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\title{Efficient V-B Block Designs for CDC Method 4}
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             \author[1]{Dr.  M.K.Sharma}

             \affil[1]{  University of Gondar}

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\date{\small \em Received: 8 April 2013 Accepted: 4 May 2013 Published: 15 May 2013}

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\begin{abstract}
        


Some optimal incomplete block designs for complete diallel cross method 4 are known in literature. These designs require several replications for each cross and thus consume more resources such as experimental units, experimental material, time etc. So, there is a need to evolve designs which require minimum possible replications of parental lines. In this paper a method of construction of these designs is proposed by using mutually orthogonal Latin squares. These designs are connected for cross effects and perform well when  compared to connected and not connected optimal designs reported by Dey and Midha (1996), Chai and Mukerjee (1999) and Gupta and Kageyama (1994), respectively.

\end{abstract}


\keywords{latin square, complete diallel cross, general combining ability, specific combining ability, mating design.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
rthogonal Latin squares are used for construction of Graeco Latin square, balanced incomplete block designs and square lattice designs. A set of p-1 orthogonal Latin square of side p can always be constructed if p is a positive prime or power of a positive prime.\par
If p = 4 t +2 and t > 1, then there exits pairs of mutually orthogonal Latin squares of order p  {\ref (Bose, Shrikhande and Parker (1960)}). From a practical view point, mutually orthogonal Latin squares are important and an exhaustive list of these squares is available in \hyperref[b7]{Fisher and Yates (1963)}. In this paper we use mutually orthogonal Latin squares in construction of mating designs for the diallel cross method 4 referred to \hyperref[b8]{Griffing (1956)}.\par
A diallel cross is a type of mating design used in plant breeding and animal breeding to study the genetic properties and potential of inbred lines or individuals. Let p denote the number of lines and let a cross between lines i and j be denoted by i × j, where i<j = 0, 1, ? , p-1 and p(p-1)/2 possible crosses.\par
Among the four types of diallel discussed by \hyperref[b8]{Griffing (1956)}, method 4 is the most commonly used diallel in plant breeding. This type of diallel crossing includes the genotypes of one set of F 1 ,S means of the type (i × j ) = ( j × i) , but neither the parents nor the reciprocals with all possible v = p(p-1)/2 crosses. This is sometimes referred to as the modified diallel. We shall refer to it as a complete diallel cross (CDC).\par
The problem of finding optimal mating designs for complete diallel cross experiments has received Author : Department of Statistics, University of Gondar, Gondar, Ethiopia. E-mail : mk\textunderscore subash@yahoo.co.in attention in recent years; see \hyperref[b6]{Gupta and Kageyama (1994)}, \hyperref[b5]{Dey and Midha (1996)} and \hyperref[b1]{Chai and Mukerjee (1999)}. Most of the results on optimal block designs for diallel crosses have been derived for the general combining ability (gca) under the assumptions that the model does not include parameters representing the specific combining ability (sca) \hyperref[b6]{Gupta and Kageyama (1994)} and \hyperref[b5]{Dey and Midha (1996)} but with few exceptions Chai and Mukerjee (1999) and \hyperref[b2]{Choi et al. (2002)}. The designs of these authors can be used to estimate specific combining ability (sca) but they demand more resources in terms of experimental units and experimental material. In such a situation there is need for designs which require minimum possible number of experimental units in conducting CDC experiments and are equally efficient in comparison to optimal block designs and randomized block designs when the model, in addition to the block effects and general combining ability, includes specific combining ability.\par
In the present paper we are proposing efficient variance balanced incomplete block designs for CDC experiments through mutually orthogonal Latin squares under the assumption that the model includes the parameter of specific combining ability. 
\section[{II.}]{II.} 
\section[{Method of Design Construction}]{Method of Design Construction}\par
It is known that when p is a prime positive integer or a power of prime positive integer, it is possible to construct (p-1) orthogonal Latin squares in such a way that they differ only in a cyclical interchange of the rows from 2 nd to p th . Such squares are taken for the construction of incomplete block designs for diallel crosses. For p =6, such squares cannot be constructed.\par
Assume that there are p inbred lines and it is desired to find an incomplete block design for a mating design involving p (p-1)/2 crosses. Out of (p-1) mutually orthogonal Latin square (MOLS), consider any two MOLS of semi-standard form of order p and superimposed one square over the other. We obtain one Graeco Latin square in which each cell contains ordered pairs of integers (i, j) taking values from 0 to p -1 . These ordered pairs of integers occur once in a square. From Graeco Latin square remove the pairs of the type with i = j and considering other ordered pairs of integers as crosses between lines i and j and the columns as blocks. By doing so we get an incomplete block design d for diallel cross experiment method 4 with parameters After superimposition L 2 over L 1 and removing cross of the type i = j and considering columns as blocks, we obtain design d as given below:\par
Design dB 1 B 2 B 3 B 4 B 5 1×2 2×3 3×4 4×0 0×1 2×4 3×0 4×1 0×2 1×3 3×1 4×2 0×3 1×4 2×0 4×3 0×4 1×0 2×1 3×2 III. 
\section[{Analysis}]{Analysis}\par
For the analysis of data obtained from design d , we will follow Singh and Hinkelmann-(1998) two stage procedures for estimating gca and sca effects. The first stage is to consider the proposed designs to estimate cross effects, say, ? = (? 01 , ? 02 , ?, ? (p-2)(p-1)/2 ) for design d by the following model.y = µ1 + X ? + D ? + e (3.1)\par
Where y is an n × 1 vector of observations, 1 is the n × 1 vector of ones, X is the n × v design matrix for treatments and D is an n × b design matrix for blocks, that is, the (h,u) th ((h,l) th ) element of X (respectively, of D) is 1 if the h th observation pertains to the u th cross ( to l th block), and is zero otherwise (h = 1, ?, n; u = 1, ?, v ; and 1, ? , b), µ is a general mean, ? is a v × 1 vector of treatment parameters, ? is a b × 1 vector of block parameters and e is an n × 1 vector of residuals. It is assumed that vector ? is fixed and e is normally distributed with E(e ) = 0, V(e) = ? 2 I and Cov (? , e´ ) = (0), , where I is the identity matrix of conformable order.\par
Following \hyperref[b14]{Tocher (1952)}, \hyperref[b10]{Raghavarao (1971)} and \hyperref[b4]{Dey (1986)}, the least square method for the analysis of a proposed designs leads to the following reduced normal equations for the model  {\ref (3.1)}.C d ? = Q d (3.2) Where C d = r ? -N k -1 N´ and Q d = (Q 1d ,?., Q vd ) = T -N k -? B\par
In the above expressions above r ? and k ? are diagonal matrices of order v × v and b × b with E (Q d ) = C d ? and V (Q d ) = ? 2 C d (3.4)\par
Now we will utilize the above equations to estimate the genetic parameters in the proposed design. The second stage is to utilize the fact that the cross effects can be expressed in terms of gca and sca effects. So we can write ? i j = g i + g j + s ij  {\ref (3.5)} Where g i (g j ) is the gca for the i th (j th ) parent, s i j (s ij = s ji ) is the sca for the cross between the i th and the j th parent ( i< j =0, 1, ? , p-1). In matrix notation equation (3.5) can be written as? = Z g + s (3.6)\par
Where Z = ( z u i ) (u = 1, 2, ?, n : i = 0, 1, ?, p-1) is the cross and gca relation matrix. z ui = 2, if the u th cross has both parents i.\par
= 1, if the u th cross has only one parent i .\par
= 0, otherwise. Following the approach used in Kempthorne and Curnow (1961), equation (3.2) can then be written asC d ? = C d Z g + C d s or E ( Q d ) = C d Z g + C d s (3.7)\par
Since the matrix C is singular, we use the unified theory of least square due to \hyperref[b12]{Rao (1973)}. So we get estimator of g as  Since the covariance matrix of g ? is a constant times the identity matrix, therefore the proposed design d is variance-balanced for general combining ability effects. We thus have the following results.g ? = (Z ´ C d C d ? C d Z )? Z ´ Q d = ( Z ´ C d Z ) ? Z ´ Q d (3. 
\section[{Theorem}]{Theorem}\par
For a positive prime p>3, if there exits a mutually orthogonal Latin square of order p, then there always exist variance-balanced incomplete block design for CDC experiment method 4. Now substituting the estimate of g in equation (3.6), we obtain the estimator of s. 
\section[{s ?=}]{s ?=}(C d ? -(p-1)/2 p (p-3) Z Z ´) Q d = (C d ? -(p-1)/2 p (p-3) Z Z´) C d ? = H 2 ? (3.11)\par
WhereH 2 = (C d ? -(p-1)/2 p (p-3) Z Z ´ ) C d Var ( s ?) = H 2 C d H´2 ? 2 (3.12)\par
Since H 1 1 v = 0, H 2 1 v = 0, H 1 H 2 ´ = 0, rank (H 1 ) = p-1 and rank (H 2 ) = v-p.\par
It follows that g and s represented by treatment contrasts that carry p-1 and v-p degrees of freedom respectively and that contrasts representing g are orthogonal to those representing s. It means the proposed design d allows for gca and sca effects to be estimated independently.\par
The sum of squares due to gca and sca for d are given bySS (gca) = Q´d Z (Z´ C d Z) ? Z´ Q d (3.13) SS (sca) = Q d ´ (C d ? -(p-1)/2 p (p-3) Z Z ´ ) Q d (3.14)\par
The ANOVA is then given in Table \hyperref[tab_0]{1}. Crosses (adjusted for blocks)rank (C d ) Q ´d C d ? Q d gca rank (H 1 ) Q´d Z (Z´ C d Z) ? Z´ Q d sca rank (H 2 ) Q d ´(C d ? -(p-1)/2p(p-3) Z Z´) Q d Residual (n-1) -rank (C d ) -rank (H 1) -rank (H 2 ) y ´y -G 2 / p(p-1) -B ´ B/p - Q ´d C d ? Q d Total n-1 y ´y -G 2 / p(p-1)\par
G = grand total of all n observations IV. 
\section[{Efficiency Factor}]{Efficiency Factor}\par
If instead of the proposed design d, one adopts a randomized complete block design with 2 blocks and each block contains p (p-1)/2 crosses, the C R -matrix can easily shown to beC R = 2 (p -2) ( I p -1/p J p ) (4.1)\par
Where I p is a identity matrix of order p and J p is a matrix of 1' s . So that the variance of best linear unbiased estimate (BLUE) of any elementary contrast among the gca effects is ? 1 2 / (p-2), where ? 1 2 is the per observation variance in the case of randomized block experiment. It is clear from (3.10) that using design d each BLUE of any elementary contrast among gca effects is estimated with variance ? 2 (p-1) / p (p-3). . Hence efficiency factor E of design d as compared to randomized block design under the assumption of equal intra block variances isE = ( ? 1 2 = ? 2 ) is p (p-3)/(p-1) (p-2) (4.2)\par
In Tables \hyperref[tab_2]{2, 3}, and 4 , we are presenting the efficiency factors of CDC by \hyperref[b6]{Gupta and Kageyama (1994)} , universally optimal and efficient block designs reported by \hyperref[b5]{Dey and Midha (1996)} and design d in relation to randomized block design, respectively. V. 
\section[{Discussion}]{Discussion}\par
In Table \hyperref[tab_1]{2}, we find that for p = 4, 5, 8, 9, 10, 11, 12, 13, and 15 parental lines , the design d perform well in comparison to optimal diallel cross Gupta and Kageyama (1994). In Table \hyperref[tab_2]{3}, for p = 5, 7 and 9 the performance of design d is more or less same in comparison to optimal design \hyperref[b5]{Dey and Midha (1996)}. In Table \hyperref[tab_3]{4}, for p = 5, 7, 8 and 10 the design perform well in comparison to efficient designs. Since design d requires minimum possible experimental units, therefore, design d can be used in place of GK and DM designs for estimating gca and sca effects. 
\section[{VI.}]{VI.} 
\section[{Illustration}]{Illustration}\par
We show the essential steps of analysis of a diallel cross experiment, using an incomplete block design proposed in this paper. For this purpose, we take data from an unpublished experiment conducted by Dr. Terumi Mukai on Drosophila melanogaster \hyperref[b3]{Cockerham and Weir (1977)} on page 203. For the purpose of illustration, we take data of relevant crosses from this experiment. Each cross is replicated twice. The layout and observations in parentheses are given below.  The following are the vector of treatment total, block total and adjusted treatment total, respectively. T = (31.9, 62.  \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}D}\end{figure}
      \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.2518518518518518\textwidth}P{0.22037037037037036\textwidth}P{0.37777777777777777\textwidth}}
Source of variation\tabcellsep Degrees of Freedom\tabcellsep Sum of squares\\
Block\tabcellsep p-1\tabcellsep B ´ B/p -G 2 / p (p-1)\end{longtable} \par
 
\caption{\label{tab_0}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.07083333333333333\textwidth}P{0.04869791666666666\textwidth}P{0.15494791666666666\textwidth}P{0.044270833333333336\textwidth}P{0.07526041666666666\textwidth}P{0.159375\textwidth}P{0.2966145833333333\textwidth}}
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep 013\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep 2\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep Year\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep D D D D D D D D ) D\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep (\\
\tabcellsep \tabcellsep \multicolumn{4}{l}{comparison to RBD}\tabcellsep \\
\multicolumn{2}{l}{S.No. p}\tabcellsep n\tabcellsep r\tabcellsep \multicolumn{2}{l}{2k E GK}\tabcellsep E d\\
1\tabcellsep 4\tabcellsep 6\tabcellsep 3\tabcellsep 4\tabcellsep 1.00\tabcellsep 0.66\\
2\tabcellsep 5\tabcellsep 10\tabcellsep 4\tabcellsep 4\tabcellsep 0.83\tabcellsep 0.83\\
3\tabcellsep 7\tabcellsep 21\tabcellsep 6\tabcellsep 6\tabcellsep 0.93\tabcellsep 0.93\\
4\tabcellsep 8\tabcellsep 28\tabcellsep 7\tabcellsep 8\tabcellsep 1.00\tabcellsep 0.95\\
5\tabcellsep 8\tabcellsep 28\tabcellsep 7\tabcellsep 4\tabcellsep 0.66\tabcellsep 0.95\\
6\tabcellsep 9\tabcellsep 36\tabcellsep 8\tabcellsep 8\tabcellsep 0.96\tabcellsep 0.96\\
7\tabcellsep 9\tabcellsep 36\tabcellsep 8\tabcellsep 6\tabcellsep 0.85\tabcellsep 0.96\\
8\tabcellsep 10\tabcellsep 45\tabcellsep 9\tabcellsep 10\tabcellsep 1.00\tabcellsep 0.98\\
9\tabcellsep 10\tabcellsep 45\tabcellsep 9\tabcellsep 6\tabcellsep 0.83\tabcellsep 0.98\end{longtable} \par
 
\caption{\label{tab_1}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.09842105263157895\textwidth}P{0.2192105263157895\textwidth}P{0.035789473684210524\textwidth}P{0.09394736842105263\textwidth}P{0.16105263157894736\textwidth}P{0.085\textwidth}P{0.15657894736842104\textwidth}}
\tabcellsep \multicolumn{4}{l}{d in comparison to RBD}\tabcellsep \tabcellsep \\
\multicolumn{2}{l}{S.No. Ref. No.}\tabcellsep p\tabcellsep n 1\tabcellsep E DM\tabcellsep n 2\tabcellsep E d\\
1\tabcellsep T2\tabcellsep 5\tabcellsep 30\tabcellsep 0.83\tabcellsep 20\tabcellsep 0.83\\
2\tabcellsep T3\tabcellsep 5\tabcellsep 60\tabcellsep 0.83\tabcellsep 20\tabcellsep 0.83\\
3\tabcellsep T4\tabcellsep 5\tabcellsep 90\tabcellsep 0.83\tabcellsep 20\tabcellsep 0.83\\
4\tabcellsep T8\tabcellsep 7\tabcellsep 210\tabcellsep 0.70\tabcellsep 42\tabcellsep 0.93\\
5\tabcellsep T22\tabcellsep 7\tabcellsep 210\tabcellsep 0.93\tabcellsep 42\tabcellsep 0.93\\
6\tabcellsep T40\tabcellsep 8\tabcellsep 280\tabcellsep 1.00\tabcellsep 56\tabcellsep 0.95\\
7\tabcellsep T41\tabcellsep 9\tabcellsep 252\tabcellsep 0.96\tabcellsep 72\tabcellsep 0.96\\
8\tabcellsep \multicolumn{3}{l}{T54 10 315}\tabcellsep 1.00\tabcellsep 90\tabcellsep 0.98\end{longtable} \par
 
\caption{\label{tab_2}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.1528776978417266\textwidth}P{0.11618705035971223\textwidth}P{0.21402877697841724\textwidth}P{0.026498800959232613\textwidth}P{0.2466426858513189\textwidth}P{0.09376498800959232\textwidth}}
\multicolumn{5}{l}{S.No. Ref. p n 1 E DM n 2 E d S.No. Ref p}\tabcellsep n 1 E DM n 2 E d\\
1\tabcellsep T12 5\tabcellsep 60 0.84 20 0.83\tabcellsep 9\tabcellsep T58 5\tabcellsep 60 0.84 20 0.83\\
2\tabcellsep T13 5\tabcellsep 90 0.92 20 0.83\tabcellsep 10\tabcellsep T60 5\tabcellsep 60 0.97 20 0.83\\
3\tabcellsep T33 5\tabcellsep 40 0.94 20 0.83\tabcellsep 11\tabcellsep \multicolumn{2}{l}{T94 7 210 0.84 42 0.93}\\
4\tabcellsep T34 5\tabcellsep 80 0.80 20 0.83\tabcellsep 12\tabcellsep \multicolumn{2}{l}{T95 7 210 0.91 42 0.93}\\
5\tabcellsep \multicolumn{2}{l}{T37 5 100 0.87 20 0.83}\tabcellsep 13\tabcellsep \multicolumn{2}{l}{T77 8 196 0.98 56 0.95}\\
6\tabcellsep T44 5\tabcellsep 30 1.00 20 0.83\tabcellsep 14\tabcellsep \multicolumn{2}{l}{T85 9 252 1.00 72 0.96}\\
7\tabcellsep T45 5\tabcellsep 60 0.84 20 0.83\tabcellsep 15\tabcellsep \multicolumn{2}{l}{T91 10 405 0.92 90 0.98}\\
8\tabcellsep T57 5\tabcellsep 30 0.84 20 0.83\tabcellsep \tabcellsep \tabcellsep \\
\multicolumn{2}{l}{DM denotes Dey and Midha ,}\tabcellsep \tabcellsep \tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_3}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.27960526315789475\textwidth}P{0.265625\textwidth}P{0.02796052631578947\textwidth}P{0.12861842105263158\textwidth}P{0.10345394736842105\textwidth}P{0.04473684210526315\textwidth}}
\tabcellsep \multicolumn{2}{l}{Efficient V-B Block Designs for CDC Method 4}\tabcellsep \tabcellsep \tabcellsep \\
013\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
2\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Year\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
20\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Volume XIII Issue I Version I\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
D D D D ) D\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
(\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Global Journal of Computer Science and Technology\tabcellsep Source\tabcellsep D.F\tabcellsep Sum of squares\tabcellsep square Mean sum of\tabcellsep F\\
\tabcellsep Blocks\tabcellsep 4\tabcellsep 137.81\tabcellsep \tabcellsep \\
\tabcellsep Crosses\tabcellsep 9\tabcellsep 418.92\tabcellsep 46.54\tabcellsep 53.70\\
\tabcellsep g.c.a\tabcellsep 4\tabcellsep 341.70\tabcellsep 85.42\tabcellsep 98.56\\
\tabcellsep s.c.a\tabcellsep 5\tabcellsep 77.20\tabcellsep 15.44\tabcellsep 17.81\\
\tabcellsep Intra block error\tabcellsep 6\tabcellsep 5.2\tabcellsep 0.86\tabcellsep \\
\tabcellsep Total\tabcellsep 19\tabcellsep 561.93\tabcellsep \tabcellsep \end{longtable} \par
  {\small\itshape [Note: © 2013 Global Journals Inc. (US)]} 
\caption{\label{tab_5}Table 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6} \par 
\begin{longtable}{P{0.12277777777777776\textwidth}P{0.39666666666666667\textwidth}P{0.33055555555555555\textwidth}}
Parent\tabcellsep Estimates of (gca )\tabcellsep ± S E\\
0\tabcellsep -1.24\tabcellsep 0.4147\\
1\tabcellsep -0.65\tabcellsep 0.4147\\
2\tabcellsep -2.16\tabcellsep 0.4147\\
3.\tabcellsep 2.58\tabcellsep 0.4147\\
4.\tabcellsep 1.47\tabcellsep 0.4147\end{longtable} \par
 
\caption{\label{tab_6}Table 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7} \par 
\begin{longtable}{P{0.1628504672897196\textwidth}P{0.08738317757009345\textwidth}P{0.13901869158878505\textwidth}P{0.09135514018691589\textwidth}P{0.1588785046728972\textwidth}P{0.21051401869158878\textwidth}}
\multicolumn{2}{l}{SCA Estimate of (sca)}\tabcellsep ± S E\tabcellsep SCA\tabcellsep Estimate of (sca)\tabcellsep ± S E\\
s 01\tabcellsep -0.63\tabcellsep 0.4818\tabcellsep s 13\tabcellsep -5.29\tabcellsep 0.4818\\
s 02\tabcellsep 8.65\tabcellsep 0.4818\tabcellsep s 14\tabcellsep -5.61\tabcellsep 0.4818\\
s 03\tabcellsep 3.13\tabcellsep 0.4818\tabcellsep s 23\tabcellsep -1.26\tabcellsep 0.4818\\
s 04\tabcellsep -3.04\tabcellsep 0.4818\tabcellsep s 24\tabcellsep 0.52\tabcellsep 0.4818\\
s 12\tabcellsep 2.58\tabcellsep 0.4818\tabcellsep s 34\tabcellsep 0.95\tabcellsep 0.4818\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep D D D D D D D D )\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep (\end{longtable} \par
 
\caption{\label{tab_7}Table 7 :}\end{figure}
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\end{document}
