application of the solutions obtained, the definition of a regular graph design given in Mitchell and John (1977) is extended to that of a semi-regular graph design and some sufficient conditions are derived for the existence of a semi-regular graph design which is optimal under a given type 1 criterion. A result is also given which shows how the sufficient conditions derived can be used to establish the optimality under a specific type 1 criterion of some particular types of semi- regular graph designs having both equal and unequal numbers of replicates. Finally,some sufficient conditions are obtained for the dual of an A- or D-optimal design to be A- or D-optimal within an appropriate class of dual designs.     

D-optimal minimax regression designs on discrete design space

One classical design criterion is to minimize the determinant of the covariance matrix of the regression estimates, and the designs are called D-optimal designs. To reflect the nature that the proposed models are only approximately true, we propose a robust design criterion to study response surface designs. Both the variance and bias are considered in the criterion. In particular, D-optimal minimax designs are investigated and constructed. Examples are given to compare D-optimal minimax designs with classical D-optimal designs.     

Some d-optimal weighing designs for n≡ (mod 4)

A number of D-optimal weighing designs are constructed with the help of block matrices. The D-optimal designs (n,k,s)=(19,13,10), (19,14,7), (19,14,8), (19,15,7), (19,15,8), (19,17,6), (19,18,6), (23,16,8), (23,17,8), (23,18,8), (4n?1,2n+3,(3n+4)/2), (4n?1,2n+4,n+3), (4n?1,2n+4,n+2) where n≡0 mod 4 and a skew H_n exists, (31,24,8), (31,25,8) and many others are constructed. A computer routine leading to locally D-optimal designs is presented.     

Some methods for constructing efficiency-balanced block designs

Three methods are presented for constructing connected efficiency-balanced block designs from other block designs with the same properties. The resulting designs differ from the original ones in the number of blocks and/or in the number of experimental units and their arrangement, while the number of treatments remains unaltered. Some remarks on the proposed methods of construction refer also to variance-balanced block designs.     

D-optimal weighing designs and optimal chemical balance weighing designs for estimating the total weight

This paper provides D-optimal spring balance designs for estimating individual weights when the number of objects to be weighed in each weighing, B, is fixed. D-optimal chemical balance designs for estimating total weight under both homogeneous and nonhomogeneous error variances are found when the number of objects weighed in each weighing is ≥ B, a fixed number.

We indicate the restriction used in Chacko & Dey(1978) and Kageyama(1988), i.e. that chemical designs X be restricted to designs in which exactly “a” objects are replaced on the left pan and exactly “b” on the right pan in each of the weighings for a, b > 0, is unnecessary.     

Connectedness of complete block designs under an interference model

We consider an experiment with fixed number of blocks, in which a response to a treatment can be affected by treatments from neighboring units. For such experiment the interference model with neighbor effects is studied. Under this model we study connectedness of binary complete block designs. Assuming the circular interference model with left-neighbor effects we give the condition for minimal number of blocks necessary to obtain connected design. For a specified class of binary, complete block designs, we show that all designs are connected. Further we present the sufficient and necessary conditions of connectedness of designs with arbitrary, fixed number of blocks.     

A note on trend-free block designs

Trend-free and nearly trend-free block designs were developed to eliminate polynomial trends across the plots of experimental designs. Yeh, Bradley and Notz (1985) proved that certain nearly trend-free designs are A- and D-optimal in a subclass of all competing designs. This article extends that result by enlarging the class of designs for which the optimality holds, and by increasing the class of optimality criteria from A- and D-optimality to the class of all Schur-convex nonincreasing functions.     

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.     

Near-Balanced Incomplete Block Designs,With an Application to Poster Competitions

Judging scholarly posters creates a challenge to assign the judges efficiently. If there are many posters and few reviews per judge, the commonly used balanced incomplete block design is not a feasible option. An additional challenge is an unknown number of judges before the event. We propose two connected near-balanced incomplete block designs that both satisfy the requirements of our setting: one that generates a connected assignment and balances the treatments and another one that further balances pairs of treatments. We describe both fixed and random effects models to estimate the population marginal means of the poster scores and rationalize the use of the random effects model. We evaluate the estimation accuracy and efficiency, especially the winning chance of the truly best posters, of the two designs in comparison with a random assignment via simulation studies. The two proposed designs both demonstrate accuracy and efficiency gain over the random assignment.     

D-optimal statistical designs with restricted string property

This paper applies Frechet derivatives to derive asymp-totically D-optimal statistical designs where the designmatrix is a (O,l)-matrix having exactly one run [of length at most k(, the number of parameters) of l's in each row. These asymptotic results have been utilized in dealing with the more intractable design problem with a finite number of observations. The problemof E-optimality has also been considered.     

New classes of D-optimal edge designs

Edge designs are screening experimental designs that allow a model-independent estimate of the set of relevant variables, thus providing more robustness than traditional designs. In this paper we construct new classes of D-optimal edge designs. This construction uses weighing matrices of order n and weight k together with permutation matrices of order n to obtain D-optimal edge designs. One linear and one quadratic simulated screening scenarios are studied and compared using linear regression and edge designs analysis.     

Optimal and efficient designs for 2-parameter nonlinear models

By Carathéodory's theorem, for a k-parameter nonlinear model, the minimum number of support points for any D-optimal design is between k and k(k+1)/2. Characterizing classes of models for which a D-optimal design sits on exactly k support points is of great theoretical interest. By utilizing the equivalence theorem, we identify classes of 2-parameter nonlinear models for which a D-optimal design is precisely supported on 2 points. We also introduce the theory of maximum principle from differential equations into the design area and obtain some results on characterizing the minimally supported nonlinear designs. Examples are given to demonstrate our results. Designs with minimum number of support points may not always be suitable in practice. To alleviate this problem, we utilize some geometric and analytical methods to obtain some efficient designs which provide more opportunity for the model checking and prevent biases due to mis-specified initial parameters.     

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.