D-Optimal Designs for Covariate Models
02 Jan 2014-Communications in Statistics-theory and Methods (Taylor & Francis Group)-Vol. 43, Iss: 1, pp 165-174
TL;DR: In this article, an alternative upper bound to the determinant of the information matrix has been found through completely symmetric C-matrices for the regression coefficients; this upper bound includes the upper bound given in Dey and Mukerjee (2006) obtained through diagonal C-Matrices.
Abstract: The problem of finding D-optimal designs in the presence of a number of covariates has been considered in the one-way set-up. This is an extension of Dey and Mukerjee (2006) in the sense that for fixed replication numbers of each treatment, an alternative upper bound to the determinant of the information matrix has been found through completely symmetric C-matrices for the regression coefficients; this upper bound includes the upper bound given in Dey and Mukerjee (2006) obtained through diagonal C-matrices. Because of the fact that a smaller class of C-matrices was used at the intermediate stage where the replication numbers were fixed, ultimately some optimal designs remained unidentified there. These designs have been identified here and thereby the conjecture made in Dey and Mukerjee (2006) has been settled.
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TL;DR: In this paper, the authors extended these results and proposed an extended mixed orthogonal array (EMOA) for the multi-factor set-up where the factorial effects involving at most t (≤m) factors are orthogonally estimable.
Abstract: The use of covariates in block designs is necessary when the experimental errors cannot be controlled by using only the qualitative factors. The choice of the values of the covariates for a given set-up ensuring minimum variance for the estimators of the regression parameters has attracted attention in recent times. Rao et al. (2003) proposed optimum covariate designs (OCD) through mixed orthogonal arrays for set-ups involving at most two factors where the analysis of variance (ANOVA) effects are orthogonally estimable. In this article, we extended these results and proposed OCDs for the multi-factor set-ups where the factorial effects involving at most t (≤m) factors are orthogonally estimable. It is seen that optimum designs can be obtained through extended mixed orthogonal arrays (EMOA, Dutta et al., 2009a) which reduce to mixed orthogonal arrays for the particular set-ups of Rao et al. (2003). We also proposed constructions of such arrays.
5 citations
Cites methods from "D-Optimal Designs for Covariate Mod..."
...For one-way set-up, D-optimal designs were proposed by Dey and Mukerjee (2006)....
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TL;DR: In this paper, optimum covariate designs have been considered for the set-up of the balanced treatment incomplete block (BTIB) designs, which form an important class of test-control designs.
Abstract: The use of covariates in block designs is necessary when the experimental errors cannot be controlled using only the qualitative factors. The choice of values of the covariates for a given set-up attaining minimum variance for estimation of the regression parameters has attracted attention in recent times. In this paper, optimum covariate designs (OCD) have been considered for the set-up of the balanced treatment incomplete block (BTIB) designs, which form an important class of test-control designs. It is seen that the OCDs depend much on the methods of construction of the basic BTIB designs. The series of BTIB designs considered in this paper are mainly those as described by Bechhofer and Tamhane (1981) and Das et al. (2005). Different combinatorial arrangements and tools such as Hadamard matrices and different kinds of products of matrices viz Khatri-Rao product and Kronecker product have been conveniently used to construct OCDs with as many covariates as possible.
3 citations
TL;DR: This paper attempts to find an optimal design which performs uniformly better with respect to multi-design-optimality criteria and demonstrates the method of comparison with an example along with its demerits in terms of design efficiency.
Abstract: In an industrial set-up, conditions of orthogonality and optimality of a statistical experimental design often get violated. This paper attempts to find an optimal design which performs uniformly better with respect to multi-design-optimality criteria. Initially, using non-dominated sorting genetic algorithm (NSGA-II) with single D-optimality criterion, a near-optimal design is searched for linear, interaction, quadratic, pure quadratic, and some pre-specified models. The solutions obtained are verified with the existing upper bounds. In the second phase, Pareto-optimal solutions are obtained with multi-design-optimality criteria. The method of comparison is illustrated with an example along with its demerits in terms of design efficiency.
2 citations
TL;DR: In this article, the authors propose to discuss several examples from standard text books and re-visit these examples with a view to suggest optimal/nearly optimal designs for estimation of the covariate parameter(s).
Abstract: We propose to discuss at length several examples from standard text books. All of these examples deal with analysis of covariance (ANCOVA) models and related analyses of data. We intend to capitalize on our understanding of optimal covariate designs (OCDs) in different ANCOVA models and re-visit these examples with a view to suggest optimal/nearly optimal designs for estimation of the covariate parameter(s). As we will see, for some examples our task is very much routine but for others, it is indeed a highly non trivial exercise. We intent to cover a total of six examples—divided in two parts. This is Part I—dealing with two examples.
1 citations
TL;DR: In this article, two strategies for specifying additional data to be included with the data of a non-orthogonal design are presented, which increase the magnitude of the information matrix X and the orthogonality of the design matrix.
Abstract: Two strategies for specifying additional data to be included with the data of a non-orthogonal design are presented. The additional data increase the magnitude of the information matrix X′X and the orthogonality of the design matrix. Sequentially, the new points are augmented to the original design, such that each new point optimally increases the smallest eigenvalue of X′X. The new runs are created in a predefined spherical region and a rectangular region. Optimum number of additional observations is presented in order to orthogonalize the design matrix X and optimize some functions of the information matrix X′X. Comparisons of the results acquired with the proposed methods versus the most commonly used procedures for data augmentation are carried out. In addition, the advantages of the use of our techniques over the studied methods to solve the augmenting data problems are discussed.
1 citations
Cites background from "D-Optimal Designs for Covariate Mod..."
...[11], Dey and Mukerjee [12] and several other authors....
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...The concept of D-optimal criterion has been advocated by Mood [6], Kiefer [7], Wynn [8], Box and Draper [9], Mitchell [10], Li et al. [11], Dey and Mukerjee [12] and several other authors....
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References
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Book•
01 Jan 1965TL;DR: Algebra of Vectors and Matrices, Probability Theory, Tools and Techniques, and Continuous Probability Models.
Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.
8,300 citations
4,122 citations
TL;DR: Rao's Linear Statistical Inference and Its Applications as discussed by the authors is one of the earliest works in statistical inference in the literature and has been translated into six major languages of the world.
Abstract: "C. R. Rao would be found in almost any statistician's list of five outstanding workers in the world of Mathematical Statistics today. His book represents a comprehensive account of the main body of results that comprise modern statistical theory." -W. G. Cochran "[C. R. Rao is] one of the pioneers who laid the foundations of statistics which grew from ad hoc origins into a firmly grounded mathematical science." -B. Efrom Translated into six major languages of the world, C. R. Rao's Linear Statistical Inference and Its Applications is one of the foremost works in statistical inference in the literature. Incorporating the important developments in the subject that have taken place in the last three decades, this paperback reprint of his classic work on statistical inference remains highly applicable to statistical analysis. Presenting the theory and techniques of statistical inference in a logically integrated and practical form, it covers: * The algebra of vectors and matrices * Probability theory, tools, and techniques * Continuous probability models * The theory of least squares and the analysis of variance * Criteria and methods of estimation * Large sample theory and methods * The theory of statistical inference * Multivariate normal distribution Written for the student and professional with a basic knowledge of statistics, this practical paperback edition gives this industry standard new life as a key resource for practicing statisticians and statisticians-in-training.
1,669 citations
TL;DR: In this article, the authors presented a list of the optimal designs for the problem of weighting k objects in chemical balance problems, and showed that for the most difficult case (n \equiv 3 (operatorname{mod} 4) = (n − 2k - 5) the optimum is known in all cases (n = (9, 11), (11, 15) and (12, 15).
Abstract: For the problem of weighing $k$ objects in $n$ weighings $(n \geq k)$ on a chemical balance, and certain related problems, we obtain new results and list the designs which have been proved $D$-optimum up to this time. While some of these optimality results have been known for some time, others are fairly recent. In particular, in the most difficult case $n \equiv 3(\operatorname{mod} 4)$ we prove a result characterizing optimum designs when $n \geq 2k - 5$. In addition, by a combination of theoretical bounds and computer search we find previously unknown optimum designs in the cases $(k, n) = (9, 11), (11, 15)$, and (12, 15), and establish the optimality of Mitchell's (10, 11) design. In some cases the optimum $X'X$ is not unique. Thus, we find two optimum $X'X$'s for the (6, 7), (8, 11), (10, 11), and (10, 15) cases. As a consequence of these results and other constructions, $D$-optimum designs are now known in all cases $k \leq 12$ (for all $n \geq k$), and in many other cases. Essentially complete listings for all $n \geq k$ had been given previously only for $k \leq 5$.
100 citations
"D-Optimal Designs for Covariate Mod..." refers background in this paper
...It is known that (cf. Galil and Kiefer, 1980; Wojtas, 1964) det X∗′X∗) is maximum at the extreme entries of X∗....
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90 citations
"D-Optimal Designs for Covariate Mod..." refers background in this paper
...(2.20) (cf. Wojtas, 1964)....
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...It is known that (cf. Galil and Kiefer, 1980; Wojtas, 1964) det X∗′X∗) is maximum at the extreme entries of X∗....
[...]