use of nested designs in diallel cross experiments - IASRI

USE OF NESTED DESIGNS IN DIALLEL CROSS EXPERIMENTS Rajender Parsad I.A.S.R.I., Library Avenue, New Delhi - 110 012 1.

Introduction

The term diallel is a Greek word and implies all possible crosses among a collection of male and female animals. Hayman (1954a, 1954b) defined diallel cross as the set of all possible crosses between several genotypes, which may be individuals, clones, homozygous lines, etc. Diallel cross is most balanced and systematic experiment to examine continuous variation. The genetic information related to parental population is available in early generation (F1 itself). Thus, it is useful to breeding strategy without losing much time. Many improvements with respect to generalization of diallel crosses enlarge its scope and utility. Diallel crosses are used mainly 1) 2)

to estimate the genetic components of variation of a quantitative character and to estimate the combining abilities of different inbred lines involved in the crosses.

The concept of combining ability is a measure of gene action and helps in the evaluation of inbreds in terms of their genetic value and in the selection of suitable parents for hybridization. Superior cross combinations can also be identified 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 inbred line is the average performance of the hybrids that this line produces with other lines chosen from a random mating population. It is analogous to main effect of a factorial experiment. It is estimated from half-sib families. Specific combining ability refers to a pair of inbred lines involved in a cross. It indicates cases in which certain combinations do relatively better or worse than would be expected on the basis of gca effects of two lines involved in it. It is the deviation of a particular cross from the expectation on the basis of average gca effects of the two lines involved. It is analogous to an interaction effect of a factorial experiment. Let there be p inbred lines. The diallel cross of these p lines results in p2 progeny families. These include 1) p inbred lines p 2) C2 F1 hybrids and p C2 F1 reciprocal hybrids 3) Depending upon which of the progeny families is included for analysis, four methods of diallel analysis have been proposed. They are

Designs for Diallel Cross Experiments

1) 2) 3) 4)

comprising of all the p2 progeny families. Including p parents and pC2 F1 hybrids, i.e., a total of p(p+1)/2 families. Including pC2 F1 hybrids and pC2 F1 reciprocals, i.e., p(p-1) combinations p C2 F1' s only.

Here, we shall consider only the designs used for the 4th type of analysis. That is, designs for diallel cross experiment involving p (p-1)/2 crosses of type (i x j) for i < j and i, j = 1,2, ..., p. This is called Type IV mating design of Griffing. It is also known as modified diallel system or half-diallel. To obtain unbiased estimates of the population parameters 2 2 of σ gca and σ sca , it is necessary to employ modified diallel system. The inclusion of

inbreds vitiates this property of the estimates obtained from experimental data. This method could be chosen for genetic investigation when maternal inheritance is not suspected, i.e., reciprocal crosses gives the same result. Also this method requires lesser number of experimental units. This has received much attention now days because gca’s are important while selecting inbred lines for hybridization programme. It may be worthwhile to note here that the diallel analysis proposed by Hayman is based on a fixed effect model. In the fixed effect model, the interest is primarily in the comparisons of the combining abilities of parents employed for diallel mating. The analysis proposed by Griffing (1956) takes care of both the fixed effects as well as the random effect situation. Random effects model is chosen when inferences are to be drawn about the base population from which the inbred lines have been sampled. Most common diallel cross experiments have been evaluated using a completely randomized design (CRD) or a randomized complete block design (RCBD) with suitable number of replications. But when the number of lines increases, the number of crosses increases very rapidly. For example, with p = 5 lines there are only 10 crosses. While for p = 10 the number of crosses is 45 and when p = 15 it becomes 105. Laying out the design, as a randomized complete block design, even with a moderately large number of lines, will, however, result into large blocks and consequently large intra-block variances. It results into high coefficient of variation (CV) and hence reduced precision on the comparisons of interest. In order to overcome this problem, one may use incomplete block designs like balanced incomplete block (BIB) designs, partially balanced incomplete block (PBIB) designs with two associate classes, cyclic designs, etc. by treating the crosses as treatments for one way elimination of heterogeneity settings. For instance, a BIB design has been used by identifying crosses as treatments [see e.g., Das and Giri (1986, pp441-442); Ceranka and Mejza, (1988)]. These designs have interesting optimality properties when making inferences on a complete set of orthonormalised treatment contrasts. However, in diallel cross experiments the interest of the experimenter is in making comparisons among general combining ability (gca) effects of lines and not of crosses and, therefore, using these designs as mating designs may result into poor precision of the comparisons among lines. Further, the analysis of a diallel cross experiment in incomplete blocks depends on

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the incidence of lines in blocks, rather than the incidence of the treatments, or crosses, in blocks. To be more clear consider the following example: Example 1.1: An experimenter is interested in generating a mating design for comparing 4-inbred lines on the basis of their gca effects. For a complete diallel cross experiment the number of crosses is 6, denoted by 1 x 2 →A, 1 x 3 →B, 1 x 4 →C, 2 x 3 → D, 2 x 4 → E and 3 x 4 → F. An incomplete block design for diallel cross experiment, D0 considering 6 crosses as treatments denoted as A, B, C, D, E and F, is a BIB design with parameters v = 6 ,b = 15 ,r = 5 ,k = 2 ,λ = 1 . The above design requires 30 experimental units and each cross is replicated 5 times.

1 J 4 ) , and the variance of 4 the Best Linear Unbiased Estimator (B.L.U.E.) of any elementary contrast among lines 2 (gca) is σ 2 . Here C is the coefficient matrix of reduced normal equations for 6 estimating linear functions of gca effects, G is a matrix with diagonal elements as replication number of lines and off-diagonal elements as replication number of crosses, N is the incidence matrix of lines vs blocks, K is diagonal matrix with elements as block sizes, Iv , Jv is an identity matrix of order v and a vxv matrix of all elements ones, The C = G − NK −1N′ matrix of the design D0 is C = 6 ( I 4 −

respectively, and σ 2 is the per plot variance. Another mating design generated through a different method is D0* . The design can be obtained by taking 5-copies of the block design with block contents as Block 1: {(1 x 4), (2 x3)}; Block 2: {(2 x 4), (3 x 1)}; Block 3: {(3 x 4), (1 x 2)}. This design also requires 30 experimental units and each cross is replicated 5 times. The 1 C = G − NK −1N′ matrix of the design D0* is C = 10( I 4 − J 4 ) , and the variance of the 4 2 2 σ . Thus, one can B.L.U.E. of any elementary contrast among line (gca) effects is 10 see that the design D0* estimates the elementary contrasts among gca's with more precision than the design D0 although both the designs are variance balanced for estimating any normalized contrast of gca effects. Another approach advocated in the literature is to start with an incomplete block design, write all the pairs of treatments within a block, identify these pairs of treatments as crosses by treating treatments of the original incomplete block design as lines and use the resulting design as a design for diallel crosses. Sharma (1996) used this approach for complete diallel cross experiments by using balanced lattice designs. This was, however, also advocated by Das and Giri (1986), in the context of BIB designs, and a balanced lattice is also a BIB design. Ghosh and Divecha (1997) used this for PBIB designs to obtain designs for partial diallel crosses and Sharma (1998) obtained designs for partial

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diallel crosses through circular designs. However, this approach also does not seem to do well as will become clear through the following examples:

Example 1.2: An experimenter is interested in generating a mating design for comparing 7-inbred lines on the basis of their gca effects. A mating design for diallel cross experiment, D, with 21 crosses can be obtained by writing all possible pairs of treatments within a block of the BIB design, Da, with parameters v = b = 7, r = k = 3, λ = 1 and treating the treatments as lines and paired treatments as crosses. In design D, the number of crosses is vc = 21 that are arranged in b = 7 blocks of size k = 3 each. Another mating design D* can also be generated through a different method in 7 lines with 21 crosses arranged in 7 blocks of size 3 each. The designs, with rows as blocks, are

1 2 3 4 5 6 7

Da 2 3 4 5 6 7 1

4 5 6 7 1 2 3

1x2 2x3 3x4 4x5 5x6 6x7 1x7

D 1x4 2x5 3x6 4x7 1x5 2x6 3x7

2x4 3x5 4x6 5x7 1x6 2x7 1x3

1x7 1x2 2x3 3x4 4x5 5x6 6x7

D* 2x6 3x7 1x4 2x5 3x6 4x7 1x5

3x5 4x6 5x7 1x6 2x7 1x3 2x4

7 1 ( I7 − J7 ) , and the variance of 3 7 the Best Linear Unbiased Estimator (B.L.U.E.) of any elementary contrast among line 6 (gca) effects is σ 2 . 7 14 1 ( I7 − J7 ) , and the variance of The C = G − NK −1N′ matrix of the design D* is C = 3 7 3 the B.L.U.E. of any elementary contrast among line (gca) effects is σ 2 . Thus, one can 7 see that the design D* estimates the elementary contrasts among gca effects with twice the precision as obtained through the design D although both the designs are variance balanced for estimating any normalized contrast of gca effects. The C = G − NK −1N′ matrix of the design D is C =

Example 1.3: Consider another situation when the experimenter is interested in designing an experiment with p = 9 lines. Sharma (1998) generated a mating design D1 from a cyclic design with parameters v = b = 9, r = k = 3, by developing the initial block ⎛k ⎞ (1, 2, 3) mod 9 and then taking all the possible ⎜⎜ ⎟⎟ crosses from each block. The ⎝2⎠ variances of the B.L.U.E. of any elementary contrast among line (gca) effects is

0.81045σ 2 , 1.04574σ 2 , 1.29411σ 2 and 1.39866σ 2 . Each of these variances is for 9 different B.L.U.E. of the elementary contrasts among gca effects and the average variance is given by 1.13724σ 2 . A similar type of mating design D1* can be obtained from a cyclic design with parameters v = b = 9, r = k = 3 by developing the initial block

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(1, 2, 4) mod 9. The variances of the B.L.U.E. of any elementary contrast among line (gca) effects is 0.87543σ 2 , 0.89409σ 2 , 0.89718σ 2 and 1.02492σ 2 , and the average variance is given by 0.92291σ 2 .

The design D1* has smaller average variance of

B.L.U.E. of elementary contrasts of gca effects as compared to D1. The design D1* seems to have an intuitive appeal also as it contains more number of distinct crosses as compared to the design D1 although the size of both the designs is the same in terms of the total number of observations. Hence, this design is more useful for same number of experimental units.

Example 1.4: Consider another situation when the experimenter is interested in designing an experiment with p = 12 lines. Ghosh and Divecha (1997) generated a mating design D2 from a group divisible design with parameters v = 12, b = 9, r = 3, k = ⎛k ⎞ 4, λ1 = 0 , λ2 = 1 , m = 4, n = 3(Clatworthy, 1973; SR41), by taking all possible ⎜⎜ ⎟⎟ ⎝2⎠ crosses of treatments within each block, by treating treatments in the original design as lines. The variances of the B.L.U.E. of elementary contrasts among gca effects are 0.44444σ 2 , for the first associates (12 in number) and 0.40741σ 2 , for the second

associates (54 in number). The average variance is 0.41414σ 2 . A similar type of mating design D*2 is obtained from a different method. The variances of the B.L.U.E. of any elementary contrast among gca effects is 0.22222σ 2 , for the first associates (12 in number) and 0.25926σ 2 for the second associates (54 in number). The average variance is 0.25253σ 2 . One can easily see that in the two mating designs the precision of the B.L.U.E. of gca effects is different and, therefore, the choice of an appropriate mating design is important. The design D*2 with rows as blocks is

D*2 1x2 1x3 1x4 1x6 1x7

5x6 5x7 5x8 5x10 5x11

Blocks 9x10 3x4 9x11 2x4 9x12 2x3 9x2 3x8 9x3 2x8

7x8 6x8 6x7 7x12 6x12

11x12 10x12 10x11 11x4 10x4

1x8 1x10 1x11 1x12

5x12 5x2 5x3 5x4

Blocks 9x4 2x7 9x6 3x12 9x7 2x12 9x8 2x11

6x11 7x4 6x4 6x3

It is clear from the above discussions that for making comparisons of gca effects of pinbred lines, the choice of an appropriate design is important. This talk addresses this and similar problems. The problem of generating optimal mating designs for experiments with diallel crosses has been recently investigated by several authors [see e.g., Gupta and Kageyama (1994), Dey and Midha(1996), Mukerjee(1997), Das, Dey and Dean(1998), Parsad, Gupta and Srivastava (1999), Chai and Mukerjee (1999)]. These authors used nested balanced incomplete block (NBIB) designs of Preece (1967) for this purpose. This paper derives 309

10x3 11x8 10x8 10x7


general methods of construction of mating designs, essentially generated from nested variance balanced block (NBB) designs. The optimality aspects have also been investigated under a non-proper setting. The model considered here involves only the gca effects, the specific combining ability effects being excluded from the model. The designs obtained are variance balanced in the sense that the variances of the B.L.U.E. of elementary contrasts among gca effects are all same.

2.

Nested Designs and Optimality results

Let d be a block design for a diallel cross experiment of the type mentioned in Section 1 involving p-inbred lines, b blocks such that the jth block is of size k j . This means that there are kj crosses or 2kj lines, respectively in each block of d. It may be mentioned here that the designs for diallel crosses have two types of block sizes, k1@ , the block sizes with respect to crosses and k2@ , the block sizes with respect to the lines and k2@ = 2 k1@ . It, therefore, follows that the block designs for diallel crosses may also be viewed as nested block designs with sub blocks of size 2 each and the pair of treatments in each sub block form the crosses, the treatments being the lines. Further, let rdl denote the number of times the lth cross appears in d, l = 1, 2,.., p(p-1)/2 and similarly sdi denotes the number of times the ith line occurs in the crosses in the whole design d, i = 1,2,…,p. Then it is easy to see that p( p −1 ) / 2

∑ rdl =

l =1

b

∑k j

= n , the total number of observations, and

j =1

p

b

i =1

j =1

∑ sdi = 2 ∑ k j , (because in every cross there are two lines).

For the data obtained from the design d, we postulate the model

Y = µ 1n + ∆1′ g + ∆2′ β + e (2.1) where Y is the nx1 vector of observed responses, µ is a general mean effect, 1n denotes an n - component column vector of all ones, g and β are vectors of p gca effects and b block effects, respectively. ∆1′ and ∆2′ are the corresponding nxp and nxb design th th matrices respectively, i.e., the (s, t) element of ∆1′ is 1 if the s observation pertains to th the t line and is zero otherwise.

Similarly (s, t)th element of ∆′2 is 1 if the sth

observation comes from the tth block and is zero otherwise. e is the random error which

(

)

follows a Nn 0, σ 2I n . In the model (2.1) we have not included the specific combining ability effects. Under this model, it can be shown that the coefficient matrix for reduced normal equations for estimating linear functions of gca effects using a design d is

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Cd = G d − N d K −d 1N′d

((

))

where G d = (( g dii′ )), N d = ndij , g dii = sdi and for i ≠ i′, g dii′ is the number of times the cross (ixi′) appears in d; ndij is the number of times line i occurs in the block j . A design d is said to be connected if and only if Rank (Cd ) = p − 1, or equivalently, if and only if all elementary contrasts among the gca effects are estimable using d. A connected design d is variance balanced if and only if all the diagonal elements of the matrix Cd are equal and all the off diagonal elements are also equal. In other words, the matrix Cd is completely symmetric. In particular, a variance balanced block design for diallel crosses is said to be a generalized binary variance balanced block (GBBB) design if in addition to completely symmetric information matrix, ndij = x j or x j + 1 and is said to be binary variance balanced block (BBB) design, if ndij = 0 or 1. For given positive integers p, b, n, D0 ( p ,b ,n ) will denote the class of all connected block designs d with p lines, b blocks and n experimental units. Here the block sizes are arbitrary but for a given design d ∈ D0 ( p ,b ,n ) , the block sizes are kd 1 ,kd 2 ,L ,kdb . Similarly, D* ( p ,b ,k1 ,...,kb ) will denote the class of all connected block designs d with p lines, b blocks such that jth block is of size k j . We may allow 2 k j > p for some or all

j = 1,2 ,L , b . Now using the Proposition 1 of Kiefer (1975) and the definitions of GBBB (BBB) designs for diallel crosses, we have the following results. Result 3.1: A GBBB design for diallel crosses, whenever existent, is universally optimal over D* ( p ,b ,k1 ,...,kb )

Result 3.2: A BBB design for diallel crosses, whenever existent, is universally optimal over D0(p, b, n). It may be noted that a design d* that is universally optimal over D* ( p ,b ,k1 ,...,kb ) is also universally optimal over D0(p, b, n) provided all 2 k j ≤ p for all j = 1,2 ,L , b . Similarly, a design d* that is universally optimal over D0(p, b, n) is also universally optimal over D( p ,b ,k1 ,...,kb ) provided kd* j = k j for all j = 1,2 ,L ,b . As a consequence of the results 3.1 and 3.2, all the designs known already in the literature as universally optimal over D(p, b, k), the class of connected block designs with p lines, b blocks such that each block contains k experimental units and 2 k ≤ p , are also universally optimal over D0(p, b, n). It can easily be seen that for a binary balanced block design for diallel crosses d* ∈ D0 (p,b,n ) , the information matrix for estimating the gca effects is

C d* = ( p − 1 ) −1 2(n - b)( I p − p −1 J p ),

311

(2.2)


where Ip is an identity matrix of order p and Jp is a p x p matrix of all ones and 2(n-b)/(p1) is the unique non-zero eigenvalue of C d* . It can easily be shown that the unique nonzero eigenvalue of a BBB design for diallel crosses is greater than or equal to 2. Clearly Cd* given by (2.2) is completely symmetric and trace (Cd*) =2(n-b). A generalized inverse of Cd* in (2.2) is C-d* = [( p − 1 ) /{ 2( n − b )}] I . It is also easy to see that using d*, each elementary contrast among gca effects is estimated with a variance

( p − 1 )σ 2 / (n - b). (2.3) Instead of the binary balanced block design for diallel crosses d* ∈ D0 (p, b, n), if one adopts a randomized complete block design with r = 2 n / {p( p − 1)} blocks, each block having all the p(p-1)/2 crosses, the C-matrix can easily be shown to be C R = r( p − 2 )( I p − p −1J p ), (2.4) so that the variance of the B.L.U.E. of any elementary contrast among the gca effects is

2σ 12 /{ r( p − 2 )}, where σ 12 is the per observation variance in the case of randomized block experiment. Thus the efficiency factor of the design d* ∈ D0(p, b, n) , relative to a randomized complete block design under the assumption of equal intra block variances ( σ 12 = σ 2 ) is given by

2( n − b ) . (2.5) r( p − 2 )( p − 1 ) Several methods of construction of optimal block designs both for proper and non-proper block design settings are available in literature. In this talk we shall restrict ourselves to the universally optimal proper block designs for diallel crosses. e=

Before establishing a connection between nested balanced incomplete block (NBIB) designs of Preece (1967) and optimal designs for diallel crosses, it will not be out of place to state that a proper BBB design for diallel crosses is equireplicated with respect to lines and also equireplicated with respect to crosses. Consider a nested balanced incomplete block design d with parameters v = p ,b1 , k1 , k2 = 2 , r* ,λ1 ,λ2 satisfying the parametric relationship

vr* = b1k1 = mb1k2 = b2 k2 , (v - 1)λ1 = (k1 − 1)r* ,(v − 1)λ2 = (k2 − 1)r* . If we identify the treatments of d as lines of a diallel experiment and perform crosses among the lines appearing in the same sub-block of d, we get a block design d* for a diallel experiment involving p lines with vc= p(p-1)/2 crosses, each replicated r = 2b2/{p(p-1)} times, and b = b1 blocks, each of size k = k1 / 2. Such a design d* ∈ D(p, b, k) is universally optimal over D and also, for such a design, nd* ij = 0 or 1 for i = 1,2, …,

p, j = 1,2,…,b. Further, if the NBIB design with parameters v = p, b1, k1, b2 = b1k1 / 2, k2 = 2 is such that λ2 = 1 or equivalently b1k1 = p(p - 1), then the optimal design d* for diallel crosses derived from this design has each cross replicated just once and hence uses the minimal number of experimental units. Keeping in view the above, we can say that the existence

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of a NBIB design d with parameters v = p, b1 = b, b2 = bk; k1 = 2k, k2 = 2 implies the existence of a universally optimal incomplete block design d* for diallel crosses. In the next section, we give the methods of construction of optimal proper block designs obtainable from NBIB designs.

3.

Methods of Construction of Optimal Block Designs

We now give some methods of construction of optimal proper block designs for diallel cross experiments. Gupta and Kageyama (1994) obtained two families of NBIB designs, leading to optimal designs for diallel crosses. These families have the following parameters:

Series 1.: v = p = 2t, b1 = 2t - 1, b2 = t (2t - 1), k1 = 2t, k2 = 2, r=2t-1,λ1=2t-1, λ2=1 Series 2.: v = p = 2t + 1 = b1, b2 = t(2t+1), k1 = 2t, k2 = 2, r=2t,λ1=2t-1, λ2=1. Here t >1 is any integer. In these two series of NBIB designs, if we identify the treatments of the NBIB design as lines of the diallel cross experiment and perform the crosses among the lines appearing in the same sub-block of size 2, we get a universally optimal block design for diallel crosses with parameters p=2t, b = 2t-1, k = t and p=2t+1, b = 2t+1, k = t, respectively. It is easy to verify that the designs in Series 1 and Series 2 use the minimal number of experimental units. Henceforth, we denote the parameters of the design for diallel 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 1: [Family 1, Parsad, Gupta and Srivastava (1999)]. Let v = p = mt+1 be a prime or prime power and x be a primitive element of the Galois field of order p, GF (p), where m = 2u for u ≥ 2 and t ≥ 1. Consider t initial blocks

{(xi , xi +ut ); (xi +t , xi +( u +1 )t );...; (xi +( u −1 )t , xi +( 2u −1 )t )} ∀

i = 0 ,1,...,t − 1 .

These initial blocks when developed mod p, give rise to a NBIB design with parameters v=p = mt + 1, b1 = t(mt + 1), b2 = ut(mt + 1), k1 = m = 2u, k2 = 2, r = mt, λ1 = m - 1, λ2 = 1, n = 2ut(mt + 1). If we identify the treatments of d as lines of a diallel crosses experiment and perform crosses among the lines appearing in the same sub-block of of size 2 in d, we get a universally optimal block design for diallel crosses over D0(p, b, n) with minimal number of experimental units and with parameters as p = (mt + 1), b = t(mt + 1), k = u such that each of the crosses is replicated once in the design. For m = 4 and m = 6, we get respectively Family 1 and Family 2 designs of Das, Dey and Dean (1998). For t = 1, we get the same designs as reported by Gupta and Kageyama (1994). These particular cases are given in the sequel.

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Family 1.1: [Family 1, Das, Dey and Dean (1998)]. Let v = p = 4t + 1, t ≥ 1 be a prime or a prime power and x be a primitive element of the Galois field of order v, GF(v). Consider the t initial blocks {( xi , xi + 2t ),( xi +t , xi + 3t )}, i = 0,1,2,..., t - 1. As shown by Dey, Das and Banerjee (1986), these initial blocks, when developed in the sense of Bose (1939), give rise to a NBIB design with parameters v = p = 4t+1, k1 = 4, b1 = t (4t+1), k2 = 2. Using this design, one can get an optimal design for diallel crosses with minimal number of experimental units and parameters p = (4t+1), b = t (4t+1), k = 2. It is interesting to note that this family of designs has the smallest block size, k = 2.

Example 3.1: Taking t =2 in Family 1.1, a NBIB design with parameters v = p = 9, b1 = 18, k1 = 4, k2 = 2, λ2 = 1 can be constructed by developing the following initial blocks over GF(32): {(1, 2),(2x+1, x+2)}; {(x, 2x),(2x + 2, x+1)}, where x is a primitive element of GF(32) and the elements of GF(32) are 0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2. Adding successively the non-zero elements of GF(32) to the contents of the initial blocks, the full nested design is obtained. The design for diallel crosses is exhibited below, where the lines have been relabelled 1 through 9, using the correspondence 0→1, 1→ 2, 2 → 3, x→4, x+1 → 5, x+2 → 6, 2x → 7, 2x+1 →8, 2x+2 → 9:

[2 x 3, 6 x 8]; [1 x 3, 4 x 9]; [1 x 2, 5 x 7]; [5 x 6, 2 x 9]; [4 x 6, 3 x 7]; [4 x 5, 1x 8]; [8 x 9, 3 x 5]; [7 x 9, 1 x 6]; [7 x 8, 2 x 4]; [4 x 7, 5 x 9]; [5 x 8,6 x 7]; [6 x 9, 4 x 8]; [1 x 7, 3 x 8]; [2 x 8, 1 x 9]; [3 x 9, 2 x 7]; [1 x 4, 2 x 6]; [2 x 5, 3 x 4]; [3 x 6,1 x 5]. This is a design for a diallel cross experiment for p = 9 lines in 18 blocks each of size two; each cross appears in the design just once. Two designs for p = 9 have been reported by Gupta and Kageyama (1994); both these designs have blocks of size larger than two. Further, no nested design listed by Preece (1967) leads to an optimal design for diallel crosses with p = 9 lines in blocks of size two.

Family 1.2: [ Family 2, Das, Dey and Dean (1998)]. Let v = p = 6t +1, t ≥ 1 be a prime or prime power and x be a primitive element of GF(v). Consider the initial blocks {( xi , xi + 3t ),( xi +t , xi + 4t ), (xi + 2t , xi + 5t )}, i = 0,1,2,...,t - 1. Dey, Das and Banerjee (1986) show that these initial blocks, when developed give a solution of a nested balanced incomplete block design with parameters v = p = 6t +1, b1 = t(6t+1), k1 = 6, k2 = 2, λ2 = 1. Hence, using this series of nested balanced incomplete block designs, we get a solution for an optimal design for diallel crosses with parameters p = (6t +1), b = t(6t +1), k = 3.

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Example 3.2: Let t = 2 in Family 2. Then a nested balanced incomplete block design with parameters v = p = 13, b1 = 26, k1 = 6, k2 = 2, λ2 = 1 is obtained by developing over GF(13) the following two initial blocks: {(1, 12), (4, 9), (3, 10)}: {2, 11), (8, 5), (6, 7)}. Using this nested design, an optimal design for diallel crosses with minimal number of experimental units and parameters vc = 78, b = 26, k = 3 can be constructed.

Example 3.3: For m = 8 and t = 2, i.e., v = p = 17, the primitive root of GF(17) is 3. Therefore developing the initial blocks [ (1, 16) ; (9, 8) ; (13, 4) ; (15, 2)] [ (3, 14) ; (10, 7) ; (5, 12) ; (11, 6)] mod 17, we get a universally optimal diallel cross design over D0(p, b, n) with p = 17, b = 34, k = 4, n = 136.

Family 2: [Family 2, Parsad, Gupta and Srivastava (1999)]. Suppose there exists a BIB design with parameters v = p, b, r, k, λ and there also exists an NBIB design with parameters k, b1, b2, k1, k2 = 2, r*, λ1, λ2. Then writing each of the block contents of BIB design as NBIB design, we get a NBIB design with parameters p, b1* = bb1, b2* = bb2, k1* = k1, k2* = 2, r** = rr*, λ1* = λλ1, λ2* = λλ2 and hence a universally optimal design for diallel crosses over D0(p, b*, n), and with parameters p, b* = bb1, k* = k1/2, n = bb1k * . Now if λ2 = λ = 1, then we get a design in minimal number of observations. This is a fairly general method of construction and the existence of any NBIB design and a BIB design satisfying the conditions mentioned above implies the existence of a NBIB design for diallel crosses. Some particular cases of interest are:

Particular Cases Case I: Suppose there exists a BIB design v = p, b, r, k = 2t, λ and a NBIB design with parameters v1 = 2t, b1 = 2t-1, b2 = t(2t - 1), r = 2t-1, k1 = 2t, k2 = 2, λ1 = 2t - 1, λ2 = 1 always exists. Therefore, we can always get a universally optimal design for diallel crosses over D with parameters p , b* = b(2t - 1), k* = t, n = b * k * . Example 3.4: Consider a BIB design with parameters p = 16, b = 20, r = 5, k = 4, λ = 1 and a NIB design with parameter, v* = k = 4, b1* = 3, b2* = 6, k1* = 4, k2* = 2, r = 3, λ1 = 3, λ2 = 1. Then we get a universally optimal design for diallel crosses D0 (16, 60, 120) with parameters p = 16 ,b = 60 ,k = 2 , n = 120. This design is not obtainable by the methods given by Gupta and Kageyama (1994), Dey and Midha (1996), Das, Dey and Dean (1998) for these values of p = 16 and k = 2.

315


Case II: If there exists a BIB design with parameters p ,b ,r ,k = 2t + 1,λ , where t is a positive integer, and an NBIB design with parameters v* = 2t + 1, b1* = 2t + 1, b*2 = t (2t + 1), k1* = 2t , k*2 = 2 , r* = 2t, λ1 = 2t − 1, λ2 = 1, we can always get a universally optimal design over D0 (p, b*, n) for diallel crosses with

parameters p, b* = b(2t + 1), k* = t , n = bt(2t + 1) .

Example 3.5: Consider a BIB design with parameters p = b = 6, r = k = 5, λ = 4 and a corresponding NBIB design with parameters v* = 2t + 1 = 5, b1* = 5 ,b*2 = 10 , k1* = 4 ,

k*2 = 2 , r = 4 , λ1 = 3 ,λ2 = 1 . Following the above procedure, we get a universally optimal design for diallel crosses over D0 (p = 6, b = 30, n = 60) with parameters as p = 6, b = 30, k = 2, n=60.

Remark 3.1: Agarwal and Das (1987) gave an application of balanced n-ary designs in the construction of incomplete block designs for evaluating the gca effects from complete diallel system IV of Griffing (1956) using BIB designs with v = p, b = p(p - 1)/2, r = p 1, k = 2, λ = 1 and triangular designs with parameters v = p(p - 1)/2, b, r, k, λi , ni , p ijk , (i, j, k =1, 2) . Although the authors do not discuss the optimality aspects of these designs, indeed some of their designs are universally optimal. In fact the design obtained in the Example given by the authors is universally optimal using the conditions of Das, Dey and Dean (1998). Some more class of designs obtained by Das, Dey and Dean (1998) are given below:

Family 3: Let 12t + 7, t ≥ 0 be a prime or a prime power and suppose x = 3 is a primitive element of GF(12t + 7). Then, as shown by Das, Dey and Dean (1998), one can get a NBIB design with parameters v = p = 12t + 8, b1 = (3t+2) (12t+7), k1 = 4, k2 = 2 by developing the following 3t+2 initial blocks.

{( 1, ∞ ),(x3t +2 , x6 t +3 )}, {( xi , xi+3t +1 ),( xi+3t +2 , xi+6 t +3 )}, i = 1,2,L,3t + 1; here ∞ is an invariant variety. Using this family of nested designs, one can get a family of optimal designs for diallel crosses with minimal number of experimental units and parameters p = 12 t + 7, b = (3t+2) (12t+7), k = 2. The next family of nested designs has λ2 = 2 and hence in the design for diallel crosses derived from this family, each cross is replicated twice. However, this family of designs is of practical utility as the optimal designs for diallel crosses derived from this family of NBIB designs have a block size, k = 2.

Family 4: Let v = p = 2t + 1, t ≥ 1 be a prime or a prime power and x be a primitive element of GF(2t + 1). Then as shown by Dey, Das and Banerjee (1986), a nested

316


balanced incomplete block design with parameters v = 2t+1, b1 = t(2t+1), k1 = 4, k2 = 2, λ2 = 2 can be constructed by developing the following initial blocks over GF(2t+1):

{( 0 , xi −1 ),(xi , xi +1 )}, i = 1,2,L ,t. Using this family of nested designs, a family of optimal designs for diallel crosses with parameters p = 2 t + 1, b = t (2t+1), k = 2 can be constructed. In particular, for t = 3, 5 we get optimal designs for diallel crosses with parameters

p = 7 ,b = 21,k = 2 , and p = 11, b = 55 ,k = 2. For these values of p, no designs with block size two are available in Gupta and Kageyama (1994).

Example 3.6: Let t = 3 in Family 4. Then a NBIB design with parameters v = p = 7, b1 = 21, k1 = 4, k2 = 2, λ2 = 2 is obtained by developing over GF (7) the following three initial blocks: {(0, 1), (3, 2)}; {(0, 3), (2, 6)}; {(0, 2), (6, 4)}. Using this nested design, an optimal design for diallel crosses with parameters p=7, b= 21, k =2. can be constructed and is shown below:

[0x1, 2x3]; [1x2, 3x4]; [2x3, 4x5]; [3x4, 5x6]; [4x5, 0x6]; [5x6, 0x1]; [0x6, 1x2];

[0x3, 2x6]; [1x4, 0x3]; [2x5, 1x4]; [3x6, 2x5]; [0x4, 3x6]; [1x5, 0x4]; [2x6, 1x5];

[0x2, 4x6]; [1x3, 0x5]; [2x4,1x6]; [3x5,0x2]; [4x6, 1x3]; [0x5, 2x4]; [1x6, 3x5];

Here the lines are number 0 through 6. A catalogue of designs obtained through these methods with p ≤ 30 and n ≤ 1000 is reported in Table 1.

Remark 3.2: In Section 2, a connection between NBIB designs and optimal designs for diallel crosses was shown. NBIB designs can be generalized to a wider class of nested designs, which may be called nested balanced block designs in the same manner as balanced incomplete block designs have been generalized to balanced block designs. Nested balanced block designs with sub-block size two can be used to derive optimal block designs for diallel crosses. One such family of designs, leading to optimal designs with minimal number of experimental units is reported below. Family 5: Let p = 2t +1, where t ≥ 1 is an integer. Then a nested balanced block design with parameters v = p = 2t+1, k1 = 2(2t+1), b1 = t, k2 = 2, λ2 = 1 can be constructed. The blocks are 317


{(j, 2t +1- j), (1+j, 1 - j), (2 + j, 2 - j),…, (2t + j, 2t - j)}, j = 1,2,…,t, where parentheses include sub-blocks, and the symbols are reduced module p. Making crosses among lines appearing in the same sub-block, one gets a solution of a block design for diallel crosses with parameters p = 12 t + 1, b = t, k = 2t +1. If d is a design for diallel crosses derived from this family of nested designs, then the C -matrix of d can be shown to be

Cd = ( 2t − 1 )( I p − p −1J p ) Clearly, Cd given above is completely symmetric. Also, trace(Cd) = 2t(2t-1). It can easily be seen that the above design is variance balanced block design for diallel crosses and in each block each of the lines appear twice, therefore, following results from section 2, the design d is optimal and has each cross replicated just once. A catalogue of designs obtainable through this method with p ≤ 30 is given in Table 2.

4.

Optimal Designs Based on Triangular PBIB Designs

It has been shown by Dey and Midha (1996) that triangular partially balanced incomplete block designs with two associate classes can be used to derive block designs for diallel crosses. To begin with let us recall the definition of a triangular design.

Definition 4.1. A binary block design with v = p (p-1) / 2 treatments and b blocks, each of size k is called a triangular design if (i) each treatment is replicated r times, (ii) the treatments can be indexed by a set of two labels (i, j), i < j, i, j = 1,2,…, p; two treatments, say (α, β) and (γ, δ) occur together in λ1 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 λ2 blocks. Observe that all triangular designs with parameters v = p (p-1) / 2, b, r, k, λ1, λ2 and treatments indexed by (i, j) can be viewed as nested incomplete block designs with p treatments, b blocks of size 2k and sub-blocks of size two. Now, following Dey and Midha (1996), we derive a block design d ∈ D(p, b, k) for diallel crosses from a triangular design d1 with parameters v = p(p -1) / 2, b, r, k, λ1, λ2 , by replacing a treatment (i, j) in d1 with the cross (i x j), i < j, i, j = 1,2,…, p. Then, it can easily be shown that

Cd = θ ( I p − p −1J p )

(4.1)

where θ = pk −1{ r( k − 1 ) − ( p − 2 )λ1 }. Therefore, using the design d, any elementary comparison among general combining ability effects is estimated with a variance 2σ2 / θ, and the efficiency factor of the design relative to a randomized complete block design is θ,/{r(p - 2)}. Further, from (4.1), it follows that

318


trace( Cd ) = k −1 p( p − 1 ){ r( k − 1 ) − ( p − 2 )λ1 }.

(4.2)

In the sequel, we give a general parametric condition on triangular designs, leading to optimal block designs for diallel crosses. This condition includes the condition of Dey and Midha (1996) as a special case and helps in setting the question of optimality of some designs left open by them.

Result 4.1: A block design for diallel crosses derived from a triangular design with parameters v = p (p-1)/2, b, r, k, λ1, λ2 is universally optimal over D(p, b, k) if p (p-1)(p-2)λ1 = bx {4k - p(x + 1)}

(4.3)

where x = [2k / p]. Further, when the condition in (4.3) holds, the efficiency factor is given by e = p{2k(k - 1 - 2x) + px(x +1)}/{2k2(p - 2)}. (4.4) We now give a result for a triangular design with λ1 = 0 and satisfying the inequality 2k ≤ p.

Result 4.2: A triangular design with parameters v = p (p - 1)/2, b, r, k, λ1, λ2 satisfying λ1 = 0 leads to a universally optimal design for diallel crosses. The optimal block designs obtainable from triangular PBIB designs given in Clatworthy (1973) are given in Tables 3 and 4.

5. Analysis of Block designs for Diallel Crosses Under the model, the reduced normal equations for estimating linear functions of gca effects, using the design d, are Cd g = Q d , where C d = G d − N d K −d 1N ′d , and

Q d = Td − N d K d−1B d .

Here, Td is the vector corresponding to line totals and B d is the vector of block totals. Q d is known as the vector of adjusted line totals. The ith element of Q d is b

Qi = Ti − ∑ nij B j / k j .For d ∈ D 0 ( p, b, n ), the adjusted sum of squares for gca effects j =1

is Q ′d C d− Q d , where C −d is a g-inverse of C d . For a BBB design for complete diallel crosses, the adjusted sum of squares due to gca effects is given by

1 p

θ

∑ Qi2 , where i =1

2(n − b) θ= . For the designs, where each line appears in each of the blocks a constant p −1 number of times, say a and the design is variance balanced, the treatment sum of squares 319


p − 1 ⎛⎜ p 2 4G 2 is ∑ Ti − p 2(n − ba ) ⎜⎝ i =1

⎞ ⎟ . If we take, a = p − 1 , then it is same as that of RCB design ⎟ ⎠

for CDC. Also, the unadjusted block sum of squares is B ′d K -d1 B d − b

B 2j

∑k j =1

j

−

(B ′d 1b )2 n

=

G2 , where G is the grand total. The analysis of variance table for a diallel n

cross design is as follows: Source gca effects

ANOVA d.f. p −1

Block effects

b −1

SS Q′d C−d Q d

(B′ 1 )2 B′d K -d1B d − d b n

Error

n -b-p-1

Total

n −1

=

b

B 2j

j =1

kj

∑

−

G2 n

By subtraction Y′Y −

(B′d 1b )2 = n

v

b

∑ ∑ yij2 −

i −1 j =1

G2 n

We now show the essential steps of the analysis of a diallel cross experiment, using a proper incomplete block design using the illustration given in Dey and Midha (1996). For this purpose we take the data from an experiment on height of sunflowers two weeks after germination, reported by Ceranka and Mejza (1988). These authors used a balanced incomplete block design with v = 25 as they considered all possible p 2 crosses, including selfings and reciprocal crosses, among p = 5 inbred lines. For the purpose of illustration,we take the data of relevant crosses from this experiment. There are 10 crosses and the design has 15 blocks, each of size 2. Each cross is replicated thrice. The layout and observations are given in Table below:

Block No. 1 2 3 4 5 6 7 8

Table: Design and Observation Crosses and Observation Block Crosses and Observation No. (1,2)6.5 (3,4)9.9 9 (1,4)9.0 (3,5)5.5 (1,2)5.3 (3,5)8.8 10 (1,5)11.0 (2,3)8.2 (1,2)6.5 (4,5)8.2 11 (1,5)7.9 (2,4)6.9 (1,3)9.2 (2,4)6.8 12 (1,5)10.6 (3,4)9.2 (1,3)7.0 (2,5)7.8 13 (2,3)8.5 (4,5)8.2 (1,3)8.1 (4,5)5.2 14 (2,4)6.8 (3,5)6.0 (1,4)9.2 (2,3)8.3 15 (2,5)6.4 (3,4)8.0 (1,4)9.6 (2,5)7.0

320


The unadjusted block sum of squares is 28.58. The line totals (Ti) and adjusted totals (Qi) for i = 1,2,…,5 are: T1 = 99.9 , T2 = 85.0 , T3 = 96.7 , T4 = 97.0 , T5 = 92.6 ; Q1 = 4.05 , Q2 = −9.00 , Q3 = 1.95 , Q4 = 3.25 , Q5 = −0.25. The value of θ =

15 and thus 2

the adjusted sum of squares for general combining ability effects is

(

)

2 2 Q1 + Q22 + Q32 + Q42 + Q52 = 14.91. 15 The total sum of squares is 66.29. Therefore, the error sum of squares on 11 degrees of freedom is 66.29-28.58-14.91 = 22.80. The estimated variance of the best linear unbiased 4 2 s , estimator of an elementary contrast among the general combining ability effects is 15 22.80 = 2.07 and s 2 is an unbiased estimator of σ 2 . For testing H 0 : all where s 2 = 11 gca effects are equal against H 1 : at least two of the gca effects are equal we make use of Mean Squares due to gca effects (14.91) / 4 F= = = 1.798. The tabulated value of F4,11 Mean Square Error (22.8) / 11 at 5% level of significance is 3.36. Therefore, we may conclude that gca effects are not significantly different. Hence, the pairwise comparison of gca effects is not required.

6.

Some Open Problems

The designs discussed above are suitable only for estimating gca effects under a fixed effects model. Some efforts are needed to obtain the optimal designs when sca effects are also included in the model. Further, in such experiments, the experimenter is also interested in estimating the variance component with resect to lines by considering the effects of crosses as random. Therefore, there is a need to obtain the optimal designs under the mixed effects/fixed effects models.

References Agarwal, S.C. and Das, M.N. (1987). A note on construction and application of balanced n-ary designs. Sankhya, B49(2), 192-196. Bose, R.C. (1939). On the construction of balanced incomplete block designs. Ann. Eugen., 9, 353-399. Ceranka, B. and Mejza, S.(1988). Analysis of diallel table for experiments carried out in BIB designs-mixed model. Biomet. J., 30, 3-16. Chai, F.S. and Mukerjee, R. (1999). Optimal designs for diallel crosses with specific combining abilities. Biometrika, 86(2), 453-458.

321


Clatworthy, W.H.(1973). Tables of two-associate-class partially balanced designs, Applied Maths. Ser.No.63, National Bureau of Standards, Washington D.C. Das, A., Dey, A. and Dean, A.M.(1998). Optimal designs for diallel cross experiments. Statistics and Probability Letters, 36, 427-436. Das, M.N. and Giri, N.C.(1986). Design and analysis of experiments, 2nd Edition. Wiley Eastern Limited, New Delhi. Dey, A., Das, U.S. and Banerjee, A.K.(1986). Construction of nested balanced incomplete block designs. Calcutta Statist. Assoc. Bull., 35, 161-167. Dey, A. and Midha, C.K. (1996). Optimal block designs for diallel crosses. Biometrika, 83, 484-489. Ghosh, D.K. and Divecha, J. (1997). Two associate class partially balanced incomplete block designs and partial diallel crosses. Biometrika, 84(1), 245-248. Griffing, B.(1956). Concepts of general and specific combining ability in relation to diallel crossing systems. Aust. J. Biol. Sci., 9, 463-493. Gupta, S. and Kageyama, S.(1994). Optimal complete diallel crosses. Biometrika, 81, 420-424. Hayman, B.I.(1954a): The analysis of variance of diallel tables. Biometrics, 10, 235-244. Hayman, B.I.(1954b): The theory and analysis of diallel crosses. Genetics, 39, 789-809. Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In A Survey of Statistical Design and Linear Models (Ed. J.N.Srivastava), pp. 333-353, North Holland, Amsterdam. Mukerjee, R.(1997). Optimal partial diallel crosses. Biometrika, 84 (4), 939-948. Parsad, R., Gupta, V.K. and Srivastava, R.(1999). Universally optimal block designs for diallel crosses. Statistics and Applications, 1(1), 35-52. Preece, D.A. (1967). Nested balanced incomplete block designs. Biometrika, 54, 479486. Sharma, M.K.(1996). Blocking of complete diallel crossing plans using balanced lattice designs. Sankhya B 58(3), 427-430. Sharma, M.K.(1998). Partial diallel crosses through circular designs. J. Ind. Soc. Ag. Statist., Vol LI, No. 1, 17-27.

322


Table 1: Universally Optimal Binary Balanced Block Designs for Diallel Cross Experiments Obtainable from NBIB Designs SL. p No. 1 4 2a,b 3 4 5 6 7a,c 8 9 10 11 12 13a 14b 15 16 17 18 19 20 21 22 23a 24 25 26 27 28 29 30 31a 32c 33b 34 35 36 37 38 39 40 41

5 5 5 6 6 7 7 7 8 8 8 9 9 9 9 9 9 10 10 10 10 11 11 11 11 12 12 12 12 13 13 13 13 13 13 13 14 14 15 15

b

k

n

3

2

6 Series 2: Gupta and Kageyama (1994)

5 10 15 5 30 7 21 35 7 42 56 9 18 36 54 60 63 9 45 75 90 11 55 55 99 11 99 110 132 13 26 39 78 117 130 143 13 182 15 105

2 2 2 3 2 3 2 3 4 2 3 4 2 2 2 3 4 5 2 3 4 5 2 3 5 6 2 3 5 6 3 2 2 4 3 6 7 3 7 2

10 20 30 15 60 21 42 105 28 84 168 36 36 72 108 180 252 45 90 225 360 55 110 165 495 66 198 330 660 78 78 78 156 468 390 858 91 546 105 210

Method of construction

Family 1 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Series 1 : Gupta and Kageyama (1994) Family 2: Parsad, Gupta and Srivastava (1999)

323

Reference Design, wherever applicable

Case I: 5,5,4,4,3 Case II: 6,6,5,5,4 Case I:7,7,4,4,2 Case I:7,7,6,6,5 Case I: 8,14,7,4,3 Case II:8,8,7,7,6

Case I:9,18,8,4,3 Case I:9,12,8,6,5 Case I:9,9,8,8,7 Case I:10,15,6,4,2 Case I:10,15,9,6,5 Case II:10,10,9,9,8 Case II:11,11,5,5,2 Case I:11,11,6,6,3 Case I:11,11,10,10,9 Case I:12,33,11,4,3 Case I:12,22,11,6,5 Case II:12,12,11,11,10

Case I:13,13,9,9,6 Case I:13,26,12,6,5 Case II:13,13,12,12,11 Case II:14,26,13,7,6 Case II:15,21,7,5,2


SL. No. 42 43 44 45 46 47 48 49 50 51 52 53a 54 55b 56 57 58 59 60 61a 62c 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79a 80 81 82 83a

p

b

k

n


15 15 15 15 16 16 16 16 16 16 16 17 17 17 17 17 17 18 18 19 19 19 19 19 19 19 20 20 20 21 21 21 21 22 22 22 22 23 23 24 24 25

105 105 175 189 15 60 80 120 144 200 210 17 34 68 136 204 238 17 255 19 57 95 171 171 171 285 19 285 380 21 105 140 210 21 231 154 231 23 253 23 414 25

3 4 3 5 8 2 3 3 5 3 4 8 4 2 2 2 4 9 3 9 3 2 2 4 5 3 10 2 2 10 2 3 3 11 2 3 4 11 2 12 2 12

315 420 525 945 120 120 240 360 720 600 840 136 136 136 272 408 952 153 765 171 171 190 342 684 855 855 190 570 760 210 210 420 630 231 462 462 924 253 506 276 828 300

Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 3 : Das, Dey and Dean (1998) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 1 : Gupta and Kageyama (1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999)

324

Reference Design, wherever applicable Case II:15,15,7,7,3 Case I:15,15,8,8,4 Case I:15,35,14,6,5 Case I:15,21,14,10,9 Case I:16,20,5,4,1 Case I:16,16,6,6,2 Case I:16,24,9,6,3 Case I:16,16,10,10,6 Case I:16,40,15,6,5 Case I:16,30,15,8,7

Case I:17,68,16,4,3 Case I:17,34,16,8,7 Case I:18,51,17,6,5

Case I:19,57,12,4,2 Case II:19,19,9,9,4 Case I:19,19,10,10,5 Case I:19,57,18,6,5 Case I:20,95,19,4,3 Case II:20,76,19,5,4 Case II:21,21,5,5,1 Case I:21,28,8,6,2 Case I:21,42,12,6,3 Case I:22,77,14,4,2 Case II:22,22,7,7,2 Case I:22,33,12,8,4

Case I:24,138,23,4,3


SL. No. 84 85 86c 87b 88 89 90 91 92a 93 94 95 96 97a 98 99 100b 101 102

p

b

k

n


25 25 25 25 25 25 26 26 27 27 28 28 28 29 29 29 29 29 30

50 75 100 150 225 300 25 325 27 351 27 189 252 29 58 203 203 406 29

6 4 3 2 4 2 13 3 13 2 14 2 3 14 7 2 4 2 15

300 300 300 300 900 600 325 975 351 702 378 378 756 406 406 406 812 812 435

Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Series 2 : Gupta and Kageyama(1994) Family 2 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 1 : Parsad, Gupta and Srivastava(1999) Family 2 : Parsad, Gupta and Srivastava(1999) Family 4 : Das, Dey and Dean (1998) Series 2 : Gupta and Kageyama(1994)

Reference Design, wherever applicable

Case II:25,25,9,9,3

Case I:26,65,15,6,3

Case I:28,63,9,4,1 Case II:28,36,9,7,2

Case I:29,29,8,8,2

a

denotes that the design can also be obtained from Series 1: Gupta and Kageyama (1994) denotes that the design can also be obtained from Family 1: Das, Dey and Dean (1998) c denotes that the design can also be obtained from Family 2: Das, Dey and Dean (1998) b

Table 2: Universally Optimal Block Designs for Diallel Crosses Obtainable from Family 5 of Das, Dey and Dean (1998). S.No. 1 2 3 4 5 6 7 8 9 10 11 12 13

P 5 7 9 11 13 15 17 19 21 23 25 27 29

b 2 3 4 5 6 7 8 9 10 11 12 13 14

k 5 7 9 11 13 15 17 19 21 23 25 27 29

n 10 21 36 55 78 105 136 171 210 253 300 351 406

325


Table 3: Universally Optimal Binary Balanced Block Designs for Diallel Crosses Generated from Triangular PBIB Designs Given by Dey and Midha (1996). SL.No.

p

b

k

n

1 2 3 4 5 6 7 8 9 10 11 12

5 5 5 6 6 6 6 7 7 8 9 10

15 30 45 45 15 30 45 105 70 70 63 63

2 2 2 2 3 3 3 2 3 4 4 5

30 60 90 90 45 90 135 210 210 280 252 315

Reference Design T2 T3 T4 T6 T16 T17 T19 T8 T22 T40 T41 T54

Table 4: Universally Optimal Generalized Binary Balanced Block Designs for Diallel Crosses Generated from Triangular PBIB Designs Obtained by Theorem 4.1 of Das, Dey and Dean (1998) SL.No.

p

b

k

n

1 2 3 4 5 6 7 8 9

5 5 5 5 6 6 8 9 10

30 10 6 10 10 10 28 28 45

3 4 5 6 6 9 7 9 9

90 40 30 60 60 90 196 252 405

326

Reference Design T13 T33 T44 T60 T62 T83 T77 T85 T91