Construction of incomplete Sudoku square and partially balanced incomplete block designs

Abstract

The present article deals with the study of association among the elements of a Sudoku square. In this direction, we have defined an association scheme and constructed incomplete Sudoku square designs which are capable of studying four explanatory variables and also happen to be the designs for two-way elimination of heterogeneity. Some series of Partially Balanced Incomplete Block (PBIB) designs have also been obtained.     

On the construction of m-associate pbib designs II

Some series of m-associate triangular PBIB designs have been constructed. A table of now three associate designs in the range b, v ≦ 100; r, k ≦ 10 with their average efficiencies has been given. These designs with (v, b, r, k) are new in the sense that the existence of two associate PBIB designs (e.f. Clatworthy (1973), John & Turner (1977), Dey (1978)) with these parameters are not known. The constructions presented herein also yield partially balanced weighing designs, nested PBIB designs and PBIB designs for m11111 response exoeriments.     

REDUCED RESIDUE CLASSES CYCLIC PBIB DESIGNS

A new general class of m-class cyclic association scheme is defined for v treatments, where v is a composite number. A simple method of construction of PBIB designs having this association scheme using more than one initial block and some methods using only one initial block are proposed. A complete analysis of this type of PBIB designs is given. Also given is a list of 39 useful PBIB designs of this type having v≤15 and r≤10 and having only three associate classes together with their efficiency factors for all types of comparisons and over all efficiency factors.     

Connectedness of PBIB designs

A necessary and sufficient condition for the connectedness of an m-associate-class PBIB design is derived in a form, and using an approach, different from those in Ogawa, Ikeda, and Kageyama (1984) and Saha and Kageyama (1984). The applicability of this condition is shown in the context of group-divisible m-associate-class PBIB designs.     

Three methods of constructing pbib designs

Three construction methods of two- or three-associate partially balanced incomplete block (PBIB) designs are presented.     

On resolvable PBIB designs

Resolvable solutions for some two associate PBIB designs obtained by duplicating some non-resolvable designs are given. For the same designs 2-, 3- and 5-resolvable solutions are reported by Clatworthy (1973). A method of construction and some new resolvable PBIB designs obtained through this are given.     

Optimum covariate designs in partially balanced incomplete block (PBIB) design set-ups

The use of covariates in block designs is necessary when the covariates cannot be controlled like the blocking factor in the experiment. In this paper, we consider the situation where there is some flexibility for selection in the values of the covariates. The choice of values of the covariates for a given block design attaining minimum variance for estimation of each of the parameters has attracted attention in recent times. Optimum covariate designs in simple set-ups such as completely randomised design (CRD), randomised block design (RBD) and some series of balanced incomplete block design (BIBD) have already been considered. In this paper, optimum covariate designs have been considered for the more complex set-ups of different partially balanced incomplete block (PBIB) designs, which are popular among practitioners. The optimum covariate designs depend much on the methods of construction of the basic PBIB designs. Different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of products of matrices viz. Khatri–Rao product, Kronecker product have been conveniently used to construct optimum covariate designs with as many covariates as possible.     

Minimal augmented square designs and their extensions

Minimal square designs are proposed and compared. All treatment contrasts in both designs are estimable under the existence of two-way heterogeneity. That is, all designs are treatment-connected. Extended treatment-connected designs are generated by adding one column to minimal treatment-connected square designs. The extended designs not only have lower variances in paired comparisons of unreplicated treatments but also provide necessary degrees of freedom to estimate the process error. (M,S)-optimal extended designs are constructed systematically. Both square designs and their extensions have large numbers of unreplicated treatments.     

Pseudo New Modified Latin Square (NML m (m)) Type PBIB Designs

A New Modified Latin square [NML _i (m)] association scheme with i constraints for v = m ² treatments was introduced by Garg (2008 Garg , D. K. ( 2008 ). New modified Latin square (NMLi) type PBIB designs . J. Math. Syst. Sci. 1 ( 4 ): 83 – 89 . [Google Scholar]). In this article, a new association scheme known as Pseudo New Modified Latin square [Pseudo NML _m (m)] type association scheme is defined. The parameters of Pseudo NML _m (m) association scheme turned out to be parameters of NML _i (m) association scheme by taking i = m in NML _i (m) association scheme. The Pseudo NML _m (m) association scheme will be the usual NML _m (m) association scheme when m is a prime or a prime power. The PBIB designs following Pseudo NML _m (m) association scheme will be called the Pseudo NML _m (m) type PBIB designs. Analysis of Pseudo NML _m (m) designs along with a construction method of these designs is also given in this article.     

Patterned constructions of partially ballvced incomplete block design

Starting from the incidence matrix of a BIB design, we obtain PBIB designs with two or three associate classes following certain patterns for their block structures .     

A class of new pbib designs

Two series of PBIB designs, one with three associate classes and theother with four associate classes ars developed. Efficiency factors for two designs are computed.     

Some new group divisible designs

The paper lists fourteen new group divisible PBIB/2 designs, which were obtained using the computer program described in John (1976).     

On a class of partially balanced incomplete block designs

Bose and Clatworthy (1955) showed that the parameters of a two-class balanced incomplete block design with λ1=1,λ2=0 and satisfying r <k can be expressed in terms of just three parameters r,k,t. Later Bose (1963) showed that such a design is a partial geometry (r,k,t). Bose, Shrikhande and Singhi (1976) have defined partial geometric designs (r,k,t,c), which reduce to partial geometries when c=0. In this note we prove that any two class partially balanced (PBIB) design with r <k, is a partial geometric design for suitably chosen r,k,t,c and express the parameters of the PBIB design in terms of r,k,t,c and λ2. We also show that such PBIB designs belong to the class of special partially balanced designs (SPBIB) studied by Bridges and Shrikhande (1974).     

General upper bound for the number of blocks having a given number of treatments common with a given block

The purpose of this paper is systematically to derive the general upper bound for the number of blocks having a given number of treatments common with a given block of certain incomplete block designs. The approach adopted here is based on the spectral decomposition of N′N for the incidence matrix N of a design, where N' is the transpose of the matrix N. This approach will lead us to upper bounds for incomplete block designs, in particular for a large number of partially balanced incomplete block (PBIB) designs, which are not covered with the standard approach (Shah 1964, 1966), Kapadia (1966)) of using well known relations between blocks of the designs and their association schemes. Several results concerning block structure of block designs are also derived from the main theorem. Finally, further generalizations of the main theorem are discussed with some illustrations.     

ON RESOLVABLE INCOMPLETE BLOCK DESIGNS1

A method of constructing resolvable incomplete block designs for v(=ks, 2 ≤ k ≤ s - 1) treatments in blocks of size k using mutually orthogonal Latin squares is proposed. It has been seen in particular that when the number of replications is s — 1 (or s), which is feasible if s is a prime or a prime power, the method gives PBIB (3) (or semi-regular GD) designs. The analysis of such designs has also been discussed.     

An infinite class of PBIB designs

In this paper, an infinite class of partially balanced incomplete block (PBIB) designs of m+1 associate classes is constructed through the use of a series of row-orthogonal matrices known as partially balanced orthogonal designs (PBOD) of m-associate classes. For the purpose, a series of PBOD is obtained through a method described herein. An infinite class of regular GD designs is also reported.     

Optimal parametric design with applications to pharmacokinetic and pharmacodynamic trials

This paper considers optimal parametric designs, i.e. designs represented by probability measures determined by a set of parameters, for nonlinear models and illustrates their use in designs for pharmacokinetic (PK) and pharmacokinetic/pharmacodynamic (PK/PD) trials. For some practical problems, such as designs for modelling PK/PD relationship, this is often the only feasible type of design, as the design points follow a PK model and cannot be directly controlled. Even for ordinary design problems the parametric designs have some advantages over the traditional designs, which often have too few design points for model checking and may not be robust to model and parameter misspecifications. We first describe methods and algorithms to construct the parametric design for ordinary nonlinear design problems and show that the parametric designs are robust to parameter misspecification and have good power for model discrimination. Then we extend this design method to construct optimal repeated measurement designs for nonlinear mixed models. We also use this parametric design for modelling a PK/PD relationship and propose a simulation based algorithm. The application of parametric designs is illustrated with a three-parameter open one-compartment PK model for the ordinary design and repeated measurement design, and an Emax model for the phamacokinetic/pharmacodynamic trial design.     

Some New Series of Partially Balanced Residual Treatment Effects Designs

Blaisdell and Raghavarao (1980 Blaisdell , E. A. , Raghavarao , D. ( 1980 ). Partially balanced change-over designs based on m-associate class PBIB designs . JRSS B 42 : 334 – 338 . [Google Scholar] 1985 Blaisdell , E. A. , Raghavarao , D. ( 1985 ). Efficiency bounds for partially balanced change-over designs based on m-associate class PBIB designs . JRSS B 47 : 132 – 135 . [Google Scholar]) introduced partially balanced residual treatment effects designs and gave methods of construction of these designs. They also gave efficiencies for the estimated direct and residual elementary treatment contrasts of these designs. In this paper, we construct some new series of partially balanced residual treatment effects designs and also give their efficiencies for the estimated direct and residual elementary treatment contrasts.     

Block designs robust against the presence of positional effects

The problem considered is to find optimum designs for treatment effects in a block design (BD) setup, when positional effects are also present besides treatment and block effects, but they are ignored while formulating the model. In the class of symmetric balanced incomplete block designs, the Youden square design is shown to be optimal in the sense of minimizing the bias term in the mean squared error (MSE) of the best linear unbiased estimators of the full set of orthonormal treatment contrasts, irrespective of the value of the positional effects.     

An easy method of constructing latin square designs balanced for the immediate residual and other order effects

Bradley (1958) proposed a very simple procedure for constructing latin square designs to counterbalance the immediate sequential effect for an even number of treatments. When the number of treatments is odd, balance in a single latin square is not possible. In the present note we have developed an analogous method for the construction of such designs which may be used for an even or odd number of treatments. A proof has also been offered to assure the general validity of the procedure.