Premadhis Das

Optimum covariate designs in split-plot and strip-plot design set-ups

The problem considered is that of finding optimum covariate designs for estimation of covariate parameters in standard split-plot and strip-plot design set-ups with the levels of the whole-plot factor in r randomised blocks. Also an extended version of a mixed orthogonal array has been introduced, which is used to construct such optimum covariate designs. Hadamard matrices, as usual, play the key role for such construction.

D-Optimal Designs for Covariate Parameters in Block Design Set Up

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.

D-Optimal Designs for Covariate Models

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.

Optimum covariate designs in a binary proper equi-replicate block design set-up

The choice of covariates values for a given block design attaining minimum variance for estimation of each of the regression parameters of the model has attracted attention in recent times. In this article, we consider 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. The construction of optimum designs is based mainly on Hadamard matrices.

On Partial Balance with Orthogonal Sub-Factorial Structure

The factorial effects have been partitioned into sub-factorial effects introducing groups among the levels of the factors and the C matrix of a design has been characterised so that these sub-effects are orthogonally estimable with partial balance. The particular case when the number of levels can be factorised has been seen to be of special interest.

Designs Robust against Violation of Normality Assumption in the Standard F-test

In this article, designs are found for which the F-test of analysis of variance is insensitive to violation of normality assumption. Atiqullah (1962) proved that the F-test for treatments adjusting for blocks in the intra-block analysis of a balanced incomplete block design is robust against non-normality in the observations. Here an attempt has been made to identify other designs robust in this sense. In particular, it is observed that for testing relevant hypothesis, a partially balanced incomplete block design in block design setup, under certain conditions, is robust. Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012), in treatment-control design setup, is also studied. Moreover, a new measure of robustness is introduced for further study. The performance of the F-test in presence of non-normality in the observations for a quadratically balanced design is also examined.

Optimum Design for Estimation of Regression Parameters in Multi-Factor Set-Up

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.

Mixture designs with orthogonal blocks

ABSTRACT Constructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.

Confounded Factorial Design with Partial Balance and Orthogonal Sub-Factorial Structure

In this paper, the contrasts belonging to any effect are divided into a number of subsets and factorial designs are proposed such that these subsets (called sub-effects) are orthogonally estimated with balance. Such designs have been called ‘partially balanced design with orthogonal sub-factorial structure’. These designs are important in the sense that these allow more flexibility in the choice of the designs retaining desirable properties such as orthogonality and partial balance and also provide more insight into the nature of the contrasts belonging to any factorial effect.

OCDs in Randomized Block Design Set-Up

We have already mentioned in Chap. 2 that the question of optimal estimation of the covariate parameters under the covariate model was first studied by Troya Lopes, J Stat Plan Inference 6:373–419, (1982a); 7:49–75, (1982b) in Completely Randomized Design (CRD) set-up. Later, Das et al., J Stat Plan Inference 115:273–285, (2003) extended her work to the construction of optimum covariate designs in the set-up of Randomized Block Designs (RBD). Also, Rao et al., Electron Notes Discret Math 15:157–160, (2003) established a relationship between mixed orthogonal arrays (MOAs) and OCDs in CRD and RBD set-ups. In this chapter we discuss the construction procedures of OCDs in RBD set-up as considered by Das et al., J Stat Plan Inference 115:273–285, (2003) and Rao et al., Electron Notes Discret Math 15:157–160, (2003).