USE OF NESTED DESIGNS IN DIALLEL CROSS EXPERIMENTS

Transcription

1 USE OF NESTED DESIGNS IN DIALLEL CROSS EXPERIMENTS. Introuction Rajener Parsa I.A.S.R.I., Library Avenue, New Delhi The term iallel is a Greek wor an implies all possible crosses among a collection of male an female animals. Hayman (954a, 954b) efine iallel cross as the set of all possible crosses between several genotypes, which may be iniviuals, clones, homozygous lines, etc. Diallel cross is most balance an systematic experiment to examine continuous variation. The genetic information relate to parental population is available in early generation (F itself). Thus, it is useful to breeing strategy without losing much time. Many improvements with respect to generalization of iallel crosses enlarge its scope an utility. Diallel crosses are use mainly ) to estimate the genetic components of variation of a quantitative character an ) to estimate the combining abilities of ifferent inbre lines involve in the crosses. The concept of combining ability is a measure of gene action an helps in the evaluation of inbres in terms of their genetic value an in the selection of suitable parents for hybriization. Superior cross combinations can also be ientifie by this technique. There are two types of combining abilities: (i) general combining ability or gca (ii) specific combining ability or sca General combining ability of an inbre line is the average performance of the hybris that this line prouces with other lines chosen from a ranom mating population. It is analogous to main effect of a factorial experiment. It is estimate from half-sib families. Specific combining ability refers to a pair of inbre lines involve in a cross. It inicates cases in which certain combinations o relatively better or worse than woul be expecte on the basis of gca effects of two lines involve in it. It is the eviation of a particular cross from the expectation on the basis of average gca effects of the two lines involve. It is analogous to an interaction effect of a factorial experiment. Let there be p inbre lines. The iallel cross of these p lines results in p progeny families. These inclue ) p inbre lines ) p C F hybris an 3) p C F reciprocal hybris Depening upon which of the progeny families is inclue for analysis, four methos of iallel analysis have been propose. They are

2 ) comprising of all the p progeny families. ) Incluing p parents an p C F hybris, i.e., a total of p(p+)/ families. 3) Incluing p C F hybris an p C F reciprocals, i.e., p(p-) combinations 4) p C F ' s only. Here, we shall consier only the esigns use for the 4 th type of analysis. That is, esigns for iallel cross experiment involving p (p-)/ crosses of type (i x j) for i < j an i, j =,,..., p. This is calle Type IV mating esign of Griffing. It is also known as moifie iallel system or half-iallel. To obtain unbiase estimates of the population parameters of σ gca an σ sca, it is necessary to employ moifie iallel system. The inclusion of inbres vitiates this property of the estimates obtaine from experimental ata. This metho coul be chosen for genetic investigation when maternal inheritance is not suspecte, i.e., reciprocal crosses gives the same result. Also this metho requires lesser number of experimental units. This has receive much attention now ays because gca s are important while selecting inbre lines for hybriization programme. It may be worthwhile to note here that the iallel analysis propose by Hayman is base on a fixe effect moel. In the fixe effect moel, the interest is primarily in the comparisons of the combining abilities of parents employe for iallel mating. The analysis propose by Griffing (956) takes care of both the fixe effects as well as the ranom effect situation. Ranom effects moel is chosen when inferences are to be rawn about the base population from which the inbre lines have been sample. Most common iallel cross experiments have been evaluate using a completely ranomize esign (CRD) or a ranomize complete block esign (RCBD) with suitable number of replications. But when the number of lines increases, the number of crosses increases very rapily. For example, with p = 5 lines there are only 0 crosses. While for p = 0 the number of crosses is 45 an when p = 5 it becomes 05. Laying out the esign, as a ranomize complete block esign, even with a moerately large number of lines, will, however, result into large blocks an consequently large intra-block variances. It results into high coefficient of variation (CV) an hence reuce precision on the comparisons of interest. In orer to overcome this problem, one may use incomplete block esigns like balance incomplete block (BIB) esigns, partially balance incomplete block (PBIB) esigns with two associate classes, cyclic esigns, etc. by treating the crosses as treatments for one way elimination of heterogeneity settings. For instance, a BIB esign has been use by ientifying crosses as treatments [see e.g., Das an Giri (986, pp44-44); Ceranka an Mejza, (988)]. These esigns have interesting optimality properties when making inferences on a complete set of orthonormalise treatment contrasts. However, in iallel cross experiments the interest of the experimenter is in making comparisons among general combining ability (gca) effects of lines an not of crosses an, therefore, using these esigns as mating esigns may result into poor precision of the comparisons among lines. Further, the analysis of a iallel cross experiment in incomplete blocks epens on 306

3 the incience of lines in blocks, rather than the incience of the treatments, or crosses, in blocks. To be more clear consier the following example: Example.: An experimenter is intereste in generating a mating esign for comparing 4-inbre lines on the basis of their gca effects. For a complete iallel cross experiment the number of crosses is 6, enote by x A, x 3 B, x 4 C, x 3 D, x 4 E an 3 x 4 F. An incomplete block esign for iallel cross experiment, D 0 consiering 6 crosses as treatments enote as A, B, C, D, E an F, is a BIB esign with parameters v = 6,b = 5,r = 5,k =, λ =. The above esign requires 30 experimental units an each cross is replicate 5 times. The C = G NK N matrix of the esign D0 is C = 6( I 4 J4 ), an the variance of 4 the Best Linear Unbiase Estimator (B.L.U.E.) of any elementary contrast among lines (gca) is σ. Here C is the coefficient matrix of reuce normal equations for 6 estimating linear functions of gca effects, G is a matrix with iagonal elements as replication number of lines an off-iagonal elements as replication number of crosses, N is the incience matrix of lines vs blocks, K is iagonal matrix with elements as block sizes, I v, J v is an ientity matrix of orer v an a vxv matrix of all elements ones, respectively, an σ is the per plot variance. Another mating esign generate through a ifferent metho is D 0. The esign can be obtaine by taking 5-copies of the block esign with block contents as Block : {( x 4), ( x3)}; Block : {( x 4), (3 x )}; Block 3: {(3 x 4), ( x )}. This esign also requires 30 experimental units an each cross is replicate 5 times. The C G NK N = matrix of the esign D 0 is C = 0( I 4 J4 ), an the variance of the 4 B.L.U.E. of any elementary contrast among line (gca) effects is σ. Thus, one can 0 see that the esign D 0 estimates the elementary contrasts among gca's with more precision than the esign D0 although both the esigns are variance balance for estimating any normalize contrast of gca effects. Another approach avocate in the literature is to start with an incomplete block esign, write all the pairs of treatments within a block, ientify these pairs of treatments as crosses by treating treatments of the original incomplete block esign as lines an use the resulting esign as a esign for iallel crosses. Sharma (996) use this approach for complete iallel cross experiments by using balance lattice esigns. This was, however, also avocate by Das an Giri (986), in the context of BIB esigns, an a balance lattice is also a BIB esign. Ghosh an Divecha (997) use this for PBIB esigns to obtain esigns for partial iallel crosses an Sharma (998) obtaine esigns for partial 307

4 iallel crosses through circular esigns. However, this approach also oes not seem to o well as will become clear through the following examples: Example.: An experimenter is intereste in generating a mating esign for comparing 7-inbre lines on the basis of their gca effects. A mating esign for iallel cross experiment, D, with crosses can be obtaine by writing all possible pairs of treatments within a block of the BIB esign, D a, with parameters v = b = 7, r = k = 3, λ = an treating the treatments as lines an paire treatments as crosses. In esign D, the number of crosses is v c = that are arrange in b = 7 blocks of size k = 3 each. Another mating esign D can also be generate through a ifferent metho in 7 lines with crosses arrange in 7 blocks of size 3 each. The esigns, with rows as blocks, are D a D D 4 x x4 x4 x7 x6 3x5 3 5 x3 x5 3x5 x 3x7 4x x4 3x6 4x6 x3 x4 5x x5 4x7 5x7 3x4 x5 x x6 x5 x6 4x5 3x6 x x7 x6 x7 5x6 4x7 x3 7 3 x7 3x7 x3 6x7 x5 x4 7 The C = G NK N matrix of the esign D is C = ( I 7 J 7 ), an the variance of 3 7 the Best Linear Unbiase Estimator (B.L.U.E.) of any elementary contrast among line 6 (gca) effects is σ. 7 4 The C = G NK N matrix of the esign D is C = ( I 7 J 7 ), an the variance of the B.L.U.E. of any elementary contrast among line (gca) effects is σ. Thus, one can 7 see that the esign D estimates the elementary contrasts among gca effects with twice the precision as obtaine through the esign D although both the esigns are variance balance for estimating any normalize contrast of gca effects. Example.3: Consier another situation when the experimenter is intereste in esigning an experiment with p = 9 lines. Sharma (998) generate a mating esign D from a cyclic esign with parameters v = b = 9, r = k = 3, by eveloping the initial block k (,, 3) mo 9 an then taking all the possible crosses from each block. The variances of the B.L.U.E. of any elementary contrast among line (gca) effects is σ,.04574σ,.94σ an.39866σ. Each of these variances is for 9 ifferent B.L.U.E. of the elementary contrasts among gca effects an the average variance is given by.374σ. A similar type of mating esign D can be obtaine from a cyclic esign with parameters v = b = 9, r = k = 3 by eveloping the initial block 308

5 (,, 4) mo 9. The variances of the B.L.U.E. of any elementary contrast among line (gca) effects is σ, σ, σ an.049σ, an the average variance is given by 0.99σ. The esign D has smaller average variance of B.L.U.E. of elementary contrasts of gca effects as compare to D. The esign D seems to have an intuitive appeal also as it contains more number of istinct crosses as compare to the esign D although the size of both the esigns is the same in terms of the total number of observations. Hence, this esign is more useful for same number of experimental units. Example.4: Consier another situation when the experimenter is intereste in esigning an experiment with p = lines. Ghosh an Divecha (997) generate a mating esign D from a group ivisible esign with parameters v =, b = 9, r = 3, k = k 4, λ = 0, λ =, m = 4, n = 3(Clatworthy, 973; SR4), by taking all possible crosses of treatments within each block, by treating treatments in the original esign as lines. The variances of the B.L.U.E. of elementary contrasts among gca effects are σ, for the first associates ( in number) an, for the secon associates (54 in number). The average variance is 0.444σ. A similar type of mating esign D is obtaine from a ifferent metho. The variances of the B.L.U.E. of any elementary contrast among gca effects is number) an 0.596σ, for the first associates ( in for the secon associates (54 in number). The average variance is 0.553σ. One can easily see that in the two mating esigns the precision of the B.L.U.E. of gca effects is ifferent an, therefore, the choice of an appropriate mating esign is important. The esign D 0.σ with rows as blocks is σ D Blocks Blocks x 5x6 9x0 3x4 7x8 x x8 5x 9x4 x7 6x 0x3 x3 5x7 9x x4 6x8 0x x0 5x 9x6 3x 7x4 x8 x4 5x8 9x x3 6x7 0x x 5x3 9x7 x 6x4 0x8 x6 5x0 9x 3x8 7x x4 x 5x4 9x8 x 6x3 0x7 x7 5x 9x3 x8 6x 0x4 It is clear from the above iscussions that for making comparisons of gca effects of p- inbre lines, the choice of an appropriate esign is important. This talk aresses this an similar problems. The problem of generating optimal mating esigns for experiments with iallel crosses has been recently investigate by several authors [see e.g., Gupta an Kageyama (994), Dey an Miha(996), Mukerjee(997), Das, Dey an Dean(998), Parsa, Gupta an Srivastava (999), Chai an Mukerjee (999)]. These authors use neste balance incomplete block (NBIB) esigns of Preece (967) for this purpose. This paper erives 309

6 general methos of construction of mating esigns, essentially generate from neste variance balance block (NBB) esigns. The optimality aspects have also been investigate uner a non-proper setting. The moel consiere here involves only the gca effects, the specific combining ability effects being exclue from the moel. The esigns obtaine are variance balance in the sense that the variances of the B.L.U.E. of elementary contrasts among gca effects are all same.. Neste Designs an Optimality results Let be a block esign for a iallel cross experiment of the type mentione in Section involving p-inbre lines, b blocks such that the j th block is of size. This means that there are k j crosses or k j lines, respectively in each block of. It may be mentione here that the esigns for iallel crosses have two types of block k, the block sizes with respect to crosses an, the block sizes with respect to the lines an k =. It, therefore, follows that the block esigns for iallel crosses may also be viewe as neste block esigns with sub blocks of size each an the pair of treatments in each sub block form the crosses, the treatments being the lines. Further, let r l enote the number of times the l th cross appears in, l =,,.., p(p-)/ an similarly s i enotes the number of times the i th line occurs in the crosses in the whole esign, i =,,,p. Then it is easy to see that p( p ) / b rl = k j = n, the total number of observations, an l= j= p s i i= b = k j j=, (because in every cross there are two lines). For the ata obtaine from the esign, we postulate the moel k k k Y = µ n + g + β + e (.) where Y is the nx vector of observe responses, µ is a general mean effect, n enotes an n - component column vector of all ones, g an β are vectors of p gca effects an b block effects, respectively. an are the corresponing nxp an nxb esign matrices respectively, i.e., the (s, t) th element of is if the s th observation pertains to the t th line an is zero otherwise. Similarly (s, t) th element of is if the s th observation comes from the t th block an is zero otherwise. e is the ranom error which follows a N n (, I ) 0 σ. n In the moel (.) we have not inclue the specific combining ability effects. Uner this moel, it can be shown that the coefficient matrix for reuce normal equations for estimating linear functions of gca effects using a esign is 30

7 C = G N K N ( ) ( ) i where = g ii, N = nij, gii = s an for i i, g i i is the number of times the cross ( ix i ) appears in ; n is the number of times line i occurs in the block j. G ( ) ij A esign is sai to be connecte if an only if Rank ( ) = p, C or equivalently, if an only if all elementary contrasts among the gca effects are estimable using. A connecte esign is variance balance if an only if all the iagonal elements of the matrix C are equal an all the off iagonal elements are also equal. In other wors, the matrix C is completely symmetric. In particular, a variance balance block esign for iallel crosses is sai to be a generalize binary variance balance block (GBBB) esign if in aition to completely symmetric information matrix, n = x or x an is ij j j + sai to be binary variance balance block (BBB) esign, if n ij = 0 or. For given positive integers p, b, n, D 0 ( p,b,n ) will enote the class of all connecte block esigns with p lines, b blocks an n experimental units. Here the block sizes are arbitrary but for a given esign D p,b,n, the block sizes are k,k, L, k. 0 ( ) Similarly, D p,b,k,...,k ) will enote the class of all connecte block esigns with p ( b lines, b blocks such that jth block is of size. We may allow k j > p for some or all k j j =,, L,b. Now using the Proposition of Kiefer (975) an the efinitions of GBBB (BBB) esigns for iallel crosses, we have the following results. Result 3.: A GBBB esign for iallel crosses, whenever existent, is universally optimal over D ( p,b,k,...,k ) b Result 3.: A BBB esign for iallel crosses, whenever existent, is universally optimal over D 0 (p, b, n). ( p,b,k,...,k ) It may be note that a esign that is universally optimal over D b is also universally optimal over D 0 (p, b, n) provie all k j p for all j =,, L, b. Similarly, a esign that is universally optimal over D 0 (p, b, n) is also universally optimal over D p,b,k,..., provie k = k for all j =,, L, b. ( ) k b j j As a consequence of the results 3. an 3., all the esigns known alreay in the literature as universally optimal over D(p, b, k), the class of connecte block esigns with p lines, b blocks such that each block contains k experimental units an k p, are also universally optimal over D 0 (p, b, n). It can easily be seen that for a binary balance block esign for iallel crosses D (p,b,n ), the information matrix for estimating the gca effects is 0 C ( p p = p ) (n - b)( I p J ), (.) b 3

8 where I p is an ientity matrix of orer p an J p is a p x p matrix of all ones an (n-b)/(p- ) is the unique non-zero eigenvalue of C. It can easily be shown that the unique nonzero eigenvalue of a BBB esign for iallel crosses is greater than or equal to. Clearly C given by (.) is completely symmetric an trace (C ) =(n-b). A generalize - inverse of C in (.) is C = [( p ) /{ ( n b )}]I. It is also easy to see that using, each elementary contrast among gca effects is estimate with a variance ( p ) σ / (n - b). (.3) Instea of the binary balance block esign for iallel crosses D 0 (p, b, n), if one aopts a ranomize complete block esign with r = n /{ p( p ) } blocks, each block having all the p(p-)/ crosses, the C-matrix can easily be shown to be C R = r( p )( I p p J p ), (.4) so that the variance of the B.L.U.E. of any elementary contrast among the gca effects is σ /{ r( p )}, where σ is the per observation variance in the case of ranomize block experiment. Thus the efficiency factor of the esign D0(p, b, n), relative to a ranomize complete block esign uner the assumption of equal intra block variances ( σ = σ ) is given by ( n b ) e =. (.5) r( p )( p ) Several methos of construction of optimal block esigns both for proper an non-proper block esign settings are available in literature. In this talk we shall restrict ourselves to the universally optimal proper block esigns for iallel crosses. Before establishing a connection between neste balance incomplete block (NBIB) esigns of Preece (967) an optimal esigns for iallel crosses, it will not be out of place to state that a proper BBB esign for iallel crosses is equireplicate with respect to lines an also equireplicate with respect to crosses. Consier a neste balance incomplete block esign with parameters v = parametric relationship vr p,b,k,k =,r, λ, ( v - ) λ = ( k ) r,( v ) = ( k ) r. = b k = mb k = b k, λ λ satisfying the If we ientify the treatments of as lines of a iallel experiment an perform crosses among the lines appearing in the same sub-block of, we get a block esign for a iallel experiment involving p lines with v c = p(p-)/ crosses, each replicate r = b /{p(p-)} times, an b = b blocks, each of size k = k /. Such a esign D(p, b, k) is universally optimal over D an also, for such a esign, n ij = 0 or for i =,,, p, j =,,,b. Further, if the NBIB esign with parameters v = p, b, k, b = b k /, k = is such that λ = or equivalently b k = p(p - ), then the optimal esign for iallel crosses erive from this esign has each cross replicate just once an hence uses the minimal number of experimental units. Keeping in view the above, we can say that the existence 3

9 of a NBIB esign with parameters v = p, b = b, b = bk; k = k, k = implies the existence of a universally optimal incomplete block esign for iallel crosses. In the next section, we give the methos of construction of optimal proper block esigns obtainable from NBIB esigns. 3. Methos of Construction of Optimal Block Designs We now give some methos of construction of optimal proper block esigns for iallel cross experiments. Gupta an Kageyama (994) obtaine two families of NBIB esigns, leaing to optimal esigns for iallel crosses. These families have the following parameters: Series.: v = p = t, b = t -, b = t (t - ), k = t, k =, r=t-,λ =t-, λ = Series.: v = p = t + = b, b = t(t+), k = t, k =, r=t,λ =t-, λ =. Here t > is any integer. In these two series of NBIB esigns, if we ientify the treatments of the NBIB esign as lines of the iallel cross experiment an perform the crosses among the lines appearing in the same sub-block of size, we get a universally optimal block esign for iallel crosses with parameters p=t, b = t-, k = t an p=t+, b = t+, k = t, respectively. It is easy to verify that the esigns in Series an Series use the minimal number of experimental units. Henceforth, we enote the parameters of the esign for iallel crosses by p, b, k where p is the number of lines, b, the number of blocks, k, the number of crosses per block or the block size. Family : [Family, Parsa, Gupta an Srivastava (999)]. Let v = p = mt+ be a prime or prime power an x be a primitive element of the Galois fiel of orer p, GF (p), where m = u for u an t. Consier t initial blocks i i+ ut i+ t i+ ( u+ )t i+ ( u )t i+ ( u )t {( x,x ); ( x,x );...;( x,x )} i = 0,,...,t These initial blocks when evelope mo p, give rise to a NBIB esign with parameters v=p = mt +, b = t(mt + ), b = ut(mt + ), k = m = u, k =, r = mt, λ = m -, λ =, n = ut(mt + ). If we ientify the treatments of as lines of a iallel crosses experiment an perform crosses among the lines appearing in the same sub-block of of size in, we get a universally optimal block esign for iallel crosses over D 0 (p, b, n) with minimal number of experimental units an with parameters as p = (mt + ), b = t(mt + ), k = u such that each of the crosses is replicate once in the esign. For m = 4 an m = 6, we get respectively Family an Family esigns of Das, Dey an Dean (998). For t =, we get the same esigns as reporte by Gupta an Kageyama (994). These particular cases are given in the sequel.. 33

10 Family.: [Family, Das, Dey an Dean (998)]. Let v = p = 4t +, t be a prime or a prime power an x be a primitive element of the Galois fiel of orer v, GF(v). Consier the t initial blocks i {( x, x i+ t ),( x i+ t, x i+ 3t )},i = 0,,,...,t -. As shown by Dey, Das an Banerjee (986), these initial blocks, when evelope in the sense of Bose (939), give rise to a NBIB esign with parameters v = p = 4t+, k = 4, b = t (4t+), k =. Using this esign, one can get an optimal esign for iallel crosses with minimal number of experimental units an parameters p = (4t+), b = t (4t+), k =. It is interesting to note that this family of esigns has the smallest block size, k =. Example 3.: Taking t = in Family., a NBIB esign with parameters v = p = 9, b = 8, k = 4, k =, λ = can be constructe by eveloping the following initial blocks over GF(3 ): {(, ),(x+, x+)}; {(x, x),(x +, x+)}, where x is a primitive element of GF(3 ) an the elements of GF(3 ) are 0,,, x, x+, x+, x, x+, x+. Aing successively the non-zero elements of GF(3 ) to the contents of the initial blocks, the full neste esign is obtaine. The esign for iallel crosses is exhibite below, where the lines have been relabelle through 9, using the corresponence 0,, 3, x 4, x+ 5, x+ 6, x 7, x+ 8, x+ 9: [ x 3, 6 x 8]; [ x 3, 4 x 9]; [ x, 5 x 7]; [5 x 6, x 9]; [4 x 6, 3 x 7]; [4 x 5, x 8]; [8 x 9, 3 x 5]; [7 x 9, x 6]; [7 x 8, x 4]; [4 x 7, 5 x 9]; [5 x 8,6 x 7]; [6 x 9, 4 x 8]; [ x 7, 3 x 8]; [ x 8, x 9]; [3 x 9, x 7]; [ x 4, x 6]; [ x 5, 3 x 4]; [3 x 6, x 5]. This is a esign for a iallel cross experiment for p = 9 lines in 8 blocks each of size two; each cross appears in the esign just once. Two esigns for p = 9 have been reporte by Gupta an Kageyama (994); both these esigns have blocks of size larger than two. Further, no neste esign liste by Preece (967) leas to an optimal esign for iallel crosses with p = 9 lines in blocks of size two. Family.: [ Family, Das, Dey an Dean (998)]. Let v = p = 6t +, t be a prime or prime power an x be a primitive element of GF(v). Consier the initial blocks i {( x,x i+ 3t ),( x i+ t,x i+ 4t ),(x i+ t,x i+ 5t )}, i = 0,,,...,t -. Dey, Das an Banerjee (986) show that these initial blocks, when evelope give a solution of a neste balance incomplete block esign with parameters v = p = 6t +, b = t(6t+), k = 6, k =, λ =. Hence, using this series of neste balance incomplete block esigns, we get a solution for an optimal esign for iallel crosses with parameters p = (6t +), b = t(6t +), k = 3. 34

11 Example 3.: Let t = in Family. Then a neste balance incomplete block esign with parameters v = p = 3, b = 6, k = 6, k =, λ = is obtaine by eveloping over GF(3) the following two initial blocks: {(, ), (4, 9), (3, 0)}: {, ), (8, 5), (6, 7)}. Using this neste esign, an optimal esign for iallel crosses with minimal number of experimental units an parameters v c = 78, b = 6, k = 3 can be constructe. Example 3.3: For m = 8 an t =, i.e., v = p = 7, the primitive root of GF(7) is 3. Therefore eveloping the initial blocks [ (, 6) ; (9, 8) ; (3, 4) ; (5, )] [ (3, 4) ; (0, 7) ; (5, ) ; (, 6)] mo 7, we get a universally optimal iallel cross esign over D 0 (p, b, n) with p = 7, b = 34, k = 4, n = 36. Family : [Family, Parsa, Gupta an Srivastava (999)]. Suppose there exists a BIB esign with parameters v = p, b, r, k, λ an there also exists an NBIB esign with parameters k, b, b, k, k =, r, λ, λ. Then writing each of the block contents of BIB esign as NBIB esign, we get a NBIB esign with parameters p, b = bb, b = bb, k = k, k =, r = rr, λ = λλ, λ = λλ an hence a universally optimal esign for iallel crosses over D 0 (p, b, n), an with parameters p, b = bb, k = k/, n = bb k. Now if λ = λ =, then we get a esign in minimal number of observations. This is a fairly general metho of construction an the existence of any NBIB esign an a BIB esign satisfying the conitions mentione above implies the existence of a NBIB esign for iallel crosses. Some particular cases of interest are: Particular Cases Case I: Suppose there exists a BIB esign v = p, b, r, k = t, λ an a NBIB esign with parameters v = t, b = t-, b = t(t - ), r = t-, k = t, k =, λ = t -, λ = always exists. Therefore, we can always get a universally optimal esign for iallel crosses over D with parameters p, b = b(t - ), k = t, n = b k. Example 3.4: Consier a BIB esign with parameters p = 6, b = 0, r = 5, k = 4, λ = an a NIB esign with parameter, v = k = 4, b = 3, b = 6, k = 4, k =, r = 3, λ = 3, λ =. Then we get a universally optimal esign for iallel crosses D 0 (6, 60, 0) with parameters p = 6,b = 60,k =, n = 0. This esign is not obtainable by the methos given by Gupta an Kageyama (994), Dey an Miha (996), Das, Dey an Dean (998) for these values of p = 6 an k =. 35

12 Case II: If there exists a BIB esign with parameters p,b,r,k = t +, λ, where t is a positive integer, an an NBIB esign with parameters = v = t +, b = t +, b = t ( t + ), k = t, k =,r = t, λ = t, λ, we can always get a universally optimal esign over D0 (p, b, n) for iallel crosses with ( ) ) parameters p, b = b t +,k = t,n = bt(t +. Example 3.5: Consier a BIB esign with parameters p = b = 6, r = k = 5, λ = 4 an a = corresponing NBIB esign with parameters v = t + = 5,b = 5,b 0, k = 4, k =, r = 4, λ = 3, λ =. Following the above proceure, we get a universally optimal esign for iallel crosses over D 0 (p = 6, b = 30, n = 60) with parameters as p = 6, b = 30, k =, n=60. Remark 3.: Agarwal an Das (987) gave an application of balance n-ary esigns in the construction of incomplete block esigns for evaluating the gca effects from complete iallel system IV of Griffing (956) using BIB esigns with v = p, b = p(p - )/, r = p -, k =, λ = an triangular esigns with parameters v = p(p - )/, b, r, k,,n, p (i, j, k =, ). Although the authors o not iscuss the optimality aspects of these esigns, inee some of their esigns are universally optimal. In fact the esign obtaine in the Example given by the authors is universally optimal using the conitions of Das, Dey an Dean (998). Some more class of esigns obtaine by Das, Dey an Dean (998) are given below: Family 3: Let t + 7, t 0 be a prime or a prime power an suppose x = 3 is a primitive element of GF(t + 7). Then, as shown by Das, Dey an Dean (998), one can get a NBIB esign with parameters v = p = t + 8, b = (3t+) (t+7), k = 4, k = by eveloping the following 3t+ initial blocks. 3t + 6t+ 3 {(, ),(x,x )}, i i+ 3t+ i+ 3t+ i+ 6t+ 3 ) {( x,x ),( x,x }, i =,, L,3t + ; here is an invariant variety. Using this family of neste esigns, one can get a family of optimal esigns for iallel crosses with minimal number of experimental units an parameters p = t + 7, b = (3t+) (t+7), k =. The next family of neste esigns has λ = an hence in the esign for iallel crosses erive from this family, each cross is replicate twice. However, this family of esigns is of practical utility as the optimal esigns for iallel crosses erive from this family of NBIB esigns have a block size, k =. Family 4: Let v = p = t +, t be a prime or a prime power an x be a primitive element of GF(t + ). Then as shown by Dey, Das an Banerjee (986), a neste λ i i i jk, 36

13 balance incomplete block esign with parameters v = t+, b = t(t+), k = 4, k =, λ = can be constructe by eveloping the following initial blocks over GF(t+): i i i+ {(0, x ),(x, x )},i =,, L, t. Using this family of neste esigns, a family of optimal esigns for iallel crosses with parameters p = t +, b = t (t+), k = can be constructe. In particular, for t = 3, 5 we get optimal esigns for iallel crosses with parameters p = 7,b =,k =,an p =,b = 55,k =. For these values of p, no esigns with block size two are available in Gupta an Kageyama (994). Example 3.6: Let t = 3 in Family 4. Then a NBIB esign with parameters v = p = 7, b =, k = 4, k =, λ = is obtaine by eveloping over GF (7) the following three initial blocks: {(0, ), (3, )}; {(0, 3), (, 6)}; {(0, ), (6, 4)}. Using this neste esign, an optimal esign for iallel crosses with parameters p=7, b=, k =. can be constructe an is shown below: [0x, x3]; [0x3, x6]; [0x, 4x6]; [x, 3x4]; [x4, 0x3]; [x3, 0x5]; [x3, 4x5]; [x5, x4]; [x4,x6]; [3x4, 5x6]; [3x6, x5]; [3x5,0x]; [4x5, 0x6]; [0x4, 3x6]; [4x6, x3]; [5x6, 0x]; [x5, 0x4]; [0x5, x4]; [0x6, x]; [x6, x5]; [x6, 3x5]; Here the lines are number 0 through 6. A catalogue of esigns obtaine through these methos with p 30 an n 000 is reporte in Table. Remark 3.: In Section, a connection between NBIB esigns an optimal esigns for iallel crosses was shown. NBIB esigns can be generalize to a wier class of neste esigns, which may be calle neste balance block esigns in the same manner as balance incomplete block esigns have been generalize to balance block esigns. Neste balance block esigns with sub-block size two can be use to erive optimal block esigns for iallel crosses. One such family of esigns, leaing to optimal esigns with minimal number of experimental units is reporte below. Family 5: Let p = t +, where t is an integer. Then a neste balance block esign with parameters v = p = t+, k = (t+), b = t, k =, λ = can be constructe. The blocks are 37

14 {(j, t +- j), (+j, - j), ( + j, - j),, (t + j, t - j)}, j =,,,t, where parentheses inclue sub-blocks, an the symbols are reuce moule p. Making crosses among lines appearing in the same sub-block, one gets a solution of a block esign for iallel crosses with parameters p = t +, b = t, k = t +. If is a esign for iallel crosses erive from this family of neste esigns, then the C -matrix of can be shown to be C = ( t )( I p p J p ) Clearly, C given above is completely symmetric. Also, trace(c ) = t(t-). It can easily be seen that the above esign is variance balance block esign for iallel crosses an in each block each of the lines appear twice, therefore, following results from section, the esign is optimal an has each cross replicate just once. A catalogue of esigns obtainable through this metho with p 30 is given in Table. 4. Optimal Designs Base on Triangular PBIB Designs It has been shown by Dey an Miha (996) that triangular partially balance incomplete block esigns with two associate classes can be use to erive block esigns for iallel crosses. To begin with let us recall the efinition of a triangular esign. Definition 4.. A binary block esign with v = p (p-) / treatments an b blocks, each of size k is calle a triangular esign if (i) each treatment is replicate r times, (ii) the treatments can be inexe by a set of two labels (i, j), i < j, i, j =,,, p; two treatments, say (α, β) an (γ, δ) occur together in λ blocks if either α = γ, β δ, or α γ, β = δ, or α = δ, β γ or, α δ, β = γ; otherwise, they occur together in λ blocks. Observe that all triangular esigns with parameters v = p (p-) /, b, r, k, λ, λ an treatments inexe by (i, j) can be viewe as neste incomplete block esigns with p treatments, b blocks of size k an sub-blocks of size two. Now, following Dey an Miha (996), we erive a block esign D(p, b, k) for iallel crosses from a triangular esign with parameters v = p(p -) /, b, r, k, λ, λ, by replacing a treatment (i, j) in with the cross (i x j), i < j, i, j =,,, p. Then, it can easily be shown that C =θ ( I p p J p ) (4.) where θ = pk { r( k ) ( p ) λ }. Therefore, using the esign, any elementary comparison among general combining ability effects is estimate with a variance σ / θ, an the efficiency factor of the esign relative to a ranomize complete block esign is θ,/{r(p - )}. Further, from (4.), it follows that 38

15 = trace( C ) k p( p ){ r( k ) ( p ) λ }. (4.) In the sequel, we give a general parametric conition on triangular esigns, leaing to optimal block esigns for iallel crosses. This conition inclues the conition of Dey an Miha (996) as a special case an helps in setting the question of optimality of some esigns left open by them. Result 4.: A block esign for iallel crosses erive from a triangular esign with parameters v = p (p-)/, b, r, k, λ, λ is universally optimal over D(p, b, k) if p (p-)(p-)λ = bx {4k - p(x + )} (4.3) where x = [k / p]. Further, when the conition in (4.3) hols, the efficiency factor is given by e = p{k(k - - x) + px(x +)}/{k (p - )}. (4.4) We now give a result for a triangular esign with λ = 0 an satisfying the inequality k p. Result 4.: A triangular esign with parameters v = p (p - )/, b, r, k, λ, λ satisfying λ = 0 leas to a universally optimal esign for iallel crosses. The optimal block esigns obtainable from triangular PBIB esigns given in Clatworthy (973) are given in Tables 3 an Analysis of Block esigns for Diallel Crosses Uner the moel, the reuce normal equations for estimating linear functions of gca effects, using the esign, are C g =, where Q C = G N K N, an Q = T N K B. Here, T is the vector corresponing to line totals an B is the vector of block totals. Q Q i is known as the vector of ajuste line totals. The i th element of is = T i b n ij j= B j / k.for D ( p, b, ), the ajuste sum of squares for gca effects j 0 n is Q C Q, where C is a g-inverse of C. For a BBB esign for complete iallel crosses, the ajuste sum of squares ue to gca effects is given by θ p Q i i= Q, where ( n b) θ =. For the esigns, where each line appears in each of the blocks a constant p number of times, say a an the esign is variance balance, the treatment sum of squares 39

16 is p p 4G Ti. If we take, a = p, then it is same as that of RCB esign ( n ba) i= p ( B ) - b for CDC. Also, the unajuste block sum of squares is B K B = n b B j G, where G is the gran total. The analysis of variance table for a iallel k n j= j cross esign is as follows: ANOVA Source.f. SS gca effects p Q C Q Block effects b - ( B b ) b B j G B K B = n j= k j n Error n -b-p- By subtraction Total n ( ) B v b Y Y b G = yij n i j= n We now show the essential steps of the analysis of a iallel cross experiment, using a proper incomplete block esign using the illustration given in Dey an Miha (996). For this purpose we take the ata from an experiment on height of sunflowers two weeks after germination, reporte by Ceranka an Mejza (988). These authors use a balance incomplete block esign with v = 5 as they consiere all possible p crosses, incluing selfings an reciprocal crosses, among p = 5 inbre lines. For the purpose of illustration,we take the ata of relevant crosses from this experiment. There are 0 crosses an the esign has 5 blocks, each of size. Each cross is replicate thrice. The layout an observations are given in Table below: Table: Design an Observation Block No. Crosses an Observation Block No. Crosses an Observation (,)6.5 (3,4)9.9 9 (,4)9.0 (3,5)5.5 (,)5.3 (3,5)8.8 0 (,5).0 (,3)8. 3 (,)6.5 (4,5)8. (,5)7.9 (,4)6.9 4 (,3)9. (,4)6.8 (,5)0.6 (3,4)9. 5 (,3)7.0 (,5)7.8 3 (,3)8.5 (4,5)8. 6 (,3)8. (4,5)5. 4 (,4)6.8 (3,5)6.0 7 (,4)9. (,3)8.3 5 (,5)6.4 (3,4)8.0 8 (,4)9.6 (,5)7.0 30

17 The unajuste block sum of squares is The line totals (T i ) an ajuste totals (Q i ) for i =,,,5 are: T = 99.9, T = 85.0, T3 = 96.7, T4 = 97.0, T5 = 9.6; 5 Q = 4.05, Q = 9.00, Q3 =.95, Q4 = 3.5, Q5 = 0.5. The value of θ = an thus the ajuste sum of squares for general combining ability effects is ( Q + Q + Q + Q + Q ) = 5 The total sum of squares is Therefore, the error sum of squares on egrees of freeom is =.80. The estimate variance of the best linear unbiase 4 estimator of an elementary contrast among the general combining ability effects is s, 5.80 where s = =. 07 an s is an unbiase estimator of σ. For testing H 0 : all gca effects are equal against H : at least two of the gca effects are equal we make use of Mean Squares ue to gca effects (4.9) / 4 F = = =.798. The tabulate value of F 4, Mean Square Error (.8) / at 5% level of significance is Therefore, we may conclue that gca effects are not significantly ifferent. Hence, the pairwise comparison of gca effects is not require. 6. Some Open Problems The esigns iscusse above are suitable only for estimating gca effects uner a fixe effects moel. Some efforts are neee to obtain the optimal esigns when sca effects are also inclue in the moel. Further, in such experiments, the experimenter is also intereste in estimating the variance component with resect to lines by consiering the effects of crosses as ranom. Therefore, there is a nee to obtain the optimal esigns uner the mixe effects/fixe effects moels. References Agarwal, S.C. an Das, M.N. (987). A note on construction an application of balance n-ary esigns. Sankhya, B49(), Bose, R.C. (939). On the construction of balance incomplete block esigns. Ann. Eugen., 9, Ceranka, B. an Mejza, S.(988). Analysis of iallel table for experiments carrie out in BIB esigns-mixe moel. Biomet. J., 30, 3-6. Chai, F.S. an Mukerjee, R. (999). Optimal esigns for iallel crosses with specific combining abilities. Biometrika, 86(),

18 Clatworthy, W.H.(973). Tables of two-associate-class partially balance esigns, Applie Maths. Ser.No.63, National Bureau of Stanars, Washington D.C. Das, A., Dey, A. an Dean, A.M.(998). Optimal esigns for iallel cross experiments. Statistics an Probability Letters, 36, Das, M.N. an Giri, N.C.(986). Design an analysis of experiments, n Eition. Wiley Eastern Limite, New Delhi. Dey, A., Das, U.S. an Banerjee, A.K.(986). Construction of neste balance incomplete block esigns. Calcutta Statist. Assoc. Bull., 35, Dey, A. an Miha, C.K. (996). Optimal block esigns for iallel crosses. Biometrika, 83, Ghosh, D.K. an Divecha, J. (997). Two associate class partially balance incomplete block esigns an partial iallel crosses. Biometrika, 84(), Griffing, B.(956). Concepts of general an specific combining ability in relation to iallel crossing systems. Aust. J. Biol. Sci., 9, Gupta, S. an Kageyama, S.(994). Optimal complete iallel crosses. Biometrika, 8, Hayman, B.I.(954a): The analysis of variance of iallel tables. Biometrics, 0, Hayman, B.I.(954b): The theory an analysis of iallel crosses. Genetics, 39, Kiefer, J. (975). Construction an optimality of generalize Youen esigns. In A Survey of Statistical Design an Linear Moels (E. J.N.Srivastava), pp , North Hollan, Amsteram. Mukerjee, R.(997). Optimal partial iallel crosses. Biometrika, 84 (4), Parsa, R., Gupta, V.K. an Srivastava, R.(999). Universally optimal block esigns for iallel crosses. Statistics an Applications, (), Preece, D.A. (967). Neste balance incomplete block esigns. Biometrika, 54, Sharma, M.K.(996). Blocking of complete iallel crossing plans using balance lattice esigns. Sankhya B 58(3), Sharma, M.K.(998). Partial iallel crosses through circular esigns. J. In. Soc. Ag. Statist., Vol LI, No.,

19 Table : Universally Optimal Binary Balance Block Designs for Diallel Cross Experiments Obtainable from NBIB Designs SL. No. p b k n Metho of construction Reference Design, wherever applicable Series : Gupta an Kageyama (994) a,b Family : Parsa, Gupta an Srivastava(999) Family 4 : Das, Dey an Dean (998) Family : Parsa, Gupta an Srivastava(999) Case I: 5,5,4,4, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case II: 6,6,5,5,4 7 a,c Family : Parsa, Gupta an Srivastava(999) Family : Parsa, Gupta an Srivastava(999) Case I:7,7,4,4, Family : Parsa, Gupta an Srivastava(999) Case I:7,7,6,6, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I: 8,4,7,4, Family : Parsa, Gupta an Srivastava(999) Case II:8,8,7,7,6 3 a Family : Parsa, Gupta an Srivastava(999) 4 b Family : Parsa, Gupta an Srivastava(999) 7 Family 4 : Das, Dey an Dean (998) Family : Parsa, Gupta an Srivastava(999) Case I:9,8,8,4, Family : Parsa, Gupta an Srivastava(999) Case I:9,,8,6, Family : Parsa, Gupta an Srivastava(999) Case I:9,9,8,8, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:0,5,6,4, Family : Parsa, Gupta an Srivastava(999) Case I:0,5,9,6, Family : Parsa, Gupta an Srivastava(999) Case II:0,0,9,9,8 3 a 5 55 Family : Parsa, Gupta an Srivastava(999) Family : Parsa, Gupta an Srivastava(999) Case II:,,5,5, Family : Parsa, Gupta an Srivastava(999) Case I:,,6,6, Family : Parsa, Gupta an Srivastava(999) Case I:,,0,0, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:,33,,4, Family : Parsa, Gupta an Srivastava(999) Case I:,,,6, Family : Parsa, Gupta an Srivastava(999) Case II:,,,,0 3 a Family : Parsa, Gupta an Srivastava(999) 3 c Family : Parsa, Gupta an Srivastava(999) 33 b Family : Parsa, Gupta an Srivastava(999) Family 4 : Das, Dey an Dean (998) Family : Parsa, Gupta an Srivastava(999) Case I:3,3,9,9, Family : Parsa, Gupta an Srivastava(999) Case I:3,6,,6, Family : Parsa, Gupta an Srivastava(999) Case II:3,3,,, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case II:4,6,3,7, Series : Gupta an Kageyama (994) Family : Parsa, Gupta an Srivastava (999) Case II:5,,7,5, 33

20 SL. No. p b k n Metho of construction Reference Design, wherever applicable Family : Parsa, Gupta an Srivastava(999) Case II:5,5,7,7,3 40 Family : Parsa, Gupta an Srivastava(999) Case I:5,5,8,8,4 55 Family : Parsa, Gupta an Srivastava(999) Case I:5,35,4,6, Family : Parsa, Gupta an Srivastava(999) Case I:5,,4,0, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:6,0,5,4, Family : Parsa, Gupta an Srivastava(999) Case I:6,6,6,6, Family : Parsa, Gupta an Srivastava(999) Case I:6,4,9,6, Family : Parsa, Gupta an Srivastava(999) Case I:6,6,0,0, Family : Parsa, Gupta an Srivastava(999) Case I:6,40,5,6, Family : Parsa, Gupta an Srivastava(999) Case I:6,30,5,8,7 53 a Family : Parsa, Gupta an Srivastava(999) Family : Parsa, Gupta an Srivastava(999) 55 b Family : Parsa, Gupta an Srivastava(999) Family 4 : Das, Dey an Dean (998) Family : Parsa, Gupta an Srivastava(999) Case I:7,68,6,4, Family : Parsa, Gupta an Srivastava(999) Case I:7,34,6,8, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:8,5,7,6,5 6 a Family : Parsa, Gupta an Srivastava(999) 6 c Family : Parsa, Gupta an Srivastava(999) Family 3 : Das, Dey an Dean (998) Family : Parsa, Gupta an Srivastava(999) Case I:9,57,,4, Family : Parsa, Gupta an Srivastava(999) Case II:9,9,9,9, Family : Parsa, Gupta an Srivastava(999) Case I:9,9,0,0, Family : Parsa, Gupta an Srivastava(999) Case I:9,57,8,6, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:0,95,9,4, Family : Parsa, Gupta an Srivastava(999) Case II:0,76,9,5, Series : Gupta an Kageyama (994) Family : Parsa, Gupta an Srivastava(999) Case II:,,5,5, Family : Parsa, Gupta an Srivastava(999) Case I:,8,8,6, Family : Parsa, Gupta an Srivastava(999) Case I:,4,,6, Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:,77,4,4, Family : Parsa, Gupta an Srivastava(999) Case II:,,7,7, Family : Parsa, Gupta an Srivastava(999) Case I:,33,,8,4 79 a Family : Parsa, Gupta an Srivastava(999) Family 4 : Das, Dey an Dean (998) Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:4,38,3,4,3 83 a Family : Parsa, Gupta an Srivastava(999) 34

21 SL. No. p b k n Metho of construction Reference Design, wherever applicable c Family : Parsa, Gupta an Srivastava(999) 300 Family : Parsa, Gupta an Srivastava(999) 300 Family : Parsa, Gupta an Srivastava(999) 87 b Family : Parsa, Gupta an Srivastava(999) Family : Parsa, Gupta an Srivastava(999) Case II:5,5,9,9, Family 4 : Das, Dey an Dean (998) Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:6,65,5,6,3 9 a Family : Parsa, Gupta an Srivastava(999) Family 4 : Das, Dey an Dean (998) Series : Gupta an Kageyama(994) Family : Parsa, Gupta an Srivastava(999) Case I:8,63,9,4, Family : Parsa, Gupta an Srivastava(999) Case II:8,36,9,7, 97 a Family : Parsa, Gupta an Srivastava(999) Family : Parsa, Gupta an Srivastava(999) Family : Parsa, Gupta an Srivastava(999) 00 b Family : Parsa, Gupta an Srivastava(999) Case I:9,9,8,8, Family 4 : Das, Dey an Dean (998) Series : Gupta an Kageyama(994) a enotes that the esign can also be obtaine from Series : Gupta an Kageyama (994) b enotes that the esign can also be obtaine from Family : Das, Dey an Dean (998) c enotes that the esign can also be obtaine from Family : Das, Dey an Dean (998) Table : Universally Optimal Block Designs for Diallel Crosses Obtainable from Family 5 of Das, Dey an Dean (998). S.No. P b k n

22 Table 3: Universally Optimal Binary Balance Block Designs for Diallel Crosses Generate from Triangular PBIB Designs Given by Dey an Miha (996). SL.No. p b k n Reference Design T T T T T T T T T T T T54 Table 4: Universally Optimal Generalize Binary Balance Block Designs for Diallel Crosses Generate from Triangular PBIB Designs Obtaine by Theorem 4. of Das, Dey an Dean (998) SL.No. p b k n Reference Design T T T T T T T T T9 36