Orthogonal Arrays and Row-Column and Block Designs for CDC Systems

Orthogonal arrays of strength d were introduced and applied in the construction of confounded symmetrical and asymmetrical factorial designs, multifactorial designs (fractional replication) and so on Rao [1-4] Orthogonal arrays of strength 2 were found useful in the construction of other combinatorial arrangements. Bose, Shrikhande and Parker [5] used it in the disproof of Euler’s conjecture. Ray-Chaudhari and Wilson [6-7] used orthogonal arrays of strength 2 to generate resolvable balanced incomplete block designs. Rao [8] gave method of construction of semi-balanced array of strength 2. These arrays have been used in the construction of resolvable balanced incomplete block design. A complete diallel crossing system is one in which a set of p inbred lines, where p is a prime or power of a prime, is chosen and crosses are made among these lines. This procedure gives rise to a maximum of v =p2 combination. Griffing [9] gave four experimental methods:


Introduction
Orthogonal arrays of strength d were introduced and applied in the construction of confounded symmetrical and asymmetrical factorial designs, multifactorial designs (fractional replication) and so on Rao [1][2][3][4] Orthogonal arrays of strength 2 were found useful in the construction of other combinatorial arrangements. Bose, Shrikhande and Parker [5] used it in the disproof of Euler's conjecture. Ray-Chaudhari and Wilson [6][7] used orthogonal arrays of strength 2 to generate resolvable balanced incomplete block designs. Rao [8] gave method of construction of semi-balanced array of strength 2. These arrays have been used in the construction of resolvable balanced incomplete block design. A complete diallel crossing system is one in which a set of p inbred lines, where p is a prime or power of a prime, is chosen and crosses are made among these lines. This procedure gives rise to a maximum of v =p 2 combination. Griffing [9] gave four experimental methods: a) parental line combinations, one set of F 1 's hybrid and reciprocal F 1 's hybrid is included (all v = p 2 combination) b) parents and one set of F 1 's hybrid is included but reciprocal F 1 's hybrid is not (v = 1/2 p (p+1) combination) c) one set of F 1 's hybrid and reciprocal are included but not the parents (v =p (p-1) combination) and d) one set of F 1 's hybrid but neither parents nor reciprocals F 1 's hybrid is included (v = 1/2p (p-1) . The problem of generating optimal mating designs for CDC method D has been investigated by several authors Singh, Gupta, and Parsad [10].
For CDC method A, B and C models of Griffing [9] involves the general combining ability (g ca) and specific combining ability (s ca ) effects of lines. Let n c denote the total number of crosses involved in CDC method A, B and C and it is desired to compare the average effects or g ca effects of lines. Generally, the experiments of these methods are conducted using either a completely randomized design (CRD) or a randomized complete block (RCB) design involving n c crosses as treatments. The number of crosses in such mating design increases rapidly with an increase in the number of lines p. Thus, if p is large adoption of CRD or an RCB design is not appropriate unless the experimental units are extremely homogeneous. It is for this reason that the use of incomplete block CDC methods A, B and C which consume less experimental units in comparison to their designs and at the same time are A-optimal or optimal. We restrict here to the estimation of general combining ability (g ca ) effects only. For analysis of these designs [12][13][14]. I in the present paper, we are deriving block and row-column designs for complete diallel cross (CDC) system i.e. methods A, B, C and d through orthogonal arrays and semi balanced arrays. Block designs and row-column designs obtained for methods a consume minimum experimental units and are A-optimal. Block designs obtained for method C are optimal in the sense of Kiefer [15] and consume minimum experimental units but row-column designs are neither A-optimal nor optimal. Conversely block designs and row-column designs obtained for methods B are A-optimal. Block designs obtained for method D are optimal in the sense of Kiefer [15] but row-column designs are neither A-optimal nor optimal.
The rest of this article is organized as follows: in section B and C we have discussed universal optimality of designs for 1-way and 2-way settings. In section 4 and 5, we give some definitions of orthogonal array, semi balanced arrays and orthogonally blocked design and relation of orthogonal; array with designs for CDC system and optimality with examples and theorems. In section 6 we give relation of semi-balanced array to CDC system along with theorem and for example.

Model and Estimation in 1-Way Heterogeneity Setting
According to Sharma   where y is an n×1 vector of observations, 1 n is the n×1 vector of ones, ∆′ 1 is the n × p design matrix for lines and ∆′ 2 is an n × b design matrix for blocks, that is, the (h,λ) th element of∆′ 1 ( respectively,

Model and Estimation in 2-Way Heterogeneity Setting
Let d be a row-column design with k rows and b columns for CDC system involving p lines and n = bk experimental units. For the data obtained from d, we consider the following linear model.
Where y is an n × 1 vector of observed responses, µ is the general mean, g, βand ã are column vectors of p general combining ability (g ca ) parameters, k row effects and b column effects, respectively,  Where Now we state the following theorem of Parsad et al. [16] without proof.
Theorem: Let d* ɛ D 1 (p, b, k) be a row -column design and d* ε D (p, b, k) be a block design for diallel crosses satisfying is the largest positive integer not exceeding z, Ip is an identity matrix of order p and 1p1′p is a p × p matrix of all ones. Then according to Kiefer [15], d*ɛ D 1 (p, b, k) or d*ε D (p, b, k) is universally optimal and in particular minimizes the average variance of the best linear unbiased estimator of all elementary contrasts among the g ca effects . Furthermore, using d*ɛ D 1 (p, b, k) or d*ε D (p, b, k) all elementary contrasts among g ca effects are estimated with variance.

Relation between Orthogonal Array (p 2 , p +1 , p, 2) and Designs for CDC System
Consider an orthogonal array (p 2 , p+1, p, 2), where p is a prime or power of a prime. If we divide this array into p groups where each group contains p × (p+1) elements and identify the elements of each group as p lines of a diallel cross experiments. Now we perform crosses in any two columns of (p+1) constraints in first group and perform crosses among the lines appearing in the corresponding columns in (p-1) groups, we get p initial columns blocks as given below, which can be developed cyclically mod(p) to get design d 1 for diallel cross experiment Griffing' s method A with p 2 distinct crosses of p parental lines consisting of p self, 1/2 p (p-1) number of F 1 crosses, and the same number of reciprocal F 1 's with parameter v = p 2 , b = p, k =p , r =1. By this procedure we obtain p (p+1)/2 designs for diallel cross experiment Griffing's method A (Table1). Note: All column blocks will be developed cyclically mod (p).   Where I p is an identity matrix of order p and 1 p is a unit column vector of ones. Clearly  Remark: The variances of the best linear unbiased estimators of elementary contrasts among gca effects are equal in A-optimal designs and also in optimal designs. It means that all the designs are variance balanced, this fact is particularly attractive to the experimenter, as it enables one to carry out the analysis of the experiment in an extremely simple manner. Now we state the following theorems.

Theorem:
The existence of an Orthogonal Array (p 2 , p +1 , p, 2) implies the incomplete block designs with parameters v = p 2 , b = p, k =p, r =1.
Example: Following Rao [19] we construct an orthogonal array (25,6,5,2) of r p =5, the 4 orthogonal Latin squares with bordered elements are (Table 2). This arrangement may be expressed in five groups as given below Table 3. The above arrangement is an orthogonal array (25, 6, 5, 2). From the above array we can derive the designs for the four experimental methods described by Griffing [9]. The procedure is explained below.
Table2. For Griffing methods A and B, we can take any two columns from first group and corresponding columns from other groups i.e. 2, 3, 4, and 5 and arrange them in columns and then we obtain design for methods A and B. Thus, we may obtain 14 different layouts designs for each method A and method B. We may also obtain 10 different layouts row-column designs for each of these methods A and B.
From the above design we can derive designs for methods C, and D

Conclusion
In the present article we have given block and row-column designs for Griffing's CDC system i.e for all methods A, B, C, and D by using orthogonal array (p 2 , p +1 , p, 2) and semi -balanced array (p (p-1) , p, p, 2). Block and row-column designs for methods A and block designs for method C consume minimum experimental units and are A-optimal and optimal, respectively. These designs are.