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On the construction and existence of a certain class of complete diallel cross designs
ARTICLE IN PRESS
Statistics & Probability Letters 77 (2007) 111–115 www.elsevier.com/locate/stapro
On the construction and existence of a certain class of complete diallel cross designs Sudesh K. Srivastav, Arti Shankar Department of Biostatistics, Tulane University, 1440 Canal Street, Suite 2019, New Orleans, LA, USA Received 9 June 2005; received in revised form 2 March 2006; accepted 23 May 2006 Available online 10 July 2006
Abstract In this paper the construction and existence of complete diallel cross designs as nested balanced incomplete block (BIB) designs with sub-block size two are discussed. Two simple and general methods of constructing these designs using binary pairwise balanced designs and strongly equineighboured designs are given. Also, the necessary conditions for the existence of nested BIB designs with sub-block size two and super-block size four are found to be sufficient. r 2006 Elsevier B.V. All rights reserved. Keywords: Nested balanced incomplete block design; Complete diallel cross design; Universal optimality; Pairwise balanced design
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1. Introduction Consider an experimental situation of diallel crosses as mating designs to study the genetic properties of inbred lines in plant and animal breeding. The designs considered, compare the inbred lines with respect to their general combining abilities (not the specific combining abilities). For a more detailed discussion on these designs and for further references, see Griffing (1956), Gilbert (1958), Hinkelmann (1975), and Gupta and Kageyama (1994). Suppose there are p parental lines and let the typical cross between lines i and j be denoted by ði; jÞ for ioj ¼ 1; 2; . . . ; p. A diallel cross design is called complete if it includes all the number of distinct crosses ðnc Þ, i.e., nc ¼ pðp 1Þ=2. All diallel cross designs considered here are complete. If each cross appears l-times then the design is called cross balanced. Consider a diallel cross design with p lines and b blocks each containing konc crosses such that: (i) (ii) (iii) (iv)
each line appears in exactly r crosses, each line among crosses appears in each block at most once, each cross occurs in exactly l blocks, i.e., design is cross balanced. ignoring crosses, each pair of lines occurs in exactly l blocks.
Corresponding author. Tel.: +1 504 988 2472; fax: +1 504 988 1706.
E-mail address: [email protected] (S.K. Srivastav). 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.05.019
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
112
Clearly, this design is simply a nested balanced incomplete block (BIB) design with sub-block size of two if each diallel cross is seen as a block of size two. From now this diallel cross design will be referred as a nested BIB design with sub-block size two. The parameters of this design are given by p; b; r; k; l, and l and will be denoted by ðp; b; r; k; l; lÞ. It can easily be established that bk ¼
pr ¼ lnc ; 2
lðp 1Þ ¼ r; and l ¼
rð2k 1Þ ¼ lð2k 1Þ p1
are necessary conditions for the existence of nested BIB designs with sub-blocks size two. The universal optimality of these designs was first established by Gupta and Kageyama (1994) and some methods of constructing these designs were also discussed earlier by, Dey and Midha (1996), Choi and Gupta (2000), and Das and Ghosh (1999). In Section 2, two methods of constructing nested BIB designs with sub-block size two using pairwise balanced designs and strongly equineigboured designs are given. Finally, in Section 3, for k ¼ 2 the necessary conditions for existing nested BIB designs with sub-block size two are shown to be sufficient. 2. Construction methods for nested BIB designs with sub-block size two Consider the designs, introduced by Hanani (1961), known as B-systems. Suppose a set E having p elements are given. Further let K ¼ fk1 ; k2 ; . . . ; kn g be a finite set of integers with 3pki , i ¼ 1; . . . ; n. Let l be a positive integer. If it is possible to form a design of binary blocks (subsets of E) in such a way that every (unordered) pair of elements of E is contained in exactly l blocks, then we shall call such a design a binary pairwise balanced design and denote it by B½K; l; p. The class of all numbers p for which designs B½K; l; p exist will be denoted by BðK; lÞ. If K ¼ fkg consists of one number, say k, then the design is a balanced incomplete block design (BIBD) and shall be denoted by B½k; l; p. Next theorem illustrate the method of constructing nested BIBD with sub-block size two using binary pairwise balanced designs that is similar to the method described by Srivastav and Morgan (1996) for constructing BIBD with nested rows and columns. Theorem 2.1. The existence of a binary pairwise balanced design B0 , a B½K; l0 ; p with K ¼ fk1 ; k2 ; . . . ; kn g, and of Bi , a nested BIB design with sub-block size two with parameters pi ¼ ki , b¯ i , ri , k, l1 , and l 1 for i ¼ 1; . . . ; n, implies the existence P of a nested BIB design with sub-block size two with parameters p, b, r, k, l, and l for appropriate r, b ¼ ðb¯ i bi Þ, l ¼ l0 l1 , and l ¼ l0 l 1 . Proof. Suppose bi are the number of blocks of size ki in B0 . Let rij be the number of times that line j occurs in the bi blocks of size ki , for i ¼ 1; . . . ; n. Also assume that liða1 ;a2 Þ denotes P the number ofPtimes that lines a1 and a2 occur together among the blocks of size ki . Then l0 ðp 1Þ ¼ i rij ðki 1Þ, and i liða1 ;a2 Þ ¼ l0 . For the nested BIB design with sub-block size two Bi , if ri denotes the number of experimental units that receive a cross with one parent i then the relationship ri =ðpi 1Þ ¼ l 1 holds for each i. Ignoring crosses, each pair of lines in nested BIB design with sub-block size two Bi occurs exactly l1 times, therefore the relationship ðri ð2k 1ÞÞ=ðpi 1Þ ¼ l 1 ð2k 1Þ ¼ l1 is also true for each i. For each block of size ki in B0 , use its lines to construct the nested BIB design P with sub-block size two Bi , producing b¯ i bi blocks of the new design D. Repeating this for each i gives b ¼ ðb¯ i bi Þ blocks in which line j has replication X i
ðri rij Þ ¼
X i
X ri l0 l1 ðp 1Þ , ðki 1Þrij ¼ l 1 ðki 1Þrij ¼ l0 l 1 ðp 1Þ ¼ 2k 1 ki 1 i
where the last two expressions do not depend on j. And for D, it isPclear that the number ofPreplications of the i i cross ði; jÞ and each pair of lines, ignoring crosses, are i lða1 ;a2 Þ l 1 ¼ l0 l 1 and i lða1 ;a2 Þ l1 ¼ l0 l1 , respectively. &
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
113
Example 1. For p ¼ 7, a B[4,2;7] design 1 2
1 2
1 1 3 4
2 3
2 4
3 5
3
5
4 6
4
5
6
6
7
5 7
7
6
7
along with the nested BIB design with sub-block size two with parameters ð4; 3; 3; 2; 3; 1Þ, 12 13 34 24
14 23
gives the following nested BIB design with sub-block size two with parameters ð7; 21; 12; 2; 6; 2Þ: 12 12
13 13
14 14
15
15
16
16
17
36 57
26 45
35 67
34
27
23
47
25
17 23
24 24
25 26
27
35
36
37
46 47
37 56
46 45
34
67
57
56
Next we shall consider another method of constructing nested BIB designs with sub-block size two, using the strongly equineighboured designs as defined in Martin and Eccleston (1991) and Street (1992). Strongly equineighboured designs have p treatments and b ¼ 12 spðp 1Þ ordered blocks, each of size k. Blocks made of columns of k b array satisfy: (i) Each treatment occurs sðp 1Þ times in rows j and k þ 1 j together, ja 12 ðk þ 1Þ, and 12 sðp 1Þ times in row 12 ðk þ 1Þ if k is odd. (ii) The columns in the 2 b submatrices obtained from rows j and k þ 1 j contain each unordered pair of distinct treatments exactly s times. (iii) The columns in the 2 b submatrices obtained from rows j and m, and from rows k þ 1 j and k þ 1 m ðj þ mak þ 1Þ, together contain each unordered pair of distinct treatments exactly 2s times. These designs are denoted by SENðk; s; pÞ. Theorem 2.2. The columns of SENð2t; s; pÞ, where tX1 and pX4, form nested BIB designs with sub-block size two with parameters ðp; b; tsðp 1Þ; t; 3ts; tsÞ. Proof. On writing the blocks of a SENð2t; s; pÞ as the columns of a 2t b array, the SEN conditions immediately implies: (i) each line occurs sðp 1Þ times in rows j and 2t þ 1 j, for j ¼ 1; 2; . . . ; t, that is, each line occurs tsðp 1Þ times; (ii) the t sets of columns in the 2 b submatrices obtained from rows j and 2t þ 1 j, j ¼ 1; 2; . . . ; t, where each set contain every unordered pair of distinct lines exactly s times, and (iii) the t sets of columns in the 2 b submatrices obtained from rows 2t þ 1 j and 2t þ 1 m ðj þ ma2t þ 1Þ, where each set collectively contain each unordered pair of distinct lines exactly 2s times. Therefore, from (ii) and (iii), each unordered pair of distinct lines occurs exactly ts þ 2tsð¼ 3tsÞ times in this arrangement. So the columns of SENð2t; s; pÞ form a BIBD with k ¼ 2t and l ¼ 3ts. On writing each ordered column ðd1 ; d2 ; . . . ; d2t1 ; d2t Þ0 , say, of a SENð2t; s; pÞ designs in the form d1 d2 .. . dt
d2t d2t1
dtþ1 ;
and using SEN property (ii) each cross will occur in exactly tsð¼ lÞ blocks. Hence we have the desired nested BIB designs with sub-block size two with parameters ðp; b; tsðp 1Þ; t; 3ts; tsÞ. &
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
114
Example 2. For p ¼ 6 and t ¼ 2, a SENð4; 1; 6Þ design is 1
1 1
1
1
2 2
2
2
3
3 3
4
4
5
3
2 2
4
3
4 6
3
5
6
6 2
3
3
1
4 2
5 6 3 4
2 5
5 6
5 5 3 4
6 5
1 6
1 4
4 1 5 6
1 5
2 6
4 6
that gives the following nested BIB design with sub-block size two with parameters ð6; 15; 10; 2; 2Þ: 12 13
14
15
16
23
24
25
34 25
26
24
35
45
56
36
26 34
35
36
45
46
56
15 16
46
12
13
23
14
3. Existence of nested BIB designs with sub-block size two for k ¼ 2 In this section, for k ¼ 2 the necessary conditions for a nested BIB designs with sub-block size two are shown to be sufficient. For k ¼ 2, the necessary conditions for a nested BIB designs with sub-block size two reduce to 4b ¼ pr;
lðp 1Þ ¼ r; and l ¼ 3l.
Therefore 4 must divide lpðp 1Þ. If ðx; yÞ denotes the greatest common divisor of x and y then the restrictions on p given l can easily be stated as below: If l 1 ðmod 2Þ;
then p 0 or 1 ðmod 4Þ,
(1)
If l 0 ðmod 2Þ;
then p is unrestricted.
(2)
Also we need pX4. If ðl; 4Þ ¼ d (say) and l ¼ qd for some integer q then we can form a nested BIB designs with sub-block size two with parameters ðp; b; r; 2; 3l; lÞ simply by taking an q-multiple of a nested BIB designs with sub-block size two for k ¼ 2 with p lines in b=q blocks, that is, by taking q copies of each block. Thus to show that the necessary conditions (1) and (2) are also sufficient, it suffices to demonstrate, with pX4, the existence of nested BIB designs with sub-block size two with parameters ðp; pðp 1Þ=4; p 1; 2; 3; 1Þ for every p 0 or 1 ðmod 4Þ, and of nested BIB designs with sub-block size two with parameters ðp; pðp 1Þ=2; 2ðp 1Þ; 2; 6; 2Þ for every p 2 or 3 ðmod 4Þ. The following lemma, given by Hanani (1961), will be needed. Lemma 3.1. If p 0 or 1 ðmod 4Þ and pX4, then p 2 B½K; 1, where K ¼ f4; 5; 8; 9; 12g. Lemma 3.2. Nested BIB designs with sub-block size two with the following parameters all exist: ð4; 3; 3; 2; 3; 1Þ, ð5; 5; 4; 2; 3; 1Þ, ð8; 14; 7; 2; 3; 1Þ, ð9; 18; 8; 2; 3; 1Þ, and ð12; 33; 11; 2; 3; 1Þ. Proof. The nested BIB designs with sub-block size two are: parameters ð4; 3; 3; 2; 3; 1Þ: 12 13 34 42
14 23
parameters ð5; 5; 4; 2; 3; 1Þ: 12 13
14
15
23
35 24
25
34
45
parameters ð8; 14; 7; 2; 3; 1Þ: 14 16 32 53
17 64
18 45
12 86
15 27
13 78
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
25 28 83 74
24 26 56 37
34 36 75 48
58 67
115
parameters ð9; 18; 8; 2; 3; 1Þ: 12 13
14 15
16 17
18
19
23
57 49
26 36
79 38
45
28
68
24 25
27 29
35 37
47
48
58
78 34
39 56
89 46
49
69
67
parameters ð12; 33; 11; 2; 3; 1Þ: 12
13
14
15
16
17
18
19
1A
1B
1C
6C
5B
29
47
24
8A
3B
6A
39
5C
79
23 7C
25 26 27 3A 58 35
28 9B
2A 7B
2B 4A
2C 89
34 8C
49 57
4C AB
56 AC
38 3C 46 9A
45 9C
48 7A
59 9B
5A 68
36 4B 67 BC
37 69 6B 79
Combining Lemmas 3.1 and 3.2, and Theorem 2.1 we have the following theorem which solves the problem presented by (1) for l ¼ 3 and l ¼ 1. Theorem 3.3. Nested BIB designs with sub-block size two with parameters ðp; ðpðp 1ÞÞ=4; ðp 1Þ; 2; 3; 1Þ exists for p 0 or 1 ðmod 4Þ, where pX4. Now to see the existence of (2) for l ¼ 6 and l ¼ 2, we shall state the following lemma whose proof can be found in Street (1992). Lemma 3.4. SENð4; 1; pÞ exists for all pX4. Therefore, using Lemma 3.4 and Theorem 2.2, we have the final result of the paper.
&
Theorem 3.5. Nested BIB designs with sub-block size two with parameters ðp; ðpðp 1ÞÞ=2; 2ðp 1Þ; 2; 6; 2) exists for all pX4. Acknowledgement The authors wish to thank the referee for his helpful comments and for suggestions for an improved presentation. References Choi, K.C., Gupta, S., 2000. On constructions of optimal complete diallel crosses. Utilitas Math. 58, 153–160. Das, A., Ghosh, D.K., 1999. Balanced incomplete block diallel cross designs. Statist. Appl. 1, 1–16. Dey, A., Midha, C.K., 1996. Optimal block designs for diallel crosses. Biometrika 83, 484–489. Gilbert, N.E., 1958. Diallel cross in plant breeding. Heredity 12, 477–492. Griffing, B., 1956. Concept of general and specific ability in relation to diallel crossing systems. Aust. J. Biol. Sci. 9, 463–493. Gupta, S., Kageyama, S., 1994. Optimal complete diallel crosses. Biometrika 81, 420–424. Hanani, H., 1961. The existence and construction of balanced incomplete block designs. Ann. Math. Statist. 32, 371–386. Hinkelmann, K., 1975. Design of Genetical experiments. In: Srivastava, J.N. (Ed.), A Survey of Statistical Design and Linear Models, vol. 42. North-Holland, Amsterdam, pp. 243–269. Martin, R.J., Eccleston, J.A., 1991. Optimal incomplete block designs for general dependence structures. J. Statist. Plann. Inf. 28, 67–81. Srivastav, S.K., Morgan, J.P., 1996. On the class of 2 2 balanced incomplete block designs with nested rows and columns. Commun. Statist. Theory Methods 25 (8), 1859–1870. Street, D.J., 1992. A note on strongly equineighboured designs. J. Statist. Plann. Inf. 30, 99–105.
Statistics & Probability Letters 77 (2007) 111–115 www.elsevier.com/locate/stapro
On the construction and existence of a certain class of complete diallel cross designs Sudesh K. Srivastav, Arti Shankar Department of Biostatistics, Tulane University, 1440 Canal Street, Suite 2019, New Orleans, LA, USA Received 9 June 2005; received in revised form 2 March 2006; accepted 23 May 2006 Available online 10 July 2006
Abstract In this paper the construction and existence of complete diallel cross designs as nested balanced incomplete block (BIB) designs with sub-block size two are discussed. Two simple and general methods of constructing these designs using binary pairwise balanced designs and strongly equineighboured designs are given. Also, the necessary conditions for the existence of nested BIB designs with sub-block size two and super-block size four are found to be sufficient. r 2006 Elsevier B.V. All rights reserved. Keywords: Nested balanced incomplete block design; Complete diallel cross design; Universal optimality; Pairwise balanced design
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1. Introduction Consider an experimental situation of diallel crosses as mating designs to study the genetic properties of inbred lines in plant and animal breeding. The designs considered, compare the inbred lines with respect to their general combining abilities (not the specific combining abilities). For a more detailed discussion on these designs and for further references, see Griffing (1956), Gilbert (1958), Hinkelmann (1975), and Gupta and Kageyama (1994). Suppose there are p parental lines and let the typical cross between lines i and j be denoted by ði; jÞ for ioj ¼ 1; 2; . . . ; p. A diallel cross design is called complete if it includes all the number of distinct crosses ðnc Þ, i.e., nc ¼ pðp 1Þ=2. All diallel cross designs considered here are complete. If each cross appears l-times then the design is called cross balanced. Consider a diallel cross design with p lines and b blocks each containing konc crosses such that: (i) (ii) (iii) (iv)
each line appears in exactly r crosses, each line among crosses appears in each block at most once, each cross occurs in exactly l blocks, i.e., design is cross balanced. ignoring crosses, each pair of lines occurs in exactly l blocks.
Corresponding author. Tel.: +1 504 988 2472; fax: +1 504 988 1706.
E-mail address: [email protected] (S.K. Srivastav). 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.05.019
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
112
Clearly, this design is simply a nested balanced incomplete block (BIB) design with sub-block size of two if each diallel cross is seen as a block of size two. From now this diallel cross design will be referred as a nested BIB design with sub-block size two. The parameters of this design are given by p; b; r; k; l, and l and will be denoted by ðp; b; r; k; l; lÞ. It can easily be established that bk ¼
pr ¼ lnc ; 2
lðp 1Þ ¼ r; and l ¼
rð2k 1Þ ¼ lð2k 1Þ p1
are necessary conditions for the existence of nested BIB designs with sub-blocks size two. The universal optimality of these designs was first established by Gupta and Kageyama (1994) and some methods of constructing these designs were also discussed earlier by, Dey and Midha (1996), Choi and Gupta (2000), and Das and Ghosh (1999). In Section 2, two methods of constructing nested BIB designs with sub-block size two using pairwise balanced designs and strongly equineigboured designs are given. Finally, in Section 3, for k ¼ 2 the necessary conditions for existing nested BIB designs with sub-block size two are shown to be sufficient. 2. Construction methods for nested BIB designs with sub-block size two Consider the designs, introduced by Hanani (1961), known as B-systems. Suppose a set E having p elements are given. Further let K ¼ fk1 ; k2 ; . . . ; kn g be a finite set of integers with 3pki , i ¼ 1; . . . ; n. Let l be a positive integer. If it is possible to form a design of binary blocks (subsets of E) in such a way that every (unordered) pair of elements of E is contained in exactly l blocks, then we shall call such a design a binary pairwise balanced design and denote it by B½K; l; p. The class of all numbers p for which designs B½K; l; p exist will be denoted by BðK; lÞ. If K ¼ fkg consists of one number, say k, then the design is a balanced incomplete block design (BIBD) and shall be denoted by B½k; l; p. Next theorem illustrate the method of constructing nested BIBD with sub-block size two using binary pairwise balanced designs that is similar to the method described by Srivastav and Morgan (1996) for constructing BIBD with nested rows and columns. Theorem 2.1. The existence of a binary pairwise balanced design B0 , a B½K; l0 ; p with K ¼ fk1 ; k2 ; . . . ; kn g, and of Bi , a nested BIB design with sub-block size two with parameters pi ¼ ki , b¯ i , ri , k, l1 , and l 1 for i ¼ 1; . . . ; n, implies the existence P of a nested BIB design with sub-block size two with parameters p, b, r, k, l, and l for appropriate r, b ¼ ðb¯ i bi Þ, l ¼ l0 l1 , and l ¼ l0 l 1 . Proof. Suppose bi are the number of blocks of size ki in B0 . Let rij be the number of times that line j occurs in the bi blocks of size ki , for i ¼ 1; . . . ; n. Also assume that liða1 ;a2 Þ denotes P the number ofPtimes that lines a1 and a2 occur together among the blocks of size ki . Then l0 ðp 1Þ ¼ i rij ðki 1Þ, and i liða1 ;a2 Þ ¼ l0 . For the nested BIB design with sub-block size two Bi , if ri denotes the number of experimental units that receive a cross with one parent i then the relationship ri =ðpi 1Þ ¼ l 1 holds for each i. Ignoring crosses, each pair of lines in nested BIB design with sub-block size two Bi occurs exactly l1 times, therefore the relationship ðri ð2k 1ÞÞ=ðpi 1Þ ¼ l 1 ð2k 1Þ ¼ l1 is also true for each i. For each block of size ki in B0 , use its lines to construct the nested BIB design P with sub-block size two Bi , producing b¯ i bi blocks of the new design D. Repeating this for each i gives b ¼ ðb¯ i bi Þ blocks in which line j has replication X i
ðri rij Þ ¼
X i
X ri l0 l1 ðp 1Þ , ðki 1Þrij ¼ l 1 ðki 1Þrij ¼ l0 l 1 ðp 1Þ ¼ 2k 1 ki 1 i
where the last two expressions do not depend on j. And for D, it isPclear that the number ofPreplications of the i i cross ði; jÞ and each pair of lines, ignoring crosses, are i lða1 ;a2 Þ l 1 ¼ l0 l 1 and i lða1 ;a2 Þ l1 ¼ l0 l1 , respectively. &
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
113
Example 1. For p ¼ 7, a B[4,2;7] design 1 2
1 2
1 1 3 4
2 3
2 4
3 5
3
5
4 6
4
5
6
6
7
5 7
7
6
7
along with the nested BIB design with sub-block size two with parameters ð4; 3; 3; 2; 3; 1Þ, 12 13 34 24
14 23
gives the following nested BIB design with sub-block size two with parameters ð7; 21; 12; 2; 6; 2Þ: 12 12
13 13
14 14
15
15
16
16
17
36 57
26 45
35 67
34
27
23
47
25
17 23
24 24
25 26
27
35
36
37
46 47
37 56
46 45
34
67
57
56
Next we shall consider another method of constructing nested BIB designs with sub-block size two, using the strongly equineighboured designs as defined in Martin and Eccleston (1991) and Street (1992). Strongly equineighboured designs have p treatments and b ¼ 12 spðp 1Þ ordered blocks, each of size k. Blocks made of columns of k b array satisfy: (i) Each treatment occurs sðp 1Þ times in rows j and k þ 1 j together, ja 12 ðk þ 1Þ, and 12 sðp 1Þ times in row 12 ðk þ 1Þ if k is odd. (ii) The columns in the 2 b submatrices obtained from rows j and k þ 1 j contain each unordered pair of distinct treatments exactly s times. (iii) The columns in the 2 b submatrices obtained from rows j and m, and from rows k þ 1 j and k þ 1 m ðj þ mak þ 1Þ, together contain each unordered pair of distinct treatments exactly 2s times. These designs are denoted by SENðk; s; pÞ. Theorem 2.2. The columns of SENð2t; s; pÞ, where tX1 and pX4, form nested BIB designs with sub-block size two with parameters ðp; b; tsðp 1Þ; t; 3ts; tsÞ. Proof. On writing the blocks of a SENð2t; s; pÞ as the columns of a 2t b array, the SEN conditions immediately implies: (i) each line occurs sðp 1Þ times in rows j and 2t þ 1 j, for j ¼ 1; 2; . . . ; t, that is, each line occurs tsðp 1Þ times; (ii) the t sets of columns in the 2 b submatrices obtained from rows j and 2t þ 1 j, j ¼ 1; 2; . . . ; t, where each set contain every unordered pair of distinct lines exactly s times, and (iii) the t sets of columns in the 2 b submatrices obtained from rows 2t þ 1 j and 2t þ 1 m ðj þ ma2t þ 1Þ, where each set collectively contain each unordered pair of distinct lines exactly 2s times. Therefore, from (ii) and (iii), each unordered pair of distinct lines occurs exactly ts þ 2tsð¼ 3tsÞ times in this arrangement. So the columns of SENð2t; s; pÞ form a BIBD with k ¼ 2t and l ¼ 3ts. On writing each ordered column ðd1 ; d2 ; . . . ; d2t1 ; d2t Þ0 , say, of a SENð2t; s; pÞ designs in the form d1 d2 .. . dt
d2t d2t1
dtþ1 ;
and using SEN property (ii) each cross will occur in exactly tsð¼ lÞ blocks. Hence we have the desired nested BIB designs with sub-block size two with parameters ðp; b; tsðp 1Þ; t; 3ts; tsÞ. &
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
114
Example 2. For p ¼ 6 and t ¼ 2, a SENð4; 1; 6Þ design is 1
1 1
1
1
2 2
2
2
3
3 3
4
4
5
3
2 2
4
3
4 6
3
5
6
6 2
3
3
1
4 2
5 6 3 4
2 5
5 6
5 5 3 4
6 5
1 6
1 4
4 1 5 6
1 5
2 6
4 6
that gives the following nested BIB design with sub-block size two with parameters ð6; 15; 10; 2; 2Þ: 12 13
14
15
16
23
24
25
34 25
26
24
35
45
56
36
26 34
35
36
45
46
56
15 16
46
12
13
23
14
3. Existence of nested BIB designs with sub-block size two for k ¼ 2 In this section, for k ¼ 2 the necessary conditions for a nested BIB designs with sub-block size two are shown to be sufficient. For k ¼ 2, the necessary conditions for a nested BIB designs with sub-block size two reduce to 4b ¼ pr;
lðp 1Þ ¼ r; and l ¼ 3l.
Therefore 4 must divide lpðp 1Þ. If ðx; yÞ denotes the greatest common divisor of x and y then the restrictions on p given l can easily be stated as below: If l 1 ðmod 2Þ;
then p 0 or 1 ðmod 4Þ,
(1)
If l 0 ðmod 2Þ;
then p is unrestricted.
(2)
Also we need pX4. If ðl; 4Þ ¼ d (say) and l ¼ qd for some integer q then we can form a nested BIB designs with sub-block size two with parameters ðp; b; r; 2; 3l; lÞ simply by taking an q-multiple of a nested BIB designs with sub-block size two for k ¼ 2 with p lines in b=q blocks, that is, by taking q copies of each block. Thus to show that the necessary conditions (1) and (2) are also sufficient, it suffices to demonstrate, with pX4, the existence of nested BIB designs with sub-block size two with parameters ðp; pðp 1Þ=4; p 1; 2; 3; 1Þ for every p 0 or 1 ðmod 4Þ, and of nested BIB designs with sub-block size two with parameters ðp; pðp 1Þ=2; 2ðp 1Þ; 2; 6; 2Þ for every p 2 or 3 ðmod 4Þ. The following lemma, given by Hanani (1961), will be needed. Lemma 3.1. If p 0 or 1 ðmod 4Þ and pX4, then p 2 B½K; 1, where K ¼ f4; 5; 8; 9; 12g. Lemma 3.2. Nested BIB designs with sub-block size two with the following parameters all exist: ð4; 3; 3; 2; 3; 1Þ, ð5; 5; 4; 2; 3; 1Þ, ð8; 14; 7; 2; 3; 1Þ, ð9; 18; 8; 2; 3; 1Þ, and ð12; 33; 11; 2; 3; 1Þ. Proof. The nested BIB designs with sub-block size two are: parameters ð4; 3; 3; 2; 3; 1Þ: 12 13 34 42
14 23
parameters ð5; 5; 4; 2; 3; 1Þ: 12 13
14
15
23
35 24
25
34
45
parameters ð8; 14; 7; 2; 3; 1Þ: 14 16 32 53
17 64
18 45
12 86
15 27
13 78
ARTICLE IN PRESS S.K. Srivastav, A. Shankar / Statistics & Probability Letters 77 (2007) 111–115
25 28 83 74
24 26 56 37
34 36 75 48
58 67
115
parameters ð9; 18; 8; 2; 3; 1Þ: 12 13
14 15
16 17
18
19
23
57 49
26 36
79 38
45
28
68
24 25
27 29
35 37
47
48
58
78 34
39 56
89 46
49
69
67
parameters ð12; 33; 11; 2; 3; 1Þ: 12
13
14
15
16
17
18
19
1A
1B
1C
6C
5B
29
47
24
8A
3B
6A
39
5C
79
23 7C
25 26 27 3A 58 35
28 9B
2A 7B
2B 4A
2C 89
34 8C
49 57
4C AB
56 AC
38 3C 46 9A
45 9C
48 7A
59 9B
5A 68
36 4B 67 BC
37 69 6B 79
Combining Lemmas 3.1 and 3.2, and Theorem 2.1 we have the following theorem which solves the problem presented by (1) for l ¼ 3 and l ¼ 1. Theorem 3.3. Nested BIB designs with sub-block size two with parameters ðp; ðpðp 1ÞÞ=4; ðp 1Þ; 2; 3; 1Þ exists for p 0 or 1 ðmod 4Þ, where pX4. Now to see the existence of (2) for l ¼ 6 and l ¼ 2, we shall state the following lemma whose proof can be found in Street (1992). Lemma 3.4. SENð4; 1; pÞ exists for all pX4. Therefore, using Lemma 3.4 and Theorem 2.2, we have the final result of the paper.
&
Theorem 3.5. Nested BIB designs with sub-block size two with parameters ðp; ðpðp 1ÞÞ=2; 2ðp 1Þ; 2; 6; 2) exists for all pX4. Acknowledgement The authors wish to thank the referee for his helpful comments and for suggestions for an improved presentation. References Choi, K.C., Gupta, S., 2000. On constructions of optimal complete diallel crosses. Utilitas Math. 58, 153–160. Das, A., Ghosh, D.K., 1999. Balanced incomplete block diallel cross designs. Statist. Appl. 1, 1–16. Dey, A., Midha, C.K., 1996. Optimal block designs for diallel crosses. Biometrika 83, 484–489. Gilbert, N.E., 1958. Diallel cross in plant breeding. Heredity 12, 477–492. Griffing, B., 1956. Concept of general and specific ability in relation to diallel crossing systems. Aust. J. Biol. Sci. 9, 463–493. Gupta, S., Kageyama, S., 1994. Optimal complete diallel crosses. Biometrika 81, 420–424. Hanani, H., 1961. The existence and construction of balanced incomplete block designs. Ann. Math. Statist. 32, 371–386. Hinkelmann, K., 1975. Design of Genetical experiments. In: Srivastava, J.N. (Ed.), A Survey of Statistical Design and Linear Models, vol. 42. North-Holland, Amsterdam, pp. 243–269. Martin, R.J., Eccleston, J.A., 1991. Optimal incomplete block designs for general dependence structures. J. Statist. Plann. Inf. 28, 67–81. Srivastav, S.K., Morgan, J.P., 1996. On the class of 2 2 balanced incomplete block designs with nested rows and columns. Commun. Statist. Theory Methods 25 (8), 1859–1870. Street, D.J., 1992. A note on strongly equineighboured designs. J. Statist. Plann. Inf. 30, 99–105.