Optimal partial triallel cross designs through diallel cross designs

Abstract

In this article we are presenting method of construction of three series of optimal partial triallel cross designs through complete diallel cross designs which have been obtained by nested balanced incomplete block designs.

Keywords

Block design partial triallel cross nested balanced incomplete block design universal optimality

1. Introduction

Mating design play very important role to study the genetic properties of a set of inbred lines in plant breeding experiments. Most commonly used mating designs are diallel or a two-way cross $(m=2)$ . Suppose there are $p$ inbred lines and it is desired to perform a diallel cross experiment involving $v=p(p-1)/2$ crosses of the type $(i\times j)=(j\times i)$ for $i,j=1,2,\dots,p$ , this type of mating design is called complete diallel cross (CDC) method (4) of Griffing (1956).

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 $3^{p}C_{3}=p(p-1)(p-2)/2$ of the type $(i\times j)\times k,i\neq j\neq k=0,1,2,\ldots,p-1$ . Rawlings and Cockerham (1962) were the first to introduce mating designs for triallel crosses. Let $n$ denote the total number of crosses (experimental units) involved in a triallel experiment. It is desired to compare the lines with respect to their general combining abilities, the specific combining abilities being not included in the model.

Triallel cross (TC) experiments are generally conducted using a completely randomized design (CRD) or a randomized complete block (RCB) design as environmental design involving $3^{p}C_{3}$ crosses. Even with a moderate number of parents, say $p=10$ , in a TC experiment; the number of crosses becomes unmanageable to be accommodated in homogeneous blocks. For such situations, Hinkelmann (1965) 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 and Srinivasan (1991) and Subbarayan (1992) obtained PTC using a class of balanced incomplete block (BIB) designs.

Other research workers who contributed in this area are Arora and Aggarwal (1984, p. 89), Ceranka et al. (1990). Recently Harun et al. (2016a, 2016b) constructed efficient PTC designs using mutually orthogonal latin squares and two associate partially balanced incomplete block designs and also gave a simple method of construction of variance – balanced PTC designs for comparing set of test treatments with a control line. More details on TC experiments can be found in Hinkelmann (1975) and Narain (1990).

Following Gupta and Kageyama (1994) and Dey and Midha (1996), Das and Gupta (1997) constructed block designs for TC by using the nested balanced block design (NBIBD) with parameters $v=p,b_{1},b_{2},k_{1},k_{2}=3$ . Their method yields designs which are universally optimal in $\bm{D}(p,b,k)$ , the class of connected block designs for TC in $p$ lines with $b$ blocks each of size $k$ such that the total number of experimental units are $<3^{p}C_{3}$ . Sharma et al. (2011) and Sharma and Fanta (2012) obtained optimal designs for PTC by using mutually orthogonal Latin squares and gave their analysis along with first line effects and second line effects.

In this paper we are presenting methods of construction of block designs for TC experiments through block design of CDC experiment method (4). These designs are found to be optimal in the sense of Das and Gupta (1997). The paper is structured as: in Section 2 we gave some definitions and in Section 3 we discussed optimality tools. In Section 4 we discussed the construction and optimality of these designs. In Section 5 we have discussed the utility of these designs.

2. Some definitions

1.
Definition: The triallel cross (T.C.) has been defined by Rawlings and Cockerham (1962) as a set of all possible three-way hybrids among a group of (inbred) lines. Given three lines $i, j$ and $k$ , there are distinct triallel crosses, namely $(i\times j)k,(j\times k)i$ and $(i\times k)j$ involving these three lines.

Thus given a set of $p$ lines, the TC will consist of a set of $[p(p-1)(p-2)/2]$ three way crosses.
2.
Hinkelmann (1965) proposed the definition of PTC as given below:

Suppose we have $p$ lines which are denoted by $i=1,2,\ldots,p$ . A three way cross is then represented by a triplet $(i\times j)k$ , where $(i\times j)$ stands for an offspring of the single cross $i\times j$ . We shall call $i$ and $j$ half-parents and $k$ full-parent. The crosses $(i\times j)k,(j\times i)k,k(i\times j)$ , and $k(j\times i)$ 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:

(ii) (i)
Each line occurs exactly $r_{H}$ times as half-parent and $r_{F}$ times as full parent.
(ii)
Each cross $(i\times j)k$ occurs either once or not at all.

The total number of crosses is $pr_{F}$ and $r_{H}=2r_{F}$ . Let $r_{F}=r$ , whence $r_{H}=2r$ .

3. Optimality tool

Following Parsad et al. (2005), let $d$ be a block design for an $m$ -allel cross experiment involving $p$ inbred lines, $b$ blocks each of size $k$ , where $m=2$ or 3 in our context. This means that there are $k$ crosses in each of the blocks of $d$ . Further, let $r_{dt}$ and $s_{di}$ denote the number of replication of the $t^{\rm th}$ cross and the number of replications of the $i^{\rm th}$ line in different crosses, respectively, in $d[t=1,2,\ldots,n_{c};i=1,2,\ldots,p]$ , where $n_{c}$ means number of crosses. Evidently, $\sum r_{dt}=bk,\sum s_{di}=mbk=mn$ and $n=bk$ , the total number of observations. In a TC experiment, the genotypic effect of the hybrid consists of single line effects, two line effects and three line specific effects. However, if we assume that for a PTC experiment (in which every line appears as half parent an equal number of times, say $r_{H}$ , and every line appears as full parent an equal number of times, say $r_{F}$ , and each of the crosses $(ij)k$ appears at most once) the two line specific effects and three line specific effects are not of importance, still the line effects are of two types viz effects as half parent and effect as full parent i.e. the ordering of lines in a triallel cross is important. Some plant breeders argue that these ordering effects can also be averaged over line effects. Das and Gupta (1997) considered the situations where ordering of lines in a triallel cross is not of importance. We also considered this and we took the following additive model for the observations obtained from design $d$ .

$\displaystyle\bm{y}=\bm{\mu}\bm{1}_{n}+\bm{\Delta}^{\prime}_{1}\bm{g}+\bm{% \Delta}^{\prime}_{2}\bm{\beta}+\bm{e}$ (1)

where $\bm{y}$ is $n\times 1$ vector of observations, $\bm{1}$ is the $n\times 1$ vector of ones, $\bm{\Delta}^{\prime}_{1}$ is the $n\times p$ design matrix for lines and $\bm{\Delta}^{\prime}_{2}$ is an $n\times b$ design matrix for blocks, that is, the $(h,l)^{\rm th}$ element of $\bm{\Delta}^{\prime}_{1}$ (respectively, of $\bm{\Delta}^{\prime}_{2}$ ) is 1 if the $h^{\rm th}$ observation pertains to the $l^{\rm th}$ line (respectively, of block) and is zero otherwise. $\bm{\mu}$ is a general mean, $\bm{g}$ is a $p\times 1$ vector of line parameters, $\bm{\beta}$ is a $b\times 1$ vector of block parameters and $\bm{e}$ is an $n\times 1$ vector of residuals. It is assumed that vector $\bm{\beta}$ is fixed and $\bm{e}$ is normally distributed with $E(\bm{e})=0,V(\bm{e})=\bm{\sigma}^{2}\bm{I}$ and $Cov(\bm{\beta},\bm{e})=\bm{0}$ , where $\bm{I}$ is the identity matrix of conformable order.

The method of least squares for the analysis of proposed design $d$ leads to the following reduced normal equations for estimating the linear functions of the general combining effects of lines under model Eq. (1).

$\displaystyle\bm{C}_{d}=\bm{G}_{d}-\bm{N}_{d}\bm{K}_{d}^{-1}\bm{N}_{d}^{\prime% }=(c_{ij})\quad(i,j=1,2,\ldots,p)$ (2)

where $\bm{G}_{d}=\bm{\Delta}_{1}\bm{\Delta}^{\prime}_{1}=(g_{dii^{\prime}})$ , $g_{dii}=s_{di}$ and for $i\neq i^{\prime}$ , $g_{dii^{\prime}}$ is the number of crosses in $d$ in which the lines $i$ and $i^{\prime}$ appear together. $\bm{N}_{d}=\bm{\Delta}_{1}\bm{\Delta}^{\prime}_{2}=(n_{dij}),n_{dij}$ is the number of times the line $i$ occurs in block $j$ of $d$ and $\bm{K}_{d}=\bm{\Delta}_{2}\bm{\Delta}^{\prime}_{2}$ is the diagonal matrix of block sizes.

A design $d$ will be called connected if and only if rank $(\bm{C}_{d})=p-1$ , or equivalently, if and only if all elementary comparisons among general combining ability (gca) effects are estimable using $d$ . We denote by $D(p,b,k)$ , the class of all such connected block design $\{d\}$ with $p$ lines, $b$ blocks each of size $k$ . To prove optimality of design $d$ , we need the following well known lemma [(see, e.g., Cheng, 1978, p1246).]

Lemma 1. For given positive integers $s$ and $t$ , the minimum of $\sum_{i=1}^{s}n_{i}^{2}$ subject to $\sum_{i=1}^{s}n_{i}=t$ , where $n_{i}$ ’s are non-negative integers, is obtained when $t-s[t/s]$ of the $n_{i}$ ’s are equal to $[t/s]+1$ and $s-t+s[t/s]$ , where $[z]$ denotes the largest integer not exceeding $z$ . The corresponding minimum of $\sum_{i=1}^{s}n_{i}^{2}$ is $t(2[t/s]+1)-s[t/s]([t/s]+1)$ .

We then have:

Theorem 1. For any design $d\in D(p,b,k)$

$\displaystyle tr(\bm{C}_{d})\leqslant k^{-1}b\{mk(k-1-2x)+px(x+1)\}$

where $x=[mk/p]$ and for a square matrix $A$ , $tr(A)$ stands for the sum of the diagonal elements of $A$ .

Proof. Follows on the lines similar to that of Das and Gupta (1997), Das, Dey and Dean (1998) and Parsad, Gupta and Srivastava (2005).

For any design $d\in D(p,b,k)$ , we have

$\displaystyle tr(\bm{C}_{d})=\sum_{i=}^{p}s_{di}-k^{-1}\sum_{i=1}^{p}\sum_{j=1% }^{b}n_{dij}^{2}=mbk-k^{-1}\sum_{i=1}^{p}\sum_{j=1}^{b}n_{dij}^{2}$

Now, since $\sum_{i=1}^{p}\sum_{j=1}^{b}n_{dij}^{2}=mbk$ , using Lemma 1

$\displaystyle\sum_{i=1}^{p}\sum_{j=1}^{b}n_{dij}^{2}\geqslant b\{mk(2x+1)-px(x% +1)\}\quad\text{where $[mk/p]$.}$

Hence

$\displaystyle tr(\bm{C}_{d})\leqslant mbk-k^{-1}b\{mk(2x+1)-px(x+1)\}=k^{-1}b% \{mk(k-1-2x)+px(x+1)\}$

By Lemma 1, the above equality is attained if and only if $n_{dij}=x$ or $x+1$ , for $i=1,2,\ldots,p$ ; $j=1,2,\ldots,b$ . Further, using proposition 1 of Kiefer (1975), we have following theorem:

Theorem 2. Let $d^{*}\in D(p,b,k)$ be a block design for $m$ -allel crosses, satisfying

(ii) (i)

$tr(C_{d^{*}})\leqslant k^{-1}b\{mk(k-1-2x)+px(x+1)$ and

(ii)

$C_{d^{*}}$ is completely symmetric.

Then $d^{*}$ is universally optimal in $D(p,b,k)$

where $x=[mk/p]$ and for a square matrix $A$ , $tr(A)$ stands for the sum of the diagonal elements of $A$ .

Kiefer (1975) showed that a design is universally optimal in a relevant class of competing design if (i) the information matrix (the $\bm{C}_{d^{*}}$ $-$ matrix) of the design is completely symmetric in the sense that $\bm{C}_{d^{*}}$ has all its diagonal elements equal and all of its off-diagonal elements equal, and (ii) the matrix $\bm{C}_{d^{*}}$ has maximum trace in the class of competing designs, that is, such a design minimize the average variance of the best linear unbiased estimators of all elementary contrasts among the parameters of interest i.e. the general combining ability in our context. Such a design $d^{*}\in D(p,b,k)$ has

$\displaystyle\bm{C}_{d^{*}}=(p-1)^{-1}k^{-1}b\{mk(k-1-2x)+px(x+1)\}(\bm{I}p-% \bm{1}p\bm{1}^{\prime}p/p)\text{ is completely symmetric.}$

where $\bm{I}p$ is an identity matrix of order $p$ and $\bm{1}p\bm{1}^{\prime}p$ is a $p\times p$ matrix of all ones. Furthermore, using $d^{*}$ all elementary contrasts among gca effects are estimated with variance

$\displaystyle[2b^{-1}(p-1)k/\{mk(k-1-2x)+px(x+1)\}]\sigma^{2}.$

Now we will show a connection between CDC designs method (4) and PTC designs through NBIBDs of Preece (1967).

Definition 1. A NBIBD with parameters $(v,b_{1},k_{1},r^{*},\lambda_{1},b_{2},k_{2},\lambda_{2},m)$ is a design for $v$ treatments, each replicated $r^{*}$ times with two systems of blocks such that:

(a) (a)

The second system is nested within the first, with each block from the first system, called henceforth as ‘block’ containing exactly $m$ blocks from the second system, called hereafter as ‘sub-blocks’;

(b)

Ignoring the first system leaves a balanced incomplete block design with parameters $v,b_{2},k_{2},r^{*},\lambda_{2}$ .

The following parametric relations hold for a nested balanced incomplete block design:

$\displaystyle vr^{*}=b_{1}k_{1}=mb_{1}k_{2}=b_{2}k_{2},(v-1)\lambda_{1}=(k_{1}% -1)r^{*},(v-1)\lambda_{2}=(k_{2}-1)r^{*}.$

Since we are using diallel cross designs for PTC experiments using NBIBD, so first we will establish relation of NBIBD with diallel cross designs. Consider now a NBIBD $d$ with parameters $v=p,b_{1},b_{2},k_{1},r^{*},k_{2}=2$ . If we identify the treatments of $d$ as lines of a CDC method (4) experiment and perform crosses among the lines appearing in the same sub block of $d$ , we get a block design $d_{1}$ for a CDC method (4) experiment involving $p$ lines with $p(p-1)/2$ crosses, each replicated $r=2b_{2}/\{p(p-1)\}$ times, and $b=b_{1}/2$ blocks, each of size $k=2k_{2}$ . Such a design $d_{1}\in D(p,b,k)$ ; also, for such a design $n_{d^{*}ij}=$ 0 or 1 for $i=1,2,\ldots,p,j=1,2,\ldots,b$ . and

$\displaystyle\bm{C}_{d1}=(p-1)^{-1}p(p-3)[I_{p}-p^{-1}1_{p}1^{\prime}_{p}]$ (3)

where $I_{p}$ is an identity matrix of order $p$ and $1_{p}$ is a unit column vector of order $p$ . Clearly $\bm{C}_{d1}$ given by Eq. (3) is completely symmetric and $tr(\bm{C}_{d1})=p(p-3)$ which equals the equality given in Theorem 2. Thus the design $d_{1}$ is universally optimal in $D(p,b,k)$ and using $d_{1}$ each elementary contrast among general combining ability effects is estimated with a variance

$\displaystyle 2(p-1)\sigma^{2}/p(p-3)$ (4)

Since $n_{d^{*}ij}=$ 0 or 1 for $i=1,2,\ldots,p,j=1,2,\ldots,b$ , a PTC triallel cross designs can be derived from $d_{1}$ by attaching $i^{\rm th}$ line with each cross in $j^{\rm th}$ block provided $i^{\rm th}$ line does not appear in $j^{\rm th}$ block, where $i=1,2,\ldots,p;j=1,2,\ldots,b$ . Hence we get a block design $d_{2}$ for a PTC experiment involving $p$ lines with $p(p-1)/2$ triallel crosses, each replicated $r=2b_{2}/\{p(p-1)\}$ times, and $b=b_{1}/2$ blocks, each of size $k=2k_{2}$ . Such a design $d_{2}\in D(p,b,k)$ ; and, for a design $d_{2}n_{d^{*}ij}=3$ for $i=1,2,\ldots,p,j=1,2,\ldots,b$ . and

$\displaystyle\bm{C}_{d2}=3/2(p-3)[I_{p}-p^{-1}1_{p}1^{\prime}_{p}]$ (5)

The $\bm{C}_{d2}$ given by Eq. (5) is completely symmetric and $tr(\bm{C}_{d2})=3/2(p-1)(p-3)$ which equals the equality given in Theorem 2. Thus the design $d_{2}$ is universally optimal in $D(p,b,k)$ and using $d_{2}$ each elementary contrast among gca effects is estimated with a variance

$\displaystyle 4\sigma^{2}/3(p-3)$ (6)

Hence we can state the following theorem:

Theorem 3. The existence of a NBIBD $d$ with parameters $v=p,b_{1}=b,b_{2},=bk,k_{1}=2k,k_{2}=2$ implies the existence of two types of designs

(ii) (i)

A universally optimal incomplete block design $d_{1}$ for CDC experiment method (4);

(ii)

A universally optimal PTC design $d_{2}$ .

4. Method of construction of PTC through diallel cross designs

Series 1: Let $p=4m+1(m\geqslant 1)$ be a prime or a prime power and $x$ be a primitive element of the GF ( $p$ ). Consider the following $m$ initial blocks.

$\{(x^{i},x^{i+2m}),(x^{i+m},x^{i+3m})\},i=0,1,2,\ldots m-1.$

As shown by Dey et al. (1986), these initial blocks, when developed in the sense of Bose (1939), give rise to a nested balanced incomplete block design with parameters $v=p=4m+1,k_{1}=4,b_{1}=m(4m+1),k_{2}=2$ .

Following Sharma and Tadesse (2017) we can put the above $m$ blocks into single block and developing the single block over mod ( $p$ ), we obtained an optimal block design for diallel crosses with minimal number of experimental units with parameters $p=4m+1,b=4m+1,k=2m$ , and $r=$ 1. This diallel cross design can be converted into optimal partial triallel cross design with parameters $p=4m+1,b=2m,k=4m+1$ , by attaching $i^{\rm th}$ line with the crosses in $j^{\rm th}$ block in which $i$ th line does not appear at all, where $i,j=1,2,\ldots,4m+1$ and considering rows as blocks,

Example 1. Let $m=$ 2. We get the following two columns.

$\left[\begin{array}[]{@{}cc@{}}{(1,2)}&{(x,2x)}\\ {(2x+1,x+2)}&{(2x+2,x+1)}\end{array}\right]$

where $x$ is a primitive element of GF $(3^{2})$ and the elements of GF $(3^{2})$ are $0,1,2,x,x+1,x+2,2x,2x+1,2x+2$ .

We can convert both columns in single column and adding successively the non-zero elements of GF $(3^{2})$ to the contents of the single column, the CDC design method (4) is obtained as given below, where the lines have been relabeled 1–9, using the correspondence $0\to 1,1\to 2,2\to 3,x\to 4,x+1\to 5,x+2\to 6,2x\to 7,2x+1\to 8,2x+2\to 9$ :

CDC design method (4)

$\begin{matrix}{B_{1}}&{B_{2}}&{B_{3}}&{B_{4}}&{B_{5}}&{B_{6}}&{B_{7}}&{B_{8}}&% {B_{9}}\\ {(2\times 3)}&{(1\times 3)}&{(1\times 2)}&{(5\times 6)}&{(4\times 6)}&{(4% \times 5)}&{(8\times 9)}&{(7\times 9)}&{(7\times 8)}\\ {(6\times 8)}&{(4\times 9)}&{(5\times 7)}&{(2\times 9)}&{(3\times 7)}&{(1% \times 8)}&{(3\times 5)}&{(1\times 6)}&{(2\times 4)}\\ {(4\times 7)}&{(5\times 8)}&{(6\times 9)}&{(1\times 7)}&{(2\times 8)}&{(3% \times 9)}&{(1\times 4)}&{(2\times 5)}&{(3\times 6)}\\ {(5\times 9)}&{(6\times 7)}&{(4\times 8)}&{(3\times 8)}&{(1\times 9)}&{(2% \times 7)}&{(2\times 6)}&{(3\times 4)}&{(1\times 5)}\end{matrix}$

The above design is an optimal diallel cross design for method (4) with parameters $p=9,b=9,k=4$ and $r=$ 1.

Example 2. Now attaching the 1, 2,…, 9 elements, respectively, with the crosses of the 9 blocks because these elements do not appear in the respective blocks of the above design. Considering rows as blocks, we obtain optimal block design for triallel cross with parameters $p=9,b=4,k=9$ and $r=$ 1, which fulfill the all conditions for PTC design. The design is given below.

Partial Triallel Cross Design Series 1

$\begin{matrix}{B_{1}}&{(2\times 3)1}&{(1\times 3)2}&{(1\times 2)3}&{(5\times 6% )4}&{(4\times 6)5}&{(4\times 5)6}&{(8\times 9)7}&{(7\times 9)8}&{(7\times 8)9}% \\ {B_{2}}&{(6\times 8)1}&{(4\times 9)2}&{(5\times 7)3}&{(2\times 9)4}&{(3\times 7% )5}&{(1\times 8)6}&{(3\times 5)7}&{(1\times 6)8}&{(2\times 4)9}\\ {B_{3}}&{(4\times 7)1}&{(5\times 8)2}&{(6\times 9)3}&{(1\times 7)4}&{(2\times 8% )5}&{(3\times 9)6}&{(1\times 4)7}&{(2\times 5)8}&{(3\times 6)9}\\ {B_{4}}&{(5\times 9)1}&{(6\times 7)2}&{(4\times 8)3}&{(3\times 8)4}&{(1\times 9% )5}&{(2\times 7)6}&{(2\times 6)7}&{(3\times 4)8}&{(1\times 5)9}\end{matrix}$

Series 2: Let $p=6m+1(m\geqslant 1)$ be a prime or a prime power and $x$ be a primitive element of the Galois field of order $p$ , GF (p). Consider the initial blocks

$\{(x^{i},x^{i+3m}),(x^{i+m},x^{i+4m}),(x^{i+2m},x^{i+5m})\},i=0,1,2,\ldots m-1.$

Dey et al. (1986) showed that these initial blocks, when developed give a solution of a nested incomplete block design with parameters $v=p=6m+1,b_{1}=m(6m+1),k_{1}=6,k_{2}=2,\lambda_{2}=1$ .

Similarly arranging the above initial $m$ blocks into single block as in series 1 and developing $\bmod(p)$ , will yield an optimal CDC design method (4) with parameters $p=6m+1,b=6m+1,k=3m$ , and $r=$ 1.

$\left[\begin{array}[]{@{}c}{(x^{i},x^{i+3m)})}\\ {(x^{i+m},x^{i+4m})}\\ {(x^{i+2m},x^{i+5m)}}\end{array},i=0,1,\ldots,m-1\right]$

This design can be converted into partial triallel cross design with parameters $p=6m+1$ , $k=3m$ , $b=6m+1$ and $r=$ 1 by the procedure described above in Series 1.

Example 3. Let $m=$ 2. Then we get the following two initial blocks.

$\left[\begin{array}[]{@{}cc@{}}{(1,12)}&{(2,11)}\\ {(4,9)}&{(8,5)}\\ {(3,10)}&{(6,7)}\end{array}\right]$

Now we arrange these two blocks in a single block as given below.

$\left[\begin{array}[]{@{}c}{(1,12)}\\ {(2,11)}\\ {(4,9)}\\ {(8,5)}\\ {(3,10)}\\ {(6,7)}\end{array}\right]$

Now developing the above block mod (13), we obtain optimal CDC design with parameters $p=$ 13, $k=$ 6, $b=$ 13 and $r=$ 1. Following the procedure of example 2, we can obtain block design for PTC with parameters $p=$ 13, $k=$ 13, $b=$ 7 and $r=$ 1.

Series 3: Let $p=2m+1(m\geqslant 2)$ be a prime or a prime power, then cyclically developing the following $m$ columns

$\displaystyle(0,2m),(1,2m-1),(2,2m-2),\ldots,(m-1,m+1)\bmod(2m+1)$

yields an optimal CDC design method (4) with parameters $p=2m+1,k=m,b=2m+1$ . A PTC design with parameters $p=2m+1,k=2m+1$ , and $b=m$ can be obtained by the procedure described in Series1.

Example 4. Let $m=$ 3. Then $p=$ 7 and developing the following columns mod (7)

$\left[\begin{array}[]{@{}c}{(0,6)}\\ {(1,5)}\\ {(2,4)}\end{array}\right]$

yields optimal CDC design with parameters $p=$ 7, $k=$ 3, and $b=$ 7 and $r=$ 1. A PTC design with parameters $p=$ 7, $b=$ 3, $k=$ 7 and $r=$ 1 can be obtained by the procedure given in Example 2.

Partial Triallel Cross Design Series 2

$\displaystyle\begin{array}[]{@{}cccccccc}{B_{1}}&{(0\times 6)3}&{(1\times 0)4}% &{(2\times 1)5}&{(3\times 2)6}&{(4\times 3)0}&{(5\times 4)1}&{(6\times 5)2}\\ {B_{2}}&{(1\times 5)3}&{(2\times 6)4}&{(3\times 0)5}&{(4\times 1)6}&{(5\times 2% )0}&{(6\times 3)1}&{(0\times 4)2}\\ {B_{3}}&{(2\times 4)3}&{(3\times 5)4}&{(4\times 6)5}&{(5\times 0)6}&{(6\times 1% )0}&{(0\times 2)1}&{(1\times 3)2}\end{array}$

Note: The $m$ columns form a nested balanced incomplete block design with parameters

$\displaystyle v=p=2m+1,b_{1}=m,k_{1}=m(2m+1),k_{2}=2,\lambda_{2}=1.$

Universally optimal block designs, with $p\leqslant$ 13, obtained by the above method from NBIB designs of Morgan et al. (2001), are listed below in Table. Using these designs optimal partial triallel cross designs can be constructed.

Table 1
Universally optimal block design for complete diallel crosses with $p\leqslant$ 13 generated by using NBIB designs of Morgan, Preece and Rees (2001). The following PTC designs can be obtained using the procedures given in Section 4

S. No.	$p$	$b$	$k$	Source
1.	7	7	6	MPR2
2.	9	18	4	MPR5w
3.	9	9	8	MPR8
4.	11	11	5	MPR 14
5.	13	39	4	MPR 20w
6.	13	26	6	MPR21
7.	13	13	12	MP23

Remark: Sources can be found in Morgan, Preece and Rees (2001).

5. Conclusion

We have obtained three series of optimal PTC design using optimal CDC designs method (4). These designs require only $p(p-1)/2$ experimental units i.e. $1/(p-2)$ th crosses in comparison to complete TC designs. According to Shunmugathai and Srinivasan (2012), these designs can be used to improve the quantitative traits of economic and nutritional importance in crops and animals. It has been established that the three way hybrids are more stable than pure lines and single cross hybrids and exhibits individual as well as population buffering mechanisms because of the genetic base are replica of lines are available.

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Optimal partial triallel cross designs through diallel cross designs

Abstract

Keywords

1. Introduction

2. Some definitions

Table 1 Universally optimal block design for complete diallel crosses with p ⩽ 13 generated by using NBIB designs of Morgan, Preece and Rees (2001). The following PTC designs can be obtained using the procedures given in Section 4

References

Table 1
Universally optimal block design for complete diallel crosses with $p\leqslant$ 13 generated by using NBIB designs of Morgan, Preece and Rees (2001). The following PTC designs can be obtained using the procedures given in Section 4