CONSTRUCTION OF SEMI-REGULAR GRAPH DESIGNS

Transcription

1 CHAPTER 4 CONSTRUCTION OF SEMI-REGULAR GRAPH DESIGNS 4.1 Introduction 4.2 Historical Review 4.3 Preliminary Results 4.4 Methods of Construction of SRG Designs Construction using BIB Designs with b = r Construction using BIB Designs with b = 2r λ 4.5 Numerical Examples 4.6 Dual Designs 4.1 INTRODUCTION In this chapter, the construction of Semi-Regular Graph (SRG) Designs obtained from BIB designs, whose parameters satisfy certain conditions on the availability of SRG designs, is discussed. Two different methods for the construction of semi-regular graph designs are carried out. In the first method, the SRG design is constructed by augmenting a new treatment in each of the b blocks of the BIB design, and augmenting one more block which contains all the (v + 1) treatments. In the second method, the SRG design is constructed by augmenting a new treatment in each of the b blocks of the BIB design, and augmenting (r λ + 1) blocks which contain all the v treatments of the BIB design only. Further, it is verified that these semi-regular graph (SRG) designs belong to a particular class of partially efficiency balanced (PEB) designs with two efficiency 58

2 classes. This shows that a particular class of PEB designs with two efficiency classes is also semi-regular graph designs. 4.2 HISTORICAL REVIEW Reinforced designs (Das, 1958), augmented designs (Federer, 1961), supplemented balanced designs (Pearce, 1960), nearly balanced designs (Nigam, 1976) and orthogonally supplemented balanced designs (Calinski, 1971) are some examples of designs constructed by augmenting a set of new treatments. Jacroux (1985) extended the definition of a regular graph design given in Mitchell and John (1976) to that of a SRG design and derived some sufficient conditions for the existence of a SRG design which is optimal under a given type 1 criterion. Jacroux made use of the general result obtained by Cheng (1978) for the solutions to optimization problems involving SRG designs. 4.3 PRELIMINARY RESULTS For the evaluation of the eigen values of the M o -matrix, the following Lemma (Mukerjee and Kageyama, 1990) is useful. Lemma 4.3.1(Mukerjee and Kageyama) Let u, s 1, s 2,..., s u be positive integers, and consider the s s matrix a 1 I s1 + b 11 J s1 s 1 b 12 J s1 s 2... b 1u J s1 s u b A = 21 J s2 s 1 a 2 I s2 + b 22 J s2 s 2... b 2u J s2 s u b u1 J sus1 b u2 J sus2... a u I su + b uu Js u s u where s = s 1 + s s u and u u matrix B = (b ij ) is symmetric. Then the eigen values of A are a i with multiplicity s i 1(1 i u) and µ 1,..., µ u, where 59

3 µ 1,..., µ u are the eigen values of = D a + D 1/2 s BDs 1/2 = diag(s 1, s 2,..., s u ), D 1/2 s = diag (s 1/2 1, s 1/2 2,..., s 1/2 u )., D a = diag(a 1,..., a u ), D s 4.4 METHODS OF CONSTRUCTION OF SEMI-REGULAR GRAPH DESIGNS In this section, two methods of construction of semi-regular graph designs, which are obtained using BIB designs, are discussed. These SRG designs are also happened to be PEB designs with two efficiency classes CONSTRUCTION USING BIB DESIGNS WITH b = r + 1 Theorem Let d(v, b, k, r, λ) be a symmetrical BIB design and let its parameters b and r satisfy the relation b = r+1, where b be the number of blocks and r the number of replications of the treatments. Let N d be the incidence matrix of the BIB design d, and N d+ be the incidence matrix of the augmented design d +. The incidence matrix N d+ defined as: N v b J v 1 N d+ = J 1 b J 1 1 gives a semi-regular graph (SRG) design with parameters v + = v + 1, b + = b + 1, r + = (r + 1,..., r + 1, b + 1), k }{{} + = (k + 1,.., k + 1, v + 1), λ }{{} 1+ = λ + 1 and v times b times λ 2+ = r + 1. Proof Let the concurrence matrix of the augmented design d + be: 60

4 N d+ N d+ = (r λ)i v + (λ + 1)J vv (r + 1)J v 1 (r + 1)J 1 v (b + 1) (i) The differences of the diagonal elements of the concurrence matrix N d+ N d+ are: (b + 1) (r + 1) = b r = 1 (since b = r + 1) (ii) Now for a BIB design, λ(v 1) = r(k 1) Using v, b = r + 1 and k = r; λ = r 1 The differences of the off-diagonal elements of the concurrence matrix N d+ N d+ are: (r + 1) (λ + 1) = r λ = 1 (since λ = r 1) That is, the diagonal elements and the off-diagonal elements of the concurrence matrix N d+ N d+ are differing by at most one. Hence the design d + is a semi-regular graph (SRG) design. This completes the proof. Corollary The class of semi-regular graph (SRG) designs constructed using Theorem is in fact PEB designs with two associate classes. 61

5 Proof This Corollary is proved using the M o -matrix of the design d +, and for that, the C-matrix and M-matrix of the design d + are obtained. The C-matrix of the design d + is obtained from the expression: C = diag(r 1, r 2,..., r v+1 ) - (N d+ diag (1/k 1, 1/k 2,..., 1/k b+1 ) N d+ ) where r 1, r 2,... are the treatment replications, k 1, k 2,... are the block sizes, and is of the form: [(r + 1) ( r λ)]i k+1 v ( λ + 1 )J k+1 v+1 vv ( r + 1 )J k+1 v+1 v 1 C = ( r + 1 )J k+1 v+1 1 v (b + 1) b 1 k+1 b+1 The M-matrix of the design d + is obtained from the expression: M = diag(1/r 1, 1/r 2,..., 1/r v+1 ) (N d+ diag (1/k 1, 1/k 2,..., 1/k b+1 ) N d+ ), and is of the form: r λ I (r+1)(k+1) v + 1 ( λ + 1 )J 1 r+1 k+1 v+1 vv ( r + 1 )J r+1 k+1 v+1 v 1 M = ( 1 )( r + 1 )J 1 b+1 k+1 v+1 1 v ( b + 1 ) b+1 k+1 b+1 The M o -matrix of the design d + is obtained from the expression: M o = diag(1/r 1, 1/r 2,..., 1/r v+1 ) (N d+ diag (1/k 1, 1/k 2,..., 1/k b+1 ) N d+ ) - J (r 1, r 2,..., r v+1 ) /n where J is a column vector of one s of order v + 1 and n is the total number of units, and is of the form: 62

6 r λ I 1 (r+1)(k+1) v + [ ( λ + 1 ) r+1]j 1 r+1 k+1 v+1 m vv] [ ( r + 1 ) b+1]j r+1 k+1 v+1 m v 1 (( 1 )( r + 1 ) r+1)j 1 b+1 k+1 v+1 m 1 v ( b + 1 ) b+1 b+1 k+1 b+1 m where m = (r + 1)v + (b + 1) Using Lemma 4.3.1, the eigen values of M o are µ 1 = r λ (r+1)(k+1) with multiplicity (v 1), and µ 2 = trace(m o ) (v 1)µ 1 with multiplicity one. So, the design d + is partially efficiency balanced (PEB) with efficiency factor (1 µ 1 ) with multiplicity (v 1), and (1 µ 2 ) with multiplicity one. This completes the proof CONSTRUCTION USING BIB DESIGNS WITH b = 2r λ Theorem Let d(v, b, k, r, λ) be a symmetrical BIB design and its parameters satisfy the relation b = 2r λ. Let N d be the incidence matrix of the BIB design d, and N d+ be the incidence matrix of the augmented design d +. The incidence matrix N d+ defined as: N v b J v (r λ+1) N d+ = J 1 b 0 1 (r λ+1) gives a semi-regular graph (SRG) design with parameters v + = v + 1, b + = b + (r λ + 1), r + = (2r λ + 1,..., 2r λ + 1, b), }{{} v times k + = (k + 1, k + 1,..., k + 1, }{{} v,..., v }{{} ), λ 1+ = r + 1 and λ 2+ = r. b times (r λ+1)times 63

7 Proof Let the concurrence matrix of the augmented design d + be: (r λ)i v + (r + 1)J vv rj v 1 N d+ N d+ = rj 1 v b (i) The differences of the diagonal elements of the concurrence matrix N d+ N d+ are: (r λ) + (r + 1) b = 2r λ + 1 b = 1 (since b = 2r λ) (ii) The differences of the off-diagonal elements of the concurrence matrix N d+ N d+ are: (r + 1) r = 1 That is, the diagonal elements and the off-diagonal elements of the the concurrence matrix N d+ N d+ are differing by at most one. Hence, the design d + is a semi-regular graph (SRG) design. This completes the proof. Corollary The class of SRG designs constructed using Theorem is in fact partially efficiency balanced (PEB) designs with two efficiency classes. 64

8 Proof This Corollary is proved using the M o -matrix of the design d +, and for that, the C-matrix and M-matrix of the design d + are obtained. The C-matrix of the design d + is obtained from the expression: C = diag(r 1, r 2,..., r v+1 ) - (N d+ diag (1/k 1, 1/k 2,..., 1/k b+(r λ+1) ) N d+ ) where r 1, r 2,... are the treatment replications, k 1, k 2,... are the block sizes, and is of the form: [(2r λ + 1) ( r λ)]i k+1 v ( λ + r λ+1 )J k+1 v vv r J k+1 v 1 C = r J k+1 1 v b b k+1 The M-matrix of the design d + is obtained from the expression: M = diag(1/r 1, 1/r 2,..., 1/r v+1 ) (N d+ diag (1/k 1, 1/k 2,..., 1/k b+(r λ+1) ) N d+ ), and is of the form: r λ I (k+1)m v + 1 ( λ + r λ+1 )J m k+1 v vv ( r )( 1 )J k+1 m v 1 M = r J 1 b(k+1) 1 v k+1 where m = 2r λ + 1. The M o -matrix of the design d + is given by the following expression: M o = diag(1/r 1, 1/r 2,..., 1/r v+1 ) (N d+ diag (1/k 1, 1/k 2,..., 1/k b+(r λ+1) ) N d+ ) - J (r 1, r 2,..., r v+1 ) /n 65

9 where J is a column vector of one s of order v + 1 and n is the total number of units, and is of the form: r λ I (k+1)m v + [ 1 ( λ + r λ+1 ) m ]J m k+1 v (vm+b) vv [( r )( 1 ) b ]J k+1 m vm+b v 1 r ( m )J 1 b(k+1) vm+b 1 v b k+1 vm+b where m = 2r λ + 1. Using Lemma 4.3.1, the eigen values of M o are µ 1 = r λ (k+1)(2r λ+1) with multiplicity (v 1), and µ 2 = trace(m o ) (v 1)µ 1 with multiplicity one. So, the design d + is partially efficiency balanced (PEB) with efficiency factor (1 µ 1 ) with multiplicity (v 1), and (1 µ 2 ) with multiplicity one. This completes the proof. 4.5 NUMERICAL EXAMPLES As an application of the above two theorems, two numerical examples are given in this section. Example Consider the symmetrical BIB design d(4, 4, 3, 3, 2) such that b = r +1, shown below: Table 4.5.1: Symmetrical BIB design with blocks shown in columns d = Using Theorem 4.4.1, the augmented design d + with parameters v + = 5, b + = 5, r + = (4,..., 4, 5), k + = (4,..., 4, 5), λ 1+ = 3 and λ 2+ = 4 is obtained, and its incidence matrix N + is expressed as: 66

10 N d+ = And the concurrence matrix of the design d + is given by the expression: N d+ N d+ = This class of designs is SRG designs, because one can find that all the diagonal and the off-diagonal elements of the concurrence matrix N d+ N d+ are differing by at most one. In order to find the M o -matrix of the design d + and its eigen values, the C- matrix and M-matrix of the design d + are obtained. The C-matrix of the design d + is given by: C = The M-matrix of the design d + is given by: 67

11 M = The M o -matrix of the design d + is obtained as follows: M o = The eigen values of M o are µ 1 = with multiplicity 3 (= v 1) and µ 2 = with multiplicity one. So, the design d + is partially efficiency balanced (PEB) with efficiency factors: (i) 1 - µ 1 = with multiplicity (v - 1) = 3, and (ii) 1 - µ 2 = with multiplicity 1. The following Table gives the examples of unequi-replicated SRG designs with unequal block sizes constructed using BIB designs. 68

12 Table showing the parameters of BIB designs and SRG Designs Parameters of BIB designs Parameters of Constructed SRG designs v b r k λ v + b + r + k + λ 1+ λ (4,...,4, 5) (4,...,4, 5) (5,...,5, 6) (5,...,5, 6) (6,...,6, 7) (6,...,6, 7) (7,...,7, 8) (7,...,7, 8) (8,...,8, 9) (8,...,8, 9) (9,...,9, 10) (9,...,9, 10) (10,...,10, 11) (10,...,10, 11) (11,...,11, 12) (11,...,11, 12) Further, it is found that all the above SRG designs belong to a particular class of PEB designs with two efficiency classes. Example Consider the symmetrical BIB design d(4, 4, 3, 3, 2) such that b = 2r λ, shown below: Table 4.5.3: Symmetrical BIB design with blocks shown in columns d = Using Theorem 4.4.2, the augmented design d + with parameters v + = 5, b + = 6, r + = (5,..., 5, 4), k + = 4, λ 1+ = 3 and λ 2+ = 4 is obtained, and its incidence matrix N d+ is expressed as: 69

13 N d+ = And the concurrence matrix of the design d + is given by the expression: N d+ N d+ = This class of designs is SRG designs, because one can find that all the diagonal and the off-diagonal elements of the concurrence matrix N d+ N d+ are differing by at most one. In order to find the M o -matrix of the design d + and its eigen values, the C- matrix and M-matrix of the design d + are obtained. The C-matrix of the design d + is given by: C = The M-matrix of the design d + is given by: 70

14 M = The M o -matrix of the design d + is obtained as follows: M o = The eigen values of M o are µ 1 = 0.05 with multiplicity 3 (= v 1) and µ 2 = 0.10 with multiplicity one. So, the design d + is partially efficiency balanced (PEB) with efficiency factors: (i) 1 - µ 1 = 0.95 with multiplicity (v - 1) = 3, and (ii) 1 - µ 2 = 0.90 with multiplicity 1. The following Table gives the examples of unequi-replicated proper SRG designs constructed using BIB designs. 71

15 Table showing the parameters of BIB designs and SRG Designs Parameters of BIB designs Parameters of Constructed SRG designs v b r k λ v + b + r + k + λ 1+ λ (5,...,5, 4) (6,...,6, 5) (7,...,7, 6) (8,...,8, 7) (9,...,9, 8) (10,...,10, 9) (11,...,11, 10) (12,...,12, 11) Further, it is found that all the above SRG designs belong to a particular class of PEB designs with two efficiency classes. 4.6 DUAL DESIGNS In this section, the dual designs of the SRG designs constructed using BIB designs with b = 2r λ, in section are considered. Theorem The dual designs du of the class of SRG designs d +, constructed from a symmetrical BIB design with b = 2r λ by augmenting a new treatment in each of the b blocks of the BIB design, and augmenting (r λ + 1) blocks which contain all the v treatments of the BIB design only, by using Theorem 4.4.2, with parameters v du = b +, b du = v +, r du = k + and k du = r + are RG designs. 72

16 Proof The incidence matrix N d+ of the constructed SRG design d + is given by: N v b J v (r λ+1) N d+ = J 1 b 0 1 (r λ+1) And the concurrence matrix of the dual design of the constructed SRG design d + is defined as: (k λ)i b + (λ + 1)J bb kj b (r λ+1) N d+ N d+ = kj (r λ+1) b vj (r λ+1)(r λ+1) (i) The differences of the diagonal elements of the concurrence matrix N d+ N d+ are: v (k + 1) = b (k + 1), (since v = b) = 2r λ (r + 1), (since b = 2r λ and k = r) = r λ 1 = r (r 1) 1, (since λ = r 1) = 0 (ii) The differences of the off-diagonal elements of the concurrence matrix N d+ N d+ are: (a) v k = b r = 2r λ r, (since b = 2r λ) = r λ = r (r 1), (since λ = r 1) = 1 (b) v (λ + 1) = b (λ + 1) = 2r λ λ 1), (since b = 2r λ) 73

17 = 2r 2λ - 1 = 2r 2(r 1) 1, (since λ = r 1) = 1 That is, all the diagonal elements are equal, and the off-diagonal elements of the concurrence matrix N d+ N d+ are differing by at most one. Hence, the dual designs du of the constructed SRG designs d + are regular graph (RG) designs. This completes the proof. Corollary The dual designs du of the class of designs constructed using Theorem are in fact partially efficiency balanced (PEB) designs with 3 efficiency classes. Proof This Corollary is proved using the M o -matrix of the dual design du, and for that the C du -matrix and M du -matrix of the dual design of the design d + are obtained. The C du -matrix of the dual design du of the design d + is obtained from the expression: C du = diag(r 1, r 2,...) - (N d+ diag (1/k 1, 1/k 2,...) N d+ ) where r 1, r 2,... are the treatment replications, k 1, k 2,... are the block sizes of the dual design, and is of the form: 74

18 C du = (k + 1 k λ m )I b ( λ m + 1 b )J bb k m J b (r λ+1) k m J (r λ+1) b vi (r λ+1) v m J (r λ+1) (r λ+1) where m = 2r λ + 1. The M du -matrix of the dual design of the design d + is obtained from the expression: M du = diag(1/r 1, 1/r 2,...) (N d+ diag (1/k 1, 1/k 2,...) N d+ ), and is of the form: M du = [ k λ where m = 2r λ + 1. m(k+1) ]I b [ λ + 1 m(k+1) b(k+1) ]J bb k m(k+1) J b (r λ+1) k J 1 mv (r λ+1) b J m (r λ+1) (r λ+1) The M o -matrix of the dual design of the design d + is given by the following expression: M o = diag(1/r 1, 1/r 2,...) (N d+ diag (1/k 1, 1/k 2,...) N d+ ) - J (r 1, r 2,...)/n where J is a column vector of one s and n is the total number of units, and is of the form: ( k λ m(k+1) )I b + ( λ + 1 k+1 m(k+1) b(k+1) (k+1)b+hv )J bb ( k m(k+1) v (k+1)b+hv )J b h ( k mv k+1 (k+1)b+hv )J h b ( 1 m v (k+1)b+hv )J h h where m = 2r λ + 1 and h = r λ + 1. Using Lemma 4.3.1, the eigen values of M o are µ 1 = k λ (k+1)(2r λ+1) with multiplicity (b 1), µ 2 = 0 with multiplicity (r λ) and µ 3 = trace(m o ) (b 1)µ 1 with multiplicity one. So, the dual design is partially efficiency balanced (PEB) 75

19 with efficiency factors (1 µ 1 ) with multiplicity (b 1), (1 µ 2 ) with multiplicity (r λ) and (1 µ 3 ) with multiplicity one. This completes the proof. Example Consider the constructed SRG design d + with parameters v + = 5, b + = 6, r + = (5,..., 5, 4), k + = 4, λ 1+ = 3 and λ 2+ = 4, given in Example Using Theorem the dual RG design du with parameters v du = 6, b du = 5, r du = 4, k du = (5,..., 5, 4), λ 1du = 3 and λ 2du = 4 is obtained. The concurrence matrix of the dual design du of the constructed design d + is given by the expression: N d+ N d+ = That is, all the diagonal elements are equal, and the off-diagonal elements of the the concurrence matrix N d+ N d+ are differing by at most one. Hence, the dual design du of the SRG design d + is a regular graph (RG) design. In order to find the M o -matrix of the dual design of the design d + and its eigen values, the C du -matrix and M du -matrix of the dual design are obained. The C du -matrix of the dual design of the design d + is given by: 76

20 C du = The M du -matrix of the dual design of the design d + is given by: M du = The M o -matrix of the dual design of the design d + is obtained as follows: M o = The eigen values of M o are µ 1 = 0.05 with multiplicity 3 (= b 1), µ 2 = 0.0 with multiplicity (r λ) and µ 3 = 0.10 with multiplicity one. So, the dual design du of the design d + is partially efficiency balanced (PEB) with efficiency factors: (i) 1 - µ 1 = 0.95 with multiplicity (b 1) = 3, (ii) 1 - µ 2 = 1.00 with multiplicity (r λ) = 1, and 77

21 (iii) 1 - µ 3 = 0.90 with multiplicity 1. The following Table gives the examples of dual designs obtained from unequi-replicated proper SRG designs constructed using BIB designs. Table showing the parameters of constructed SRG designs and its dual RG designs Parameters of Constructed SRG designs Parameters of Dual RG designs v + b + r + k + λ 1+ λ 2+ v b r k λ 1 λ (5,...,5, 4) (5,...,5, 4) (6,...,6, 5) (6,...,6, 5) (7,...,7, 6) (7,...,7, 6) (8,...,8, 7) (8,...,8, 7) (9,...,9, 8) (9,...,9, 8) (10,...,10, 9) (10,...,10, 9) (11,...,11, 10) (11,...,11, 10) (12,...,12, 11) (12,...,12, 11) Further, it is found that all the above dual RG designs belong to a particular class of PEB designs with three efficiency classes. 78

22 References [1] Calinski, T. (1971). On some desirable patterns in block designs (with discussion). Biometrics, 27, [2] Cheng, C.S. (1978). Optimality of certain asymmetrical experimental designs. Ann. Statist., 6, [3] Das, M.N. (1958). On re-inforced incomplete block designs. J.Ind. Soc.Agri. Stat., 10, [4] Federer, W.T. (1961). Augmented designs with one-way elimination of heterogeneity. Biometrics, 17, [5] Jacroux, M. (1985). Some sufficient conditions for the type 1 optimality of block designs. J.Statist.Plann.Inf., 11, [6] Mukerjee, R. and Kageyama, S. (1990). Robustness of group divisible designs. Commu. Statist.-Theor. Meth., 19(9), [7] Nigam, A.K. (1976). Nearly balanced incomplete block designs. Sankhya, B 38, [8] Pearce, S.C. (1960). Supplemented balance, Biometrics, 47,

23 Index augmented design, 59, 60, 63, 64, 66, 69 BIB designs, 58, 60, 61, 63, 66, 68, 69, 71, 72, 78 symmetrical BIB design, 60, 63, 66, 69, 72 Calinski, T., 59 Cheng, C.S., 59 concurrence matrix, 60, 61, 64, 67, 70, 73, 74, 76 Das, M.N., 59 eigen values, 59, 60, 63, 66, 67, 68, 70, 71, 75, 76, 77 Federer, W.T., 59 incidence matrix, 60, 63, 66, 69, 73 Jacroux, M., 59 Kageyama, S., 59 M o -matrix, 59, 62, 63, 65, 66, 67, 68, 70, 71, 74, 75, 76, 77 Mukerjee, R., 59 Nigam, A.K., 59 Pearce, S.C., 59 PEB designs, 58, 59, 60, 61, 63, 64, 66, 68, 69, 71, 72, 74, 75, 77, 78 SRG designs, 58, 59, 60, 61, 63, 64, 67, 68, 69, 70, 71, 72, 73, 74, 76, 78 80