Balanced Designs for Partial Double Cross Volume 3 - Issue 4

Mahendra Kumar Sharma* and Mohammed Omer Ibrahim

• Addis Ababa University, Ethiopia

Received: December 03, 2021   Published: January 6, 2022

*Corresponding author: Mahendra Kumar Sharma, Addis Ababa University, Ethiopia

DOI: 10.32474/CTBB.2022.03.000166

Abstract PDF

Abstract

Hinkelmann [1] constructed optimal balanced tetra partial cross i.e. optimal balanced partial double cross using balanced incomplete block designs and partially balanced incomplete block designs. In the present article we are presenting method of construction of three series of balanced partial double cross designs using nested balanced incomplete block designs in one way set up, when p is odd. Our proposed designs for partial double cross are new. AMS classification: 62 k 05.

Keywords: Nested Balanced Incomplete Block Design; Complete diallel Cross; Balanced Partial Double Cross; Block Design; Optimality

Introduction

Mating designs involving multi–allele cross m (≥ 2) ines play very important role to study the genetic properties of a set of inbred lines in plant breeding experiments. A most common type of mating design in genetics is the diallel cross which consist of v = p ( p −1) / 2crosses of p inbred lines such that the crosses of the type (i × j ) = ( j × i ) for i, j = 1, 2,..., p . This type of mating design is called complete diallel cross (CDC) method 4 of Griffing (1956). The concept of CDC can be easily extended to double cross designs. A double cross mating design is obtained by crossing two unrelated F₁ hybrids symbolized as (i × j) and (k × l), where i ≠ j ≠ k ≠ l, are 4 parents and (i × j) and (k × l), are two F₁’s{see Singh et al.2012}. Omitting reciprocal crosses, there exist

different possible crosses from p lines. We shall consider the case where one observes only a sample of all possible crosses. Such a design we call a partial tetra-allel cross (PTAC) or partial double cross.

Let n_c denote the total number of crosses involved in a balanced partial double cross design. Generally double cross experiments are conducted using a Completely Randomized Design (CRD) or a Randomized Complete Block (RCB) design involving some or all of the nc crosses as treatments. The number of crosses in such a mating design increases rapidly with increase in the number of lines. It leads to an overall inefficient experiment. It is for this reason that the use of incomplete block design as environment design is needed for double cross experiments [2]. Constructed optimal block designs for partial double cross experiment by using balanced incomplete block designs and nested balanced incomplete block designs [3]. Sharma and [4] constructed double cross designs for even and odd value of p by using initial block of unreduced balanced incomplete block designs given [1] and initial block of row-column designs given by [5], respectively. In this paper we are using initial blocks of nested balanced incomplete block designs [6] [7] for the construction of three series of balanced partial double cross designs in one way set up, when p is odd. The parameters of our proposed balanced partial double cross designs are different from [2] designs. In our proposed designs every cross is replicated equal number of times in a design and every line occurs in single cross with every line. We have considered the model that includes the gca effects, apart from block effects, but no specific combining ability effects.

Some Definitions:

Before we describe the constructions, it might be appropriate to recall some definitions.

Definition: The double cross has been defined [8] as a cross between two unrelated F₁ hybrids, say denoted by (i × j)and (k × l ) where i ≠ j ≠ k ≠ l ≠ i, are denoting the grandparents and no two of them are same. Ignoring reciprocal crosses, with p grandparents, there will be

different possible double crosses from p lines. The number of combinations of 4 lines from p lines is

, where pi denotes ( p − i). For each combination, three different double crosses can be formed by changing the manner in which the four lines are paired to form F1 hybrid. We shall consider here the design where one observes only a sample of all possible crosses. Such a design we call a partial double cross (PDC).

Definition: According to [9], a set of mating is said to be partial double crosses, if the following conditions are satisfied:

a) Every line occurs exactly r times in the set,

b) Every four-way cross occurs either once or not at all.

Definition: According to [9], a set of mating is said to be partial double crosses in the strict sense, if it satisfies the condition of Definition 2.2 and

a) Every single cross occurs once or not at all in the set.

Definition: According to [9], a partial double cross is said to be balanced partial double crosses, if every line occurs equally often with every other line in the same single cross.

Definition: According to [10], an arrangement of p treatments each replicated r* times in two systems of blocks is said to be a nested balanced incomplete design (NBIBD) with parameters

if

a) The second system is nested within the first, with each block from the first system, called hence forth as’ block’ containing exactly m blocks from the second system, called hereafter as sub blocks.

b) Ignoring the second system leaves a balanced incomplete block design with parameters

c) Ignoring the first system leaves a balanced incomplete block design with parameters

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

Definition: A design d* ɛ D (p, b, k) is said to be A-optimal if and only if

means that trace of variance-covariance matrix d* is less than equal to trace of variance-covariance matrix d.

Method of Construction

Consider a nested balanced incomplete block design d obtained by developing t₁ initial blocks mod (p), each sub-block is divided into t₂ sub-blocks. The parameters of such a NBIB design are

. If we identify the treatments of a design d as lines of a diallel experiment and perform double crosses among the lines appearing in the same sub block of d and arrange these sub blocks into one bigger block and develop this bigger block mod (p), we get a design d₁ for a partial double cross experiment involving p lines with b_1/2 crosses arranged in p blocks. Each partial double cross is replicated once. Such a design d1 belongs to D(p,b,k).

For such a design

or 1for i ≠ j ≠ k ≠ l,

The information matrix for designd1 is given below.

given above by (3.1) is completely symmetric.

trace of

Using d₁ each elementary contrast among line effects is estimated with a variance

Now we give the method of construction of three series of balanced partial double cross designs using NBIB designs [11].

Series: Let p = 4t +1,t ≥1 be a prime or a prime power and x be a primitive element of the GF (p). Consider the following t initial blocks.

{(xⁱ , x^i+2t ),(x^i+t , x^i+3t )} ,i = 0,1, 2,...t −1.

As shown [11], these initial blocks, when developed in the sense of Bose (1939), give rise to a nested balanced incomplete block design with parameters.

v = p = 4t +1,b1 = t (4t +1), r = 4t, k1 = 4,λ1 = 3,b2 = 2t (4t +1)k2 = 2,λ 2 =1

Arrange the following m initial blocks into single block

By making pair of crosses in a single block and developing mod (p), we get balanced partial double

cross design with parameters

Remark: For construction of balanced partial double cross design it is necessary that the block size of a single block must be an even number i.e. t must be a multiple of 2. The procedure to construct the above designs, has been explained by the following illustrative examples.

Example: If we let t = 2, we get the following two blocks.

We can now write both blocks in a single block as given below.

where is a primitive element of GF(3²). Now cross the elements of the individual block and put these crosses in a single block. Adding successively the non-zero elements of GF (3²) to the contents of the single block, we obtain block design for partial double cross experiment with parameters p = 9,b = 9, k = 2. The design is exhibited below, where the lines have been relabelled 1-9, using the correspondence 0→1,1→2, 2→3, x→4, x +1→5, x + 2→6, 2x +1→8, 2x + 2→9 :

Partial Double Cross Block Design

B₁ B₂ B₃ B₄ B₅ B₆

(2×3)×(6×8) (1×3)×(4×9) ( 1 × 2 ) × ( 5 × 7 )

(5×6)×(2×9) (4×6)×(3×7) (4×5)×(1×8)

(4×7)×(5×9) (5×8)×(6×7) (6×9)×(4×8) (1×7)×(3×8)

(2×8)×(1×9) (3×9)×(2×7)

B₇ B₈ B₉

(8×9) × (3×5) (7×9) × (1×6) (7×8) × (2×4)

(1×4) × (2×6) (2×5) × (3×4) (3×6) × (1×5)

According to [2] the above design is balanced and is of minimal size, hence optimal.

Series: Let p = 6t +1,t ≥1 be a prime or a prime power and x be a primitive element of the Galois field of order p. Consider the set of initial blocks

{(xⁱ , x^i+3t ),(x^i+t , x^i+4t ),(x^i+2t , x^i+5t )},i = 0,1, 2,...t −1.

Dey, et al. (1986) showed when these initial blocks developed over mod (p), give a solution of a nested incomplete block design with parameters

.

We can then arrange the above initial blocks into a single block as given below

Example 2: Let t = 2. Then we get the following two initial blocks

We can arrange these two blocks in a single block as given below.

Now performing crosses in pairs and developing these crosses over mod (p), we obtain A- optimal block

design for partial double cross with parameters

p = 13,b = 13and k = 3.

Series: Let p = 2t +1,t ≥ 2 be a prime or a prime power and consider the following m blocks

(0, 2t),(1, 2t −1),(2, 2t −1).....(t −1,t +1)mod (2t +1)

Example 3: If we let t = 4, then the single block will be as given below.

Now applying the procedure given above in example 1, we can obtain an A-optimal block design for double cross experiment with parameters

p = 9,b = 9, k = 2

Note 2: The t blocks given in series 3 form a nested balanced incomplete block design with parameters

given [10].

Analysis of Bpdc Design

Following [9] we can assume the following reduced model for proposed incomplete balanced partial double cross (BPDC) mating designd1. For the BPDC(i× j)(k ×l), where i ≠j ≠k ≠ l, in m blocks, we adopt the following model

where y_(ij)(kl) is the observation on double cross (ij)(kl) from m blocks, m = 1,2,..., p,i, j, k,l = 1,2,..., p;i ≠ j ≠ k ≠ l,μ is the general mean, g_i is the effect of ith line, β_m is the effect mthblock and e_ijklm are random errors assumed independent and normally distributed with mean zero and variance σ².

An incomplete four - way mating design d₁ is defined such that every line occurs exactly r times and every four-way cross occurs once or not all [9]. In matrix notation (4.1) can be written as below

y is an n×1 vector of observations, 1 is an n×1 vector of ones, Ä′₁ is an n × p design matrix for lines and Ä′₂ is an n × b design matrix for blocks, that is, the (h, l)^th element of 1 Ä′₁ ( also of Ä′₂ ) is 1 if the h^th observation pertains to the l^th line (also of block) and is zero otherwise. μ is a general mean, g is a p × 1 vector of line parameters, β is a p × 1 vector of block parameters and e is an n × 1 vector of residuals. It is assumed that vector β is fixed and e is normally distributed with mean 0 and Var (e) = σ²IwithCov (β, e) = 0, where I is the identity matrix of conformable order. For the analysis of proposed design d₁, the method of least squares leads to the following reduced normal equations for estimating the linear functions of the line effects of lines under model (4.1).

where '

and for i ≠ i' ,gd1ii´ is the number of crosses in d in which the lines i and i´ appear together.

is the number of times the line i occurs in block j of d₁ and

is the diagonal matrix of block sizes? T_i is the total of i^th line in design d₁.

The sum of the elements in each row and each column of C_d1 is equal to zero. Hence rank

but for all differences

, to be estimable it is necessary that rank

solution of (4.3) is then given by

where

is the generalized inverse of C_d1 with property

The sum of squares due to crosses is

with d.f = rank (C_d1) ( and expectation and variance of Q_d1 are

From (4.7), it follows immediately that the using d₁ each elementary contrast among line effects is estimated with a variance

Now let us suppose that the proposed design d₁ is replicated u times because if we analysis the proposed design d₁ no degrees of freedom is left for the error. Now we consider the following model for replicated BPDC design.

where y(ij)(kl)v is the observation on double cross (ij)(kl) from v^th replicate?

v = 1, 2,...,u,i, j, k,l = 1, 2,..., p;i ≠ j ≠ k ≠ l ≠,μ is the general mean,

is the effect v^th replicate and e_ijklu are random errors assumed independent and normally distributed with mean zero and variance

The Analysis of variance of replicated BPDC design is given in Table 1.

Table 1: Analysis of variance of PTAC.

Optimality

Go to

• Abstract
• Introduction
• Some Definitions:
• Method of Construction
• Analysis of Bpdc Design
• Optimality
• Discussion
• Conclusion
• References

design d₁ is connected since rank

and variance balanced because the matrix C_d1 is completely symmetric. It implies that all elementary comparisons among line effects are estimable using d₁ with the same variance. We denote by D (p,b,k), the class of all such connected block design {d} with p lines, b blocks each of size k. According to [9], such a design can be considered as being optimal in the sense that it is completely balanced and of minimal size. So, our proposed design d₁ is optimal and of minimal size. Now we will use the Kiefer criteria to see the status of our proposed design d₁ for universal optimality.

Kiefer (1975) showed that a design is universally optimal in a relevant class of competing design:

a) The information matrix ( the C_d – matrix) of the design is completely symmetric in the sense that C_d has all its diagonal elements equal and all of its off- diagonal elements equal, and

b) The matrix C_d has maximum trace in the class of competing designs, that is, such a design minimizes the average variance of the best linear unbiased estimators of all elementary contrasts among the parameters of interest i.e. the line effects in our context. Now we state the Kiefer criterion of the universal optimality in sense [2] to compare proposed designs optimality with universal optimal designs.

Theorem: For any proposed design d*ε D( p,b,k) be a block design for m-allelcrosses satisfying

Where X = [2k / p,] where [z] denotes largest integer not exceeding z, Ip is an identity matrix of order p and 1p 1′p is a p × p matrix of all ones. Furthermore, using d* all elementary contrasts among line effects are estimated with variance

Then d* is universally optimal in D (p, b, k), and in particular minimizes the average variance of the best linear unbiased estimator of all elementary contrasts among the line effects. According to the definition 2.6, the trace of our proposed design

is greater than the equality given in theorem 4.1 i.e., 4p = 36 . Hence our design d₁ is also A-optimal. Hence, we state the following theorems.

Theorem: The existence of a nested balanced incomplete block design d with parameters

, where p is odd, implies the existence of A-optimal block designs for balanced partial double cross experiment.

Discussion

Go to

• Abstract
• Introduction
• Some Definitions:
• Method of Construction
• Analysis of Bpdc Design
• Optimality
• Discussion
• Conclusion
• References

A-Optimal balanced partial double cross designs with p ≤ 16, s ≤ 30 obtained by the above methods from NBIB designs [7], are listed in the following Table 2. These are the designs other than the designs catalogued [12,10,8]. These are the new designs and can be used in agricultural experimentation successfully [13,14].

Table 2: optimal block design for double cross with p ≤ 16, s ≤ 30 generated by using NBIB designs of Morgan [3].

Conclusion

Go to

• Abstract
• Introduction
• Some Definitions:
• Method of Construction
• Analysis of Bpdc Design
• Optimality
• Discussion
• Conclusion
• References

We have presented a method of construction of three series of A-optimal balanced block designs for partial double cross by using nested balanced incomplete block designs. Our proposed Aoptimal block designs for partial double crosses are new.

References

Go to

• Abstract
• Introduction
• Some Definitions:
• Method of Construction
• Analysis of Bpdc Design
• Optimality
• Discussion
• Conclusion
• References

1. Bose RC, Shrikahande SS, Bhattacharya KN (1953) On the construction of group divisible incomplete block designs. Ann Math Stat 24: 167-195.
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3. Morgan JP, Preece DA, Rees DH (2001) Nested balanced incomplete block designs. Discrete Mathematics 231(1-3): 351-389.
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5. Gupta S, Choi K C (1998) Optimal row-column designs for complete diallel crosses. Communication in Statistics-Theory and Methods 27(11): 2827-2835.
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