D-Optimal Designs for Covariate Parameters in Block Design Set Up
04 Sep 2010-Communications in Statistics-theory and Methods (Taylor & Francis Group)-Vol. 39, Iss: 19, pp 3434-3443
TL;DR: In this article, the problem of finding D-optimal design for the estimation of covariate parameters and the treatment and block contrasts in a block design set up in the presence of non-stochastic controllable covariates was considered.
Abstract: The problem considered is that of finding D-optimal design for the estimation of covariate parameters and the treatment and block contrasts in a block design set up in the presence of non stochastic controllable covariates, when N = 2(mod 4), N being the total number of observations. It is clear that when N ≠ 0 (mod 4), it is not possible to find designs attaining minimum variance for the estimated covariate parameters. Conditions for D-optimum designs for the estimation of covariate parameters were established when each of the covariates belongs to the interval [−1, 1]. Some constructions of D-optimal design have been provided for symmetric balanced incomplete block design (SBIBD) with parameters b = v, r = k = v − 1, λ =v − 2 when k = 2 (mod 4) and b is an odd integer.
Citations
More filters
TL;DR: The problem of finding the optimum covariate design (OCD) for the estimation of covariate parameters in a binary proper equi-replicate block (BPEB) design model with covariates, which cover a large class of designs in common use, is considered.
8 citations
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.
7 citations
Cites methods from "D-Optimal Designs for Covariate Par..."
...Also for incomplete block design set-up, D-optimal designs for estimation of the regression coefficients only were proposed by Dutta et al. (2010b) where the most efficient designs did not exist....
[...]
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 background from "D-Optimal Designs for Covariate Par..."
...Dutta et al. (2009b, 2010a) also considered optimal estimation of the regression coefficients under different set-ups where the ANOVA effects are non orthogonally estimable....
[...]
...The OCDs are difficult to construct here unless some patterns in the incidence matrices exist (cf. Das et al., 2003, Dutta et al., 2009b, 2010a)....
[...]
...Furthermore, Dutta et al. (2010b) considered D-optimal covariate designs for estimation of regression coefficients in incomplete block design set-up when globally optimal design did not exist....
[...]
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.
3 citations
TL;DR: In this paper, a short review of the present developments in this connection is presented, where the covariate effects were estimated with global optimality and a series of different design set-ups and proposed optimum covariate designs were considered.
Abstract: Study of optimality of designs with covariate models started with Lopes Troya (1982a, 1982b). Later on, it was considered by a number of authors. Notable among them are Das et al. (2003) and Rao et al. (2003). In a series of papers, Dutta, Das and Mandal considered different design set-ups and proposed optimum covariate designs where the covariate effects were estimated with global optimality. The paper contains a short review of the present developments in this connection.
1 citations
Cites background from "D-Optimal Designs for Covariate Par..."
...But it was observed in Dutta et al. (2010) that if b = mv, where m is a positive integer, b is the number of blocks and v is the number of treatments then an OCD accommodating (k − 1) covariates can always be constructed for the BPEBD set-up where k is the block size....
[...]
...Keeping such situations in view, Dutta et al. (2009b, 2010a) proposed optimum covariate designs for the set-ups of partially balanced incomplete block designs (PBIBD), cyclic designs and more generally the set-ups of binary proper equireplicate block designs (BPEBD)....
[...]
References
More filters
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
Book•
08 Mar 1993
TL;DR: Experimental designs in linear models Optimal designs for Scalar Parameter Systems Information Matrices Loewner Optimality Real Optimality Criteria Matrix Means The General Equivalence Theorem Optimal Moment Matrices and Optimal Designs D-, A-, E-, T-Optimality Admissibility of moment and information matrices Bayes Designs and Discrimination Designs Efficient Designs for Finite Sample Sizes Invariant Design Problems Kiefer Optimality Rotatability and Response Surface Designs Comments and References Biographies Bibliography Index as discussed by the authors
Abstract: Experimental Designs in Linear Models Optimal Designs for Scalar Parameter Systems Information Matrices Loewner Optimality Real Optimality Criteria Matrix Means The General Equivalence Theorem Optimal Moment Matrices and Optimal Designs D-, A-, E-, T-Optimality Admissibility of Moment and Information Matrices Bayes Designs and Discrimination Designs Efficient Designs for Finite Sample Sizes Invariant Design Problems Kiefer Optimality Rotatability and Response Surface Designs Comments and References Biographies Bibliography Index.
1,823 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