Chapter 6 Balanced Incomplete Block Design (BIBD) The designs like CRD and RBD are the complete block designs. We now discuss the balanced incomplete block design (BIBD) and the partially balanced incomplete block design (PBIBD) which are the incomplete block designs. A balanced incomplete block design (BIBD) is an incomplete block design in which - b blocks have the same number k of plots each and - every treatment is replicated r times in the design. - Each treatment occurs at most once in a block, i.e., nij 0 or 1 where nij is the number of times the jth treatment occurs in ith block, ibjv1,2,..., ; 1,2,..., . - Every pair of treatments occurs together is of the b blocks. Five parameters denote such design as D(,bkvr ,,; ). The parameters bkvr,,, and are not chosen arbitrarily. They satisfy the following relations: (Ibkvr ) ()II ( v 1)( r k 1) (III ) b v(and hence r k ). Hencenkij for all i j nrij for all j j n and nn nn... nn for all jj ' 1,2,..., v . Obviously ij cannot be a constant for all 1'jij 2' jij bjj b' r j. So the design is not orthogonal. Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 1 Example of BIBD In the designDbkvr ( , ; , ; ) : consider b 10 ( sayB , ,..., B ), v6 ( sayT , ,..., T ), k3, r 5, 2 110 16 Blocks Treatments BTTT1 12 3 B2 TT12 T 4 B3 TT13 T 4 B4 TT14 T 6 B5 TT15 T 6 B6 TT23 T 6 B7 TT24 T 5 B8 TT25 T 6 B9 TT34 T 5 B10 TT34 T 6 Now we see how the conditions of BIBD are satisfied. (ibk ) 10 3 30 and vr 6 5 30 bk vr ()ii ( v 1)2 5 10and( r k 1)5 2 10 (vrk 1) ( 1) ()iii b 106 Even if the parameters satisfy the relations, it is not always possible to arrange the treatments in blocks to get the corresponding design. The necessary and sufficient conditions to be satisfied by the parameters for the existence of a BIBD are not known. The conditions (I)-(III) are some necessary condition only. The construction of such design depends on the actual arrangement of the treatments into blocks and this problem is handled in combinatorial mathematics. Tables are available, giving all the designs involving at most 20 replication and their method of construction. Theorem: ()Ibk vr ()(II v 1)( r k 1) ()III b v . Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 2 Proof: (I) Let N ():nbvij the incidence matrix Observing that the quantities ENE11bvand ENE 1 v ' b 1 are the scalars and the transpose of each other, we find their values. Consider nn11 21 nb 1 1 nn n 1 ENE (1,1,...,1)12 22b 2 11bv nn12vv n bv1 n1 j j n2 j (1,1,...,1) j nbj j k k (1,1,...,1) = bk. 1b k Similarly, nn11 21 nb 1 1 nn n ENE' (1,...,1)12 22b 2 11vb nn12vv n bv1 ni i r =(1,1,...,1) (1,1,...,1) vr 1v r niv i But ENE111bv ENE v' b 1 as both are scalars. Thus bk vr. Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 3 Proof: (II) Consider nn11 21 nbv 1 nn 11 12 n 1 nn n nn n NN' 12 22bv 2 21 22 2 nn12v v n bv nn b 12 b n bv 2 nnnnniiiiiv1121 ii i 2 nn n nn ii12 i 2 i 2 iv iii 2 nniv i12 nn iv i n iv ii i r r . (1) r 2 Since nij 1 or 0 as nij 1 or 0, 2 so nij Number of times j occurs in the design i = rjfor all 1,2,..., v of times occurs in the design and nnij ij'' Number of blocks in which jand j occurs together i = for alljj '. r 1 r 1 NNE' v1 r 1 rv(1) rv(1) [(1)].rvE (2) v1 rv(1) Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 4 Also nn11 12 n 1v 1 nn n 1 NNE'' N21 22 2v v1 nnbb12 n bv1 n1 j j n2 j N 'j nbj j nn11 21 nb 1 k nn n k 12 22b 2 nniv2 v n bv k b1 ni1 i n i2 k i niv i r r k r krEv1 (3) From 2 and 3 [rvEkrE ( 1)] vv11 or rv( 1) kr or (vrk 1) ( 1) Proof: (III) From (I), the determinant of NN' is detNN ' [ r ( v 1)]( r )v1 v1 rrk(1) r rk() r v1 0 because since if r from (II) that k = v. This contradicts the incompleteness of the design. Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 5 Thus NN' is a vv nonsingular matrix. Thus rank(') N N v . We know from matrix theory result rank() N rank (') N N so rank( N ) v But rank() N b , there being b rows in N. Thus vb . Interpretation of conditions of BIBD Interpretation of (I) bk = vr This condition is related to the total number of plots in an experiment. In our settings, there are k plots in each block and there are b blocks. So the total number of plots are bk . Further, there are v treatments and each treatment is replicated r times such that each treatment occurs atmost in one block. So total number of plots containing all the treatments is vr . Since both the statements counts the total number of plots, hence bk vr. Interpretation of (II) k kk(1) Each block has k plots. Thus the total pairs of plots in a block = . 2 2 There are b blocks. Thus the total pairs of plots such that each pair consists of plots within a block = kk(1) b . 2 v vv(1) There are v treatments, thus the total number of pairs of treatment = . 2 2 Each pair of treatment is replicated times, i.e., each pair of treatment occurs in blocks. vv(1) Thus the total number of pairs of plots within blocks must be . 2 kk(1) vv 1 Hence b 22 Using bk vr in this relation, we get rk 11. v Proof of (III) was given by Fisher but quite long, so not needed here. Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 6 Balancing in designs: There are two types of balancing – Variance balanced and efficiency balanced. We discuss the variance balancing now and the efficiency balancing later. Balanced Design (Variance Balanced): A connected design is said to be balanced (variance balanced) if all the elementary contrasts of the treatment effects can be estimated with the same precision. This definition does not hold for the disconnected design, as all the elementary contrasts are not estimable in this design. Proper Design: An incomplete block design with kk12.... kkb is called a proper design. Symmetric BIBD: A BIBD is called symmetrical if the number of blocks = number of treatments, i.e., bv . Since bv , so from bk vr kr. Thus the number of pairs of treatments common between any two blocks = . The determinant of NN' is NN'[(1)]() r v r v1 v1 rrk 1 r rk(). r v1 When BIBD is symmetric, b = v and then using bk vr, we have kr . Thus NN'(), N2 r21 r v so v1 Nrr(). 2 Since N is an integer, hence when v is an even number, ()r must be a perfect square. So ' NN'() r I EEvv11 , (')NN11'1 N N 1 ' IEEvv11, rr 2 '1 1 ' NIEEvv11. rr 2 Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 7 Post-multiplying both sides by N ' , we get ' NNr'( ) IEENN vv11 '. Hence in the case of a symmetric BIBD, any two blocks have treatment in common. Since BIBD is an incomplete block design. So every pair of treatment can occur at most once is a block, we must have vk . If vk , then it means that each treatment occurs once in every block which occurs in case of RBD. So in BIBD, always assume v > k . Similarly r. [If rvrkvk then ( 1) ( 1) which means that the design is RBD] Resolvable design: A block design of - b blocks in which - each of v treatments is replicated r times is said to be resolvable if b blocks can be divided into r sets of br/ blocks each, such that every treatment appears in each set precisely once. Obviously, in a resolvable design, b is a multiple of r. Theorem: If in a BIBD Dvbrk(,,, , ), b is divisible by r, then bvr1. Proof: Let bnr (where n 1 is a positive integer). For a BIBD, (vrk 1) ( 1) because vr bk (1)v orrvrnrk or (1)k or vnk (1)nk (1)k n 1 n. k 1 Sincenk 1 and 1, so n 1 is an integer. Since r has to be an integer. (1)n is also a positive integer. k 1 Analysis of Variance | Chapter 6 | Balanced Incomplete Block Design | Shalabh, IIT Kanpur 8 Now, if possible, let bvr1 nr v r 1 orrn ( 1) v 1 rk(1) rk (1) orrn ( 1) (because v 1 ) (1)n 1which is a contradiction as integer can not be less than one k 1 bvr1 is impossible. Thus the opposite is true. bvr1 holds correct. Intrablock analysis of BIBD: Consider the model yibjvij i j ij ; 1,2,..., ; 1,2,..., , where is the general mean effect; th i is the fixed additive i block effect; th j is the fixed additive j treatment effect and 2 ij is the i.i.d.