BLOCK DESIGNS WITH NESTED ROWS AND COLUMNS Rajener Parsa I.A.S.R.I., Lirary Avenue, New Delhi 110 012 rajener@iasri.res.in 1. Introuction For experimental situations where there are two cross-classifie factors causing heterogeneity in the experimental material an are neste within the locking factor, the lock esigns with neste rows an columns have een evelope. Before giving efinition an characterization of properties of these esigns, let us have a look at the examples of some such experimental situations. Experimental Situation 1: Consier the case of Agroforestry experiments involving evaluation of crop species/varieties for their comparative performance when grown along with a given tree species. The tree species woul e aing to the heterogeneity in the growing conitions for the crop species / varieties eing evaluate since (i) their stan (oth numer an vigour) in ifferent plots (locks) will e sujecte to high variation an (ii) the tree row will have column (irectional) effects an row (tree or tree row segment) effects neste within it. These row an column effects within a lock are crosse etween themselves. Experimental Situation 2: Consier the case of animal nutrition experiments where numer of lactations has een taken as locking factor. However, age an stage of lactation within animals of same numer of lactations may contriute significantly to error variance an these two factors are crosse with each other. Therefore, for such situations, within locking factor, two cross-classifie factors, age (rows) an stage of lactation (columns) are neste. Experimental Situations 3: Consier an experiment, which was conucte using a lock esign. The experiment was harveste lockwise. To meet the ojectives of the experiment, the harveste samples are to e analyse for their contents in the laoratory y ifferent technicians over ifferent time perios. The factors, technicians an time perios are crossclassifie with each other. Therefore, to control the variations ue to technicians an time perios, this has to e taken as a situation of lock esigns with neste rows (technicians) an columns (time perios). Therefore, in this kin of situation the experimental units are roaly classifie into locks such that within each lock the experimental units can e arrange in p rows an q columns. Therefore, we have a total of pq experimental units. To cope with the aove type of situations, repeate lattice - square esigns, (Cochran an Cox, 1957, pp. 483-497,Yates1940) were introuce, where each square can e consiere a lock (complete) within which are neste two other factors, enote y rows an columns, so that one can eliminate two sources of variation within each lock. However, the repeate lattice squares have some limitations viz. 1. Availale only for v = s 2 treatments when s is a prime or power ; 2. Complete Block is require.
Block Designs With Neste Rows an Columns Srivastava (1978) generalise this iea y giving a variance alance esign with v = 5 in (2 2) squares - a esign when v s 2. v = 5, = 5, r = 2, p = 2,q = 2, n = 20 1 2 2 3 3 4 4 5 5 1 3 4 4 5 5 1 1 2 2 3 The information matrix for treatment comparisons for this esign is C = (5/4) (I - (1/5) 11 ). Singh an Dey (1979) generalize the iea of repeate lattice - square esigns to a class of esigns, calle Block Designs with Neste Rows an Columns. They terme these esigns as Balance Incomplete Block Designs with Neste Rows an Columns (BIB-RC Design). They gave efinition, properties an a metho of analysis of these esigns. Several methos of construction of these esigns were also given. 2. BIB-RC Designs Definition 1: A lock esign with neste rows an columns with v treatments an sets (locks), each lock containing p rows an q columns (pq<v) is sai to e a BIB-RC esign if the following conitions are satisfie: (i) every treatment occurs at most once in a lock; (ii) given a pair of treatments(i,j) p λ r( i, j ) + qλc( i, j ) λ( i, j ) = λ (constant) where λ r( i, j ), λ c( i, j ) an λ( i, j ) enote the numer of locks in which treatment i an j occur together in the same row, same column an elsewhere respectively an λ is a constant inepenent of i an j. It is easy to see that in a BIB-RC esign, every treatment occurs in exactly r locks, where r = λ (v)/{(p)(q)}. Two methos of construction of BIB-RC esigns have een escrie y Singh an Dey (1979). These are escrie elow. Construction of BIB-RC Designs Metho 2.1: If is a prime or prime power, a BIB esign with parameters v = 2, = (+1), r = +1, k =, λ =1 exists. This esign can e split into (+1) sets of rows an columns each such that the set of {rows} alone or {columns} alone form a BIB esign. Example 2.1: For =3, a BIB esign with parameters v= 9, = 12, r = 4, k = 3, λ = 1 exists. Then using the aove metho we get a BIB-RC Design with parameters v = 9, = 4, p = 3, q = 3, r = 4, λ = 1. The arrangement is as follows: Block Block-2 Block-3 Block-4 1 2 3 1 4 7 1 6 8 1 9 5 4 5 6 2 5 8 9 2 4 6 2 7 7 8 9 3 6 9 5 7 3 8 4 3 II-204
Block Designs With Neste Rows an Columns Consier now another BIB esign with parameters v,, r, k = 2, λ. Then, using the aove arrangement of the BIBD { 2, (+1,,1)}, we get a BIB-RC esign. This leas to the following result. Theorem 2.1: The existence of a BIB esign {v,, r, k = 2, λ }, eing a prime or prime power, implies the existence of a BIB-RC esign with parameters v = v, s = (+1), r = r (+1), p = q =, λ = () λ. To illustrate the aove result, consier the following example. Example 2.2: Let = 2. The BIB esign (v = 4, = 6, r = 3, k = 3, λ =1) can e resolve into three sets such that the {rows} an {columns}, treate as locks form a BIB esign each. The arrangement is as shown elow. 1 2 1 3 1 4 3 4 4 2 2 3 Example 2.3: Consier the BIB esign (v = 7 =, r = 4 = k, λ = 2), the locks for this esign eing otaine from the initial lock (1, 4, 6, 7) mo 7. Writing the arrangement aove with the lock contents of each of the locks of the secon BIB esign, we get the following BIB-RC esign. 1 4 1 6 1 7 2 5 2 7 2 1 3 6 3 1 3 2 6 7 7 4 4 6 7 1 1 5 5 7 1 2 2 6 6 1 4 7 4 2 4 3 5 1 5 3 5 4 6 2 6 4 6 5 2 3 3 7 7 2 3 4 4 1 1 3 4 5 5 2 2 4 7 3 7 5 7 6 5 6 6 3 3 5 The aove esign has parameters v = 7, = 21, r = 12, p = q = 2, λ = 2. Metho 2.2: Suppose there exists a BIB esign, for which a solution ase on initial locks is availale. Suppose further that these initial locks can e arrange in rows an columns in such a manner that the ifferences arising from rows an columns from all the initial locks are symmetrically repeate. Then, y eveloping these initial sets of rows an columns, we can get a BIB-RC esign. The following example illustrates this construction metho. Example 2.4: Consier a BIB esign with parameters v = 13, = 26, r = 12, k = 6, λ = 5, a solution of which can e otaine y eveloping the initial locks (1,3,9,4,10,12) an (2,5,6,7,8,11) mo 13. Now, arrange locks into 2 rows an 3 columns as uner: 1 3 9 7 8 11 12 10 4 6 5 2 Developing these two initial row-column sets, we otain a solution for the BIB-RC esign with parameters v = 13, = 26, r = 12, p = 2, q = 3, λ = 2. II-205
Block Designs With Neste Rows an Columns Several other construction methos of BIB-RC esigns have een propose y several authors in literature. For an excellent review on these methos of construction, one way refer to Sreenath (1998). The case of the complete lock esigns, with pq = v, was consiere y Cheng (1986) an were calle as Balance Complete Block Designs with neste Rows an Columns with same parameters an enote as BCBRCD (v,, r, p, q, λ). Concept of BIB-RC esigns was also extene to partially alance incomplete lock esigns with neste rows an columns as well, repeate lattice square esigns fall in this category. Some methos of construction of BIB-RC esigns have also een given where rows (alone) or columns (alone) oes not form a BIB esign. The example given y Srivastava (1978) elongs to this case. 3. Analysis of Block Designs with neste rows an columns In its most general form, suppose v treatments are to e compare via n experimental units arrange in locks such that j th lock is of size k j = p j q j ; (1), an there are p j rows an q j columns in the j th lock. Let n ijlm enote the numer of times i th treatment is applie in the m th column an l th row of j th lock; m=1(1)q j ; l =1(1)p j ; j = (1); i=1(1)v. Let us consier the following matrices for otaining reuce normal equations. N = ((n ij.. )): v incience matrix of treatments Vs locks; N 1 = ((n ijl.)) : v j= 1 N 2 =((n ij.m )):~v j= 1 q p j j incience matrix of treatments Vs rows; incience~matrix~of~treatments~vs~columns. + + Q = qji p, P = pji q enote respectively the j j= 1 j j q iagonal matrices of row sizes an column sizes. j q j K = Diag(p 1 q 1,..., p q ), is the iagonal matrix of lock sizes. R = Diag(r 1,...,r v ), where r i, i=1(1)v is the replication numer of i th treatment. p p an The reuce normal equations are Cτ = Q, where τ is the v 1 vector of treatment effects. The coefficient matrix C of reuce normal equations for estimating linear functions of treatment effects using a lock esign with neste rows an columns is C = R N Q N N P N + N K N. (3.1) 1 1 2 C = R N 2P N 2 L* ( = N 1Q N 1 N K N C is symmetric non-negative efinite matrix with rows an columns sums zero an for a connecte esign Rank (C ) = v. The v 1 vector of ajuste totals is 2 ). j II-206
Block Designs With Neste Rows an Columns 1 1 1 Q = T N Q M N P L + N K B, where T 1 2 ( v 1 ), M( p j 1), L ( q j 1), B ( 1) note respectively the vectors of totals of treatments, rows, columns an locks. The ajuste treatment sum of squares is given y Q C Q. For proper lock esign set up i.e., when k = pq C = R 1 / q ) N N (1 / p ) N N + (1 / pq ) N N. (3.2) ( 1 1 2 2 C = R 1 / p ) N N L( = (1 / q ) N N (1 / pq ) N N. (3.3) ( 2 2 2 2 λv 1 For a BIB-RC esign C = I 11. Therefore for a BIB-RC esign the ajuste pq v v pq 2 treatment sum of squares is Q i. λv i= 1 4. Variance-Balance Designs Definition 4.1: A lock esign with neste rows an columns is sai to e variance alance if the esign estimates each elementary contrast of treatment effects with the same variance. Theorem 4.1: A necessary an sufficient conition for lock esign with neste rows an columns to e variance alance is that its C matrix is of the form C = θ ( I v - v 1v 1 v ) (4.1) where θ is the unique non-zero eigenvalue of C with multiplicity (v). For a BIB-RC esign θ = (λv)/(pq) = (p)(q)/(v). In inary an proper setting c ii = r i (p- 1)(q)/pq an the conition C1 = 0 an constancy of off iagonal elements for a BIB-RC esign ensures that all r i s are equal. Therefore in the inary an proper setting c ii = r(p)(q- 1)/pq an hence trace is fixe for a BIB-RC esign. Therefore in the class of inary lock esigns with neste rows an columns, a BIB-RC esign or a BCB-RC esign, whenever existent is universally optimal. However, in a general class of lock esigns with neste rows an columns D (v,, p, q), BIB-RC esign may not e optimal. To e more clear, consier an experimental situation where 4 treatments are to e compare via 24 experimental units arrange in 6 locks such that there are 2 rows an 2 columns in each of the locks. From a numer of esigns elonging to this class of esigns let us choose the following two esigns. Design 1: BIB-RC Design (v = 4, = 6, r = 6, p = 2,q = 2, λ = 2) 1 2 1 3 1 4 1 2 1 3 1 4 3 4 4 2 2 3 3 4 4 2 2 3 C = (8/4) (I - (1/4) 11 ). Var( ˆ τ i ˆ τ j ) = σ 2 i,2,3,4 II-207
Block Designs With Neste Rows an Columns Design 2: Variance Balance Designs with neste rows an columns Design (v = 4, = 6, r = 6, p = 2,q = 2, λ = 2) 1 2 1 3 1 4 2 3 2 4 3 4 2 1 3 1 4 1 3 2 4 2 4 3 C = (4) (I - (1/4) 11 ). Var( ˆ τi ˆ τ j ) = σ 2 /2 i j =1,2,3,4 Now we see that in over class of connecte proper lock esigns with neste rows an columns BIB-RC esigns are not optimal. Various optimality aspects of lock esigns with neste rows an columns were stuie y Bagchi, Mukhopahyay an Sinha (1990), Chang an Notz (1988), Chang an Notz (1989), Chang an Notz (1990) an Morgan an Uin (1993) over D (v,, p, q). They terme variance-alance esigns with neste rows an columns as Balance Neste Row-Column Designs (BN-RC esigns). 5. Balance Neste Row -Column esign Definition 5.1: A lock esign in v treatments arrange in locks such that each lock contains p rows an q columns is sai to e a BN-RC esign if following (i) an (ii) conitions of theorem are satisfie i.e. if (i) L= 1 N ' 1 N 1 - N N = O p (ii) N 2 is the incience matrix of a Balance Block Design. Whenever a BN-RC esign D (v,, p, q) exists, it is universally optimal. If pq v i.e. BIB-RC exists, then p v an hence, for a BN-RC esign with pq v, λv 1 ' q(p 1) 1 ' C 1 = ( I v 1 v 1 v ) = ( I v 1 v 1 v ) p v v 1 v If BIB-RC esign exists λv 1 (p )(q ) 1 C 2 = ( I 11 ) = ( I 11 ) pq v v 1 v q C 1 = C2 q - 1 Relative efficiency of BIB-RC esign as compare to an BN-RC esign is = (q)/q Theorem 5.2: A lock esign with neste rows an columns for v treatments, locks of p-rows an q-columns is universally optimal if 1. the columns form a variance alance (Balance lock ) esign. 2. a treatment appears in a row of a lock it appears equally frequently. In such a setting esign is a lock esign, with columns as locks an all the results availale for lock esign setting are applicale to lock esigns with neste rows an columns provie if a treatment appears in a row of a lock, it appears equally frequently in all rows of that lock. II-208
Block Designs With Neste Rows an Columns Constructions of BN-RC Design Metho 5.3: Arrange the contents of each of the locks of a BIB esign v,, r, k, λ, in the form of a latin square. We get a BN - RC esign which is universally optimal over D (v,, p = k, q = k). Example 5.1: Consier a BIB esign D (4,6,3,2,1) with lock contents as (1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Following the proceure aove we get a BN - RC esign with parameters v = 4, = 6, p = 2, q = 2. The layout of the esign is as given elow: 1 2 1 3 1 4 2 3 2 4 3 4 2 1 3 1 4 1 3 2 4 2 4 3 Metho 5.2: Existence of a BIB esign with parameters v,, r, k, λ an a Youen Square esign YSD (k, p). Then on arranging the contents of each of the locks of BIBD in the form of a Youen Square, we get a BN - RC esign (v,, p, q = k). Example 5.3: Let a BIB esign D (4, 4, 3, 3, 2) with lock contents as (1, 2, 3); (1, 2, 4); (1, 3, 4); (2, 3, 4). Let a Youen Square esign (3, 2) exists. A B C B C A. Following the proceure of metho 3.2, we get a BN - RC esign with parameters v = 4, = 4, p = 2, q = 3. 1 2 3 1 2 4 1 3 4 2 3 4 2 3 1 2 4 1 3 4 1 3 4 2 Remark: A BN - RC esign with parameters v, = v(v)/2, p = 2 = q always exists. The BN-RC esign with parameters v = 5, = 10, r = 8, p = 2, q = 2 is 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5 2 1 3 1 4 1 5 1 3 2 4 2 5 2 4 3 5 3 5 4 The optimality aspects of these esigns uner non-proper lock esign setup have een stuie y Parsa, Gupta an Voss (2001). Several methos of construction of optimal nonproper lock esigns with neste rows an columns have een given y Chakraorty (1996) an Parsa, Gupta an Voss (2001). References Bagchi, S., Mukhopahyay, A.C. an Sinha, B.K. (1990). A search for optimal neste rowcolumn esigns. Sankhya B52, 9304. Chakraorty,A.K. (1996). Stuies on lock esigns with neste rows an columns. Unpulishe Ph.D. Thesis, IARI, New Delhi. II-209
Block Designs With Neste Rows an Columns Chang, Y.J. an Notz, W. (1988). Optimal Block esigns with neste rows an Columns. Tech. Report, 405, Department of Statistics, The Ohio State University. Chang, Y.J. an Notz, W. (1989). Some universal, Type1, Type2 an E-optimal lock esigns with neste rows an columns. Tech. Report, 435, Department of Statistics, The Ohio State University. Chang, Y.J. an Notz, W. (1990). Metho of constructing universally optimal lock esigns with neste rows an columns. Utilitas Mathematica, 38, 263-276. Cheng, C.S.(1986). A metho for constructing alance incomplete lock esigns with neste rows an columns. Biometrika, 73, 695-700. Cochran, W.. an Cox, G.M. (1957). Experimental esigns. 2 n Eition, John Wiley an Sons, New York. Morgan, J.P. an Uin, N. (1993). Optimality an construction of neste row column esigns. Journal of Statistical Planning an Inference, 37, 81-93. Parsa, R., Gupta, V.K. an Voss, D. (2001). Optimal neste row an column esigns. Journal of Inian Society of Agricultural Statistics, 54(2), 244-257. Singh,M. an Dey, A. (1979). Block esigns with neste rows an columns. Biometrika, 66, 321-366. Sreenath, P.R. (1998). Construction of alance incomplete lock esigns with neste rows an columns. IASRI Pulication. Srivastava, J.N.(1978). Statistical esign of agricultural experiments. J. Inian Soc. Agric. Statist., 30, 10. Yates, F. (1940). Lattice squares. Jour. Agr. Sci., 30, 672-687. II-210