A Note on Optimal Partial Triallel Cross Designs

Triallel crosses form an important class of mating designs, which are used for studying the genetic properties of a set of inbred lines in plant breeding experiments. For p inbred lines, the number of different crosses for a complete triallel experiment is ( )( ) 3 3 1 2 2 p C p p p = − − of the type ( ) , i j k × 0,1, 2, , 1. i j k p ≠ ≠ = ... − Rawlings & Cockerham [5] were the first to introduce mating designs for triallel crosses. Triallel cross experiments are generally conducted using a completely randomized design (CRD) or a randomized complete block (RCB) design as environmental design involving 3 3 C crosses. Even with a moderate number of parents, say 10, p = in a triallel cross experiment; the number of crosses becomes unmanageable to be accommodated in homogeneous blocks. For such situations, Hinkelmann [6] developed partial triallel crosses (PTC) involving only a sample of all possible crosses by establishing a correspondence between PTC and generalized partially balanced incomplete block designs (GPBIBD). Ponnuswamy & Srinivasan [7] and Subbrayan [8] obtained PTC using a class of balanced incomplete block (BIB) designs. Dharmlingum [9] also constructed PTC using Trojan squares. Actually Trojan squares are MOLS. Other research workers who contributed in this area are Arora & Aggarwal [10,11], Ceranka et al. [12]. More details on triallel cross experiments can be found in Hinkelmann [13] and Narain [14]. Das & Gupta [4] constructed block designs for triallel crosses by using nested balanced block design with parameters


Introduction
Triallel crosses form an important class of mating designs, which are used for studying the genetic properties of a set of inbred lines in plant breeding experiments. For p inbred lines, the number of different crosses for a complete triallel experiment is ( )( ) 3 3 1 2 2 p C p p p = − − of the type ( ) , i j k × 0,1, 2, , 1. i j k p ≠ ≠ = … − Rawlings & Cockerham [5] were the first to introduce mating designs for triallel crosses. Triallel cross experiments are generally conducted using a completely randomized design (CRD) or a randomized complete block (RCB) design as environmental design involving 3 3 p C crosses. Even with a moderate number of parents, say 10, p = in a triallel cross experiment; the number of crosses becomes unmanageable to be accommodated in homogeneous blocks. For such situations, Hinkelmann [6] developed partial triallel crosses (PTC) involving only a sample of all possible crosses by establishing a correspondence between PTC and generalized partially balanced incomplete block designs (GPBIBD). Ponnuswamy & Srinivasan [7] and Subbrayan [8] obtained PTC using a class of balanced incomplete block (BIB) designs. Dharmlingum [9] also constructed PTC using Trojan squares. Actually Trojan squares are MOLS. Other research workers who contributed in this area are Arora & Aggarwal [10,11], Ceranka et al. [12]. More details on triallel cross experiments can be found in Hinkelmann [13] and Narain [14].
Das & Gupta [4] constructed block designs for triallel crosses by using nested balanced block design with parameters designs for triallel crosses in p lines with b blocks each of size k such that the total number of experimental units are 3 3 . p C < Sharma et al. [1] and Sharma and Fanta [2] also constructed optimal block and unblocked designs for PTC using two mutually orthogonal Latin squares together. We, in this paper, are proposing unblocked and blocked designs for triallel cross experiments by using transversal and parallel transversal of a Latin square. Our methods are different from Sharma et al. [1] and Sharma & Fanta [2] methods. These designs are found to be optimal in the sense of Kiefer [14] and Das & Gupta [4].

Summary
Two simple alternative methods of construction of optimal partial triallel cross designs (unblocked and blocked) for p>3 lines are proposed by using transversal and parallel transversals of a Latin square of order p, where p is a prime or power of a prime. Sharma et al. [1] and Sharma & Fanta [2] also constructed these designs by using two mutually orthogonal Latin squares but we constructed these designs only taking one Latin square. Therefore our methods are simple in comparison to their methods. These designs are found to be optimal in the sense of Kiefer [3] and Das & Gupta [4].

Biostatistics and Biometrics Open Access Journal
× are considered to be identical in three way crosses. Then PTC can be defined as follows: A set of matings is said to be a PTC if it satisfies the following conditions:   Now we border the columns and rows of the above Latin square by the bracketed elements of the transversal. We get the following arrangement.  Now we attach border elements with each cell (Table 1). Then we get the following design Now deleting the bracketed cells, we get following unblocked design d 1 From the above mating design, we can derive block design for PTC. Consider the triplet of any column of the above mating designs as initial blocks and then developing row-wise cyclically mod (5) these blocks, we get block design for PTC. We denote this design Consider the first column triplets as 4 initial blocks and then developing cyclically mod (5), we get the following block design for PTC, which satisfy the condition of PTC.
By observing the above design we find that every triple cross occurs two times in the design. Hence this design is different from the design given in example 1. Again by taking triplets of any column as initial blocks and developing each triplet cyclically rowwise mod (5) we obtain block design 4 d with parameters 5, p = 4 b = and 5 k = in which each triple cross is also replicated two times in the design.
We see that unblocked design 1 d is different from unblocked   Now we border the columns and rows of the above Latin square by the bracketed elements of the parallel transversals following the rule given in method 2. We get the following arrangement. Now we attach border elements with each cell. Then we get the following triallel cross mating design Now deleting the bracketed cells, we get following unblocked design d 5  From the above mating design, we can derive blocked mating design for PTC. Consider any column of the above mating designs as initial blocks and then developing these blocks cyclically row wise mod (5) Sharma et al. [1] and Sharma & Fanta [2] constructed these designs by using two M.OL.S. of order p together, where p is a prime or prime power but our method needs only one Latin square. d) Remark 4: When we take parallel transversals out of 2 p − or 2 n p − Latin squares only one Latin squares will give mating design for PTC in which every triple cross occurs two times in unblocked and blocked designs.