Optimal row-column designs for CDC method (1)

Abstract

In the present article, we are presenting row-column designs for Griffing’s complete diallel cross methods (1) for $p$ parents by using a complete set of ( $p-1$ ) mutually orthogonal Latin squares, when $p$ is prime or a power of prime. The row-column designs for Griffing’s methods (1) are new and universally optimal in the sense of Kempthrone (1956) and Kiefer (1975). The row-column designs for methods (1) are orthogonally blocked designs. In an orthogonally blocked design no loss of efficiency on the comparisons of interest is incurred due to blocking. The analysis includes the analysis of variance (ANOVA), estimation of general combining ability (gca), specific combining ability (sca) and reciprocal combining ability (rca). Tables of universally optimal row-column designs have been provided.

Keywords

Mutually orthogonal Latin squares (MOLS)complete diallel cross (CDC)row-column design optimality

1. Introduction

A complete set of ( $p-1$ ) Latin squares of order $p$ are called MOLS, if they are pair-wise orthogonal. MOLS are used for the construction balanced incomplete block designs, square lattice designs and complete diallel cross designs. A set of $p-1$ MOLS of side $p$ can always be constructed if $p$ is a prime or power of a prime.

A common experimental design in genetics is the diallel cross, in which pairs of distinct lines (strains) are cross-breed in order to estimate genetic effects. Let $p$ denote the number of lines and it is desired to perform a diallel cross experiment containing $v=p^{2}$ crosses of the types ( $i\times i$ ), ( $i\times j$ ) and ( $j\times i$ ) between lines $i$ and $j$ , where $i,j=1,\ldots,p$ . This is the CDC method (1) mating design of Griffing (1956). Griffing (1956) discussed in detail the analysis of CDC method (1) in randomized block design (RBD). Later incomplete block designs were introduced by many authors {See Singh et al. (2012)}. However, this approach was not quite satisfactory if one is interested in optimal designs for diallel cross experiments. Several authors such as Gupta and Kageyama (1994), Dey and Midha (1996), Mukerjee (1997), Das et al. (1998), Parsad et al. (1999) and Sharma (2004) investigated optimal block designs either for modified diallel i.e. Griffing’s method (4) or for partial diallel crosses to estimate both general and specific combining ability or only general combining ability. Optimal block designs for CDC method (1) and (2) and variance balanced designs for CDC method (3) in 1-way elimination of heterogeneity set up have been constructed by Sharma and Fanta (2009, 2010) by using Mutually orthogonal Latin squares (MOLS).

The universal optimality and combinatorial aspects for CDC experiment methods (1), (2) and method (4) in the 1-way elimination of heterogeneity setting was studied by many authors. Gupta and Choi (1998) and Parsad et al. (2005) are the only authors who studied optimality of CDC method (4) in row-column designs. However, row-column designs for complete diallel cross methods (1) did not received any attention so far in statistical literature.

In the present paper, we are proposing row-column designs for complete diallel cross method (1) through a complete set of ( $p-1$ ) mutually orthogonal Latin squares, where p is prime or a power of prime. The rest of this article is organized as follows: in Section 2, we give some definitions and method of construction of row-column design for method (1) through complete set of MOLS with examples. In Section 3 we discuss the model and estimation of parameters. In Section 4 we discuss optimality in the sense of Kempthrone (1956) and Kiefer (1975). In Section 5 we discuss about the efficiency factor of this design.

2. Some definitions and method of construction

Definition 2.1: A Latin square is said to be in the standard form if the symbols in the first row and the first column are in natural order, and it is said to be in the semi-standard form if the first row is in natural order.

Definition 2.2: According to Gupta et al. (1995), a diallel cross design to be orthogonally blocked if each line occurs in every block $r/b$ times, where $r$ is the constant replication number of the lines and $b$ is the number of blocks in the design.

Here we take a complete set of ( $p-1$ ) semi-normalized Latin squares for the construction of row-column CDC designs for p varieties because this operation preserves the orthogonally.

By superimposition of all the ( $p-1$ ) semi-normalized orthogonal Latin squares, we obtain a composite square, say, C. Now we transpose the composite C square. The transpose composite square can be partitioned into $p$ rows and $p$ columns where each column contains ( $p-1$ ) ordered elements in $p$ rows. In the transpose composite square the entry (1, $i$ ) contains ( $p-1$ ) elements $i$ and in the entry (2, $i$ ) all the ( $p-1$ ) elements are different where $i=1,2,\ldots p$ . None of the elements of the entry (2, 1) can coincide with the elements of the entry (1, 1). Indeed, if the two elements from this entry are equal to $j$ , then the entry ( $i, j$ ) contains a pair of element $j$ , which contradicts orthogonality. Hence none of the different elements of the entry (2, 1) can coincide with the elements of the entry (1, 1).

Let us consider that the elements in $p$ rows and $p$ columns represent the $p$ varieties. Now we give the method of construction of row-column designs for Griffing’s (1956) methods (1) in two parts by using transpose composite square as follows:

Out of ( $p-1$ ) elements in the first column in each row, we perform crosses between any two elements, say, ( $i\times i$ ), where $i=1,2,\ldots,p$ and similarly perform crosses between corresponding two elements in other ( $p-1$ ) columns and ( $p-1$ ) rows, say, ( $j\times l$ ) where $i\neq j\neq l=1,2,\ldots p$ , we get mating design for CDC experimental method (1) containing $p^{2}$ crosses in $p^{2}$ experimental units. We call this mating design as the first part of CDC experimental method (1).

Similarly for second part of CDC design for method (1), we perform crosses between two different elements, other than first selected elements in first part, in the first column and corresponding two elements, say, ( $r\times s$ ) in other cells of the ( $p-1$ ) columns in each row, where $i\neq r\neq s=1,2,\ldots p$ , we get second part of mating design of CDC experimental method (1) containing $p^{2}$ crosses in $p^{2}$ experimental units. Now we juxtaposed the second part with the first part, we get mating design for CDC method (1). This mating design can be converted into row-column design $d$ for CDC method (1) by considering columns as blocks and rows as row blocks with parameters $v=p^{2},b=2p,k=p$ and $r=2$ .

Example 2.1. Let us take $p=5$ , for construction of designs for method (1), we take 4 mutually semi-normalized Latin squares of order 5. After superimposing and transposing, we get the following composite square which has been shown below.

Table 1

Columns

1 2 3 4 5

1 1111 2345 3524 4235 5432

2 2222 3451 4135 5314 1543

Rows 3 3333 4512 5241 1425 2154

4 4444 5134 1352 2531 3215

5 5555 1234 2413 3142 4321

Columns
		1	2	3	4	5
	1	1111	2345	3524	4235	5432
	2	2222	3451	4135	5314	1543
Rows	3	3333	4512	5241	1425	2154
	4	4444	5134	1352	2531	3215
	5	5555	1234	2413	3142	4321

Out of 4 elements in the first column we cross any two elements and also cross corresponding elements in five rows of the first column and similarly we cross corresponding elements in other 4 columns and 4 rows. We get first part of CDC method (1) containing 25 crosses in 25 experimental units. Now we perform crosses between any two elements in the first column, other than the first selected elements in the first column, and corresponding elements in 4 columns and 4 rows. We get the second part of CDC method (1) containing 25 crosses in 25 experimental units. By juxtaposition both the parts and considering columns as blocks and row as row blocks, we get following row-column design with parameters $v=25$ , $b=10$ , $k=5$ and $r=2$ design for CDC method (1).

Row-column design d For CDC method (1)

$\displaystyle{\begin{array}[]{*{20}c}&{B_{1}}&{B_{2}}&{B_{3}}&{B_{4}}&{B_{5}}&% {B_{6}}&{B_{7}}&{B_{8}}&{B_{9}}&{B_{10}}\\ {R_{1}}&{1\times 1}&{2\times 3}&{3\times 5}&{4\times 2}&{5\times 4}&{1\times 1% }&{4\times 5}&{2\times 4}&{5\times 3}&{3\times 2}\\ {R_{2}}&{2\times 2}&{3\times 4}&{4\times 1}&{5\times 3}&{1\times 5}&{2\times 2% }&{5\times 1}&{3\times 5}&{1\times 4}&{4\times 3}\\ {R_{3}}&{3\times 3}&{4\times 5}&{5\times 2}&{1\times 4}&{2\times 1}&{3\times 3% }&{1\times 2}&{4\times 1}&{2\times 5}&{5\times 4}\\ {R_{4}}&{4\times 4}&{5\times 1}&{1\times 3}&{2\times 5}&{3\times 2}&{4\times 4% }&{2\times 3}&{5\times 2}&{3\times 1}&{1\times 5}\\ {R_{5}}&{5\times 5}&{1\times 2}&{2\times 4}&{3\times 1}&{4\times 3}&{5\times 5% }&{3\times 4}&{1\times 3}&{4\times 2}&{2\times 1}\\ \end{array}}$

Remark 1: According to Gupta et al. (1995) these designs are orthogonally blocked. In an orthogonal design no loss of efficiency on the comparisons of interest is incurred due to blocking. A block design for which $N=\theta 1p1^{\prime}b$ is orthogonal for estimating the contrasts among gca parameters, where $N$ denotes the line versus block incidence matrix and $\theta$ is some constant.

Remark 2: Choi et al. (2002) proved that orthogonally blocked designs remain optimal for the estimation of gca comparisons even in the presence of sca effects in the model when each cross is replicated twice.

Now we state the following theorem.

Theorem 2.1: For a complete set of MOLS of order $p$ , if $p$ is prime or power of prime, there exist $\left({{\begin{array}[]{*{20}c}{(p-1)}\\ 2\\ \end{array}}}\right)=\frac{(p-1)(p-2)}{2}$ optimal row column designs for complete diallel cross methods (1) with parameters $v=p^{2},b=2p,k=p$ and $r=2$ .

3. Universal optimality of designs for 2-way heterogeneity setting

Let d be a row-column designs with $k$ rows with $b$ columns for CDC methods (1) involving $p$ lines and having n $=$ bk units, respectively. For the data obtained from this design we postulate the following model.

$\displaystyle{y}=\mu{1}_{n}+{{{\Delta}^{\prime}}}_{{1}}\tau+{{{\Delta}^{\prime% }}}_{{{2}}}{{\beta}}+{{{\Delta}^{\prime}}}_{{3}}{{\gamma}}+{{e}}$ (1)

For the analysis of data obtained from design $d$ we will follow Singh and Hinkelmann (1998) and Sharma and Fanta (2009) two stage procedures for estimating gca, sca effects and reciprocal cross effects with some modification. The first stage is to consider to estimate the cross effects, say, $\tau=(\tau_{11,}\tau_{12},\ldots,\tau_{p^{2}})^{\prime}$ for design $d$ .

Where $y$ be $n\times 1$ vectors of observations, 1 is the $n\times 1$ vectors of ones, $\mu$ is the general mean, ${{\tau}},{{\beta}}$ and ${{\gamma}}$ are column vectors of $v=p^{2}$ cross effects column effects and $k$ row effects of designs $d$ , respectively. ${{{\Delta}^{\prime}}}_{1}(n\times v)$ is the corresponding design matrix and ${{e}}$ denotes the vector of independent random errors having mean 0 and covariance matrix $\sigma^{2}{{I}}_{n}$ .

Let $N_{d1}={{\Delta}}_{1}{{{\Delta}^{\prime}}}_{2}$ be the $v\times k$ incidence matrix of crosses vs rows and $N_{d2}={{\Delta}}_{1}{{{\Delta}^{\prime}}}_{3}$ be the $v\times b$ incidence matrix of crosses vs columns and ${{\Delta}}_{2}{{{\Delta}^{\prime}}}_{3}={1}_{k}{1}_{b}$ . Let $r_{dl}$ denote the number of times the $\ell^{\text{th}}$ cross appears in the design $d$ , $\ell=1,2,\ldots,n$ . Under (4.1), it can be shown that the reduced normal equations for estimating the cross effects of lines, after eliminating the effect of rows and columns, in design $d$ are.

$\displaystyle{{C}}_{{d}}{{\hat{\tau}}}={{Q}}_{{{d1}}}$ (2)

where

$\displaystyle C_{d}=r^{{\delta}}_{{d1}}-\frac{{1}}{{b}}{{N}}_{{d1}}{{{N}^{% \prime}}}_{{d1}}-\frac{{1}}{{k}}{{N}}_{{{d}}{2}}{N^{\prime}}_{d2}+\frac{{{r}}_% {{{d}}l}{{{r^{\prime}}}}_{{dl}}}{{{{r}^{\prime}}}_{{{dl}}}{{1}}}$ $\displaystyle Q_{d}=T{-}1/bN_{d1}R{-}1/kN_{d2}C+(G/bk)r_{dl}$

Where $C_{d}$ is a $v\times v$ information matrix of the crosses of design $d$ and ${{r}}^{{\delta}}_{d}$ is a diagonal matrix of designs d of order $v\times v$ with elements number of replication of crosses in the diagonal. ${{N}}_{{d1}}=(n_{\textit{dij}\ldots}).$ , $n_{\textit{dij}}$ is the number of times cross $i\times j$ appear in row $j$ of designs $d$ . $N_{d2}=(n_{di.t})$ , $n_{i.t}$ is the number of times the cross $i$ occurs in column $t$ of designs $d$ . $r_{dl}$ is the replication vector of crosses in designs $d$ . $Q_{d}$ is a $v\times 1$ vector of adjusted crosses total. $T$ is a $v\times 1$ vector of cross totals, $R$ is a $k\times 1$ vector of rows totals, $C$ is a $b\times 1$ vector of columns totals, respectively, in design $d$ . $G$ is a grand total of all observations in design $d$ . The sum of squares due to crosses for design $d$ is ${Q}^{\prime}_{d}{{C}}^{-}_{{{d}}}{{Q}}_{{{d}}}$ with degree of freedom $=$ ( $v-1$ ), where ${{C}}^{-}_{{{d}}}$ is the generalized inverses of ${{C}}_{{{d}}}$ with property $CC^{-}C=C$ and expectation and variance of $Q_{d}$ is

$\displaystyle E(Q_{d})={{C}}_{{{d}}}\tau\text{ and }V(Q_{d})=\sigma^{2}C$

Now we will utilize the above equations to estimate the genetic parameters in proposed design. We will give below the estimation procedure of genetic parameters in design $d$ .

(A) Estimation of gca, sca and reciprocal effects in design d: The second stage is to utilize the fact that the cross effects can be expressed in terms of gca, sca and reciprocal effects. So we can write

$\displaystyle\tau_{ij}=g_{i}+g_{j}+s_{ij}+r_{ij}$ (3)

where $g_{i}(g_{j})$ is the gca for the $i^{\text{th}}$ ( $j^{\text{th}}$ ) parent, $s_{ij}$ ( $s_{ij}=s_{ji}$ ) is the sca for the cross between the $i^{\text{th}}$ and the $j^{\text{th}}$ parents and $r_{ij}$ is the reciprocal effects ( $r_{ij}=-r_{ji}$ ) where ( $i,j=1,2,\ldots,p$ for design $d$ and we also assume that $\sum\limits_{j}{s_{ij}}_{=}0$ , for all $i$ . In matrix notation model can be written as

$\displaystyle\tau=Zg+s+r,$ (4)

where $Z=(z_{ui})(u=1,2,\ldots,v;i=1,2,\ldots,p)$ is the cross and gca relation matrix.

$z_{ui}=$ 2 if the $u^{\text{th}}$ cross has both parent $i$ . $=$ 1 if the $u^{\text{th}}$ cross has only one parent $i$ . $=$ 0, otherwise.

Now multiplying Eq. (4) by $C_{d1}$ we get following equation.

$\displaystyle C_{d}\tau=C_{d}Zg+Cs+C_{d}r$ $\displaystyle\text{i.e. }E(Q_{d})=C_{d}Zg+C_{d}s+C_{d}r,V(Q_{d})=\sigma^{2}C_{d}$ (5)

Since the matrix $C_{d}$ is singular, we use the unified theory of least square due to Rao (1973). We get the estimate of general combining ability $g$ .

$\displaystyle g=(Z^{\prime}C_{d}C^{-}_{d}C_{d}Z)^{-}Z_{d}^{\prime}Q_{d}=(Z^{% \prime}C_{d}Z_{d})^{-}Z^{\prime}C_{d}\tau=H_{1}\tau$ (6)

where

$\displaystyle{H}_{1}=(Z^{\prime}C_{d}Z)^{-}Z^{\prime}C_{d}$ $\displaystyle\text{and }(Z^{\prime}C_{d}Z)=2rp\left[{I}_{p}-\frac{1}{p}{1}_{p}% {1}_{p}^{\prime}\right]$ (7)

The generalized inverse of $(Z^{\prime}C_{d}Z)$ is as given below.

$\displaystyle((Z^{\prime}C_{d}Z)^{-}=\frac{1}{2rp}I_{p}$

where, for positive integers $t^{\prime}$ , ${I}_{n}$ is a $n\times n$ identity matrix, ${1}_{n}$ is $n\times 1$ vector with all elements unity.

$\displaystyle\text{with }\textit{Var}(g)=\sigma^{2}H_{1}C^{-}_{d}{H}_{1}^{% \prime}=\sigma^{2}\frac{1}{2rp}{I}_{p}$ (8)

This shows that the all elementary contrast among general combining abilities effects through design $d$ is estimated with the same variance. Thus, the design $d$ is variance balanced. We thus have the following theorem.

Theorem 3.1: For a complete set of mutually orthogonal Latin squares of order $p$ , if $p(>3)$ is a prime or power a prime, there always exists a V-B row-column design for CDC experiment method (1) with parameters $v=p^{2},b=2p,k=p$ and $r=2$ .

4. Optimality

We now take up the optimality aspects. The optimality criterion chosen is the minimization of the average variance of the best linear unbiased estimators of all elementary comparison between general combining ability effects. According to Kiefer (1975), a design is universally optimal in a relevant class of competing designs if:

(i)
The information matrix $C_{d}$ of the design is completely symmetric means $C_{d}$ has all its diagonal elements equal and all its off-diagonal elements equal; and
(ii)
The matrix $C_{d}$ has maximal trace over all designs in the class of competing designs that is $2(n-b)$ where $n$ is the total number of experimental units in the design and $b$ is number of blocks.

In our case the $C_{d}$ matrix of designs d is ( $Z^{\prime}C_{d}Z$ ) which is completely symmetric. From Eq. (3) it is easy to see that the trace of design $d$ is $2rp(p-1)$ which is equal to $2(n-b)$ i.e $2rp(p-1)$ . Hence, we have following result.

Theorem 4.1: The designs d for diallel crossing experiment method 1 with parameters $v=p^{2},b=2p,k=p$ and $r=2$ , obtained from complete set of mutually orthogonal Latin squares of semi standard order, is universally optimal.
5. Efficiency factor

It is clear from Eq. (3) that using proposed design d each elementary contrast among gca effects are estimated with a variance $(1/rp)\sigma^{2}$ under randomized block design with equal replications, the variance of any elementary contrast among the gca effects for method 1 are estimated with variances $(1/rp)\sigma_{1}^{2}$ where $\sigma_{1}^{2}$ is the per observation variance. The efficiency factor of designs d as compared to randomized complete block design having same number of crosses is given by $E_{1}=1$ .

Table 2

Row-column designs for complete diallel crosses method (1) with $p\leqslant 17$ generated by superimposition of mutually orthogonal Latin squares

S.No.	${P}$	${b}$	${k}$
1.	5	10	5
2.	9	18	9
3.	11	22	11
4.	13	26	13
5.	17	34	17

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Optimal row-column designs for CDC method (1)

Abstract

Keywords

1. Introduction

2. Some definitions and method of construction

Table 1 Columns 1 2 3 4 5 1 1111 2345 3524 4235 5432 2 2222 3451 4135 5314 1543 Rows 3 3333 4512 5241 1425 2154 4 4444 5134 1352 2531 3215 5 5555 1234 2413 3142 4321

References

Table 1

Columns

1 2 3 4 5

1 1111 2345 3524 4235 5432

2 2222 3451 4135 5314 1543

Rows 3 3333 4512 5241 1425 2154

4 4444 5134 1352 2531 3215

5 5555 1234 2413 3142 4321