A simple construction procedure for resolvable incomplete block designs. for any number of treatments

Transcription

1 A imple contruction procedure for reolvable incomplete block deign for any number of treatment By MEENA KHARE and W. T. FEDERER Biometric Unit, Cornell Univerity, Ithaca, New York BU-666-M June, 1979 SUMMARY. A imple, traightforward procedure, which require no pecial table or generator, i preented for contructing reolvable incomplete block deign for v = pk, v = p 2 k,..., treatment, for k ; p, in incomplete block of ize k. Alo, it i hown how to obtain incomplete block deign for any v in block of ize k and k + 1. The procedure allow contruction of balanced incomplete block deign for p = k a prime number. For p = n not a prime number, incomplete block deign can be obtained by the procedure, but are not balanced. However, for p being the mallet prime multiple of n, p + 1 for v = n 2 p 2 + p + 1 for v = n 3,, arrangement can be obtained for which the ' occurrence of any treatment pair in the block i either zero or one. Thi i called a zero-one concurrence deign. Procedure are decribed for obtaining additional zero-one concurrence arrangement. It i hown that the efficiency of thee deign i maximum. Both intra-block and inter-block analye are decribed. Some key word: Zero-one concurrence; Variety cutting; Succeive diagonalizing; Efficiency.

2 -2-1. INTRODUCTION Each year, around the world everal hundred experiment on varietal trial, peticide, oil fumigant, medical trial, enory difference tet, etc. are deigned a incomplete block experiment deign. The number of entrie in an experiment i often larger than can be accommodated in the available block of relatively uniform experimental unit, and it i often deirable to have reolvable incomplete block deign, i.e., the incomplete block can be arranged in complete block for each replication of the entrie. Any attempt to create a complete file, or catalogue, of experiment deign for all ituation i doomed at the tart, owing to the ize of uch a file and in having a method for finding a deign, given that uch a file exited. The main file now available are thoe of Boe, Clatworthy, and Shrikhande (1954), Clatworthy (1973), and Cochran and Cox (1957). Source for contructing reolvable incomplete block deign with ome file are Yate (1936) for quare lattice, Harhbarger (1947, 1949, 1951) for rectangular lattice, Kempthorne (1952) and Federer (1955) for prime power lattice, Boe and Nair (1962) for two-replicate deign, and David (1967) and John, Wolock, and David (1972) on cyclic deign. Numerou other reference on variou apect of incomplete block deign may be found in Federer and Balaam (1972) under categorie E2 to E5. Since it i not feaible to contruct a complete file of incomplete block deign for all ituation, ame imple contruction procedure uable by an experimenter would be deirable. Two uch procedure are available in the literature, and a third one i preented herein. The firt one i by Patteron and William (1976); it require that a table of initial generating a-array be available; then, after ame manipulation on the a-array to produce an intermediate a-array, the incomplete block experiment deign :may be eaily obtained. The econd procedure, given by Jarrett and Hall (1978), require a et of initial

3 -3- block, but once thee are obtained, it i a imple procedure to contruct incomplete block deign. The procedure preented here require no table, array, or generator, and it lead to a wide cla of deign with efficiencie a high or higher than thoe given by Patteron and William (1976) or by Jarrett and Hall (1978). If block ize and the number of treatment are pecified, it may be neceary to ue one of thee three method in order to contruct an appropriate deign. However, if the block ize need only be in a range of value, ay a ~ k b, then the method propoed here may be ui table for mot ituation. Thi method involve firt writing down number l to n 2 in a quare array of n row and n column, number 1 to n 3 in a rectangular array of n 2 row and n column, etc. Then, ue i ucceively made of :main right diagonal, denoted a "ucceive diagonalizing" and certain number are deleted, i.e., "variety cutting", to reduce the total number of ymbol to the deired level. Both equal and unequal block ize are obtained. The method of "ucceive diagonalizing" produce a reolvable balanced incomplete block deign for n a prime number; the number of time a pair of varietie occur together, a concurrence, in thi deign i A = 1 From thee deign, and with "variety cutting", either pairwie balanced incomplete block deign or deign with zero or one concurrence, i.e., a pair of varietie occur together either zero or one time in the block, are contructed. When the number of entrie deleted from n 2, n 3, i not a multiple of n, unequal block ize reult. Variance heterogeneity may or :may not be encountered (ee, e.g., Federer and Ladipo (1978) and Shafiq and Federer (1979)) when block are of unequal ize, but it hould be negligible for block of ize k and k + 1 in everal type of experiment. In the econd ection an algorithm i given for contructing deign for v p 2 treatment, and conequently, for v = pk treatment in equal and unequal

4 -4- block ize for p a prime number. Then we dicu the contruction procedure for v = n 2 treatment, when n i a prime power a well a when n i not a prime power. Deign are contructed for any number of treatment n 2 In the third ection, a econd algorithm, which make ue of the firt algorithm, i given for contructing incomplete block deign for v = p 3 treatment, and cone~uently, for v = p 2 k treatment in e~ual and une~ual block ize, uing the "variety cutting" method. The method i then extended for n e~ual any poitive integer. Thee contruction procedure, uing the two algorithm, :may be ued to obtain deign for v = p 4, p5, p 6,, etc. treatment in block of ize p, and cone~uently, for v = p3k, p 4 k, p 5 k, etc. treatment in e~ual and une~ual block ize. Then we conider the efficiencie of ame of the contructed deign, and compare them with deign available in the literature. It i alo noted that additional replicate for n 2, ri3, treatment, n any integer, may be obtained for all ituation for which t orthogonal latin ~uare of order n exit. It i poible to obtain t + 2 replicate of a zero-one concurrence deign in thi cae. Statitical analye for the contructed deign follow directly from publihed theory. For completene, we have included the normal e~uation and olution of effect in matrix form. Both intra-block and inter-block e~uation and olution are given. 2. CONSTRUCTION PROCEDURES FOR A ZERO-ONE CONCURRENCE CLASS OF INCOMPLETE BLOCK DESIGNS FOR v TREATMENTS In thi ection, we give an algorithm for contructing reolvable balanced and partially balanced incomplete block deign for v = pk treatment in p block of ize k and for which the concurrence of pair of treatment are either zero or one. Then, we preent a cla of pairwie balanced (i.e., every pair of

5 -5- treatment occur together exactly A = 1 time and there i only one concurrence type) incomplete block deign for v = pk treatment in p 2 block of ize k and in k block of ize p Alo, deign with two block ize k and k + 1, for non-prime number, and additional plan for v = n 2 deign are preented in the lat part of thi ection. The procedure involve firt contructing a balanced incomplete block deign for v = p 2 treatment in block of p by a method called "ucceive diagonalizing 11 and then uing a method called "variety cutting" (an early reference to the method i given by Rao (1947); alo ee Federer (1955), page 424)) which involve deleting treatment from the et v = p 2 to obtain the reulting incomplete block deign For equal block ize k The "ucceive diagonalizing" method and conequent "variety cutting" method ha been ued and taught by the econd author ince he wa a graduate tudent at Iowa State Univerity in the late 1940'. It i a method for contructing reolvable balanced incomplete block (BIB) deign for v = p 2, p a prime number, in b = p(p + 1) block of ize p, for the number of replicate r = p +1, and for f.. = 1 Thi method i formalized below in Algorithm 2 1 and i exemplified in Example 2 1. ALGORITHM 2 1. The tep in contructing BIB deign with parameter v =p2, k=p, b=p(p+l) =p 2 +p, r=p+l, and f...=l for p ~prime number, by the method of "ucceive diagonalizing", are: (1) Write the number 1, 2,.., p 2 conecutively in~ quare array of p row and p column beginning in the left-hand corner of the firt row and ubequently continuing at the beginning of each row. Thi i quare :!:-_with row being the block.

6 -6- ( 2) Tranpoe the row and column of quare _! to obtain quare ~. ( 3) Take the :main right diagonal of quare ~ a the firt row of quare.1. Then, write the element of each column of quare g in ~ cyclic order in the ame column for quare J. (4) Repeat the proce in tep J on quare J to obtain quare 4. (p+l) Repeat the proce in tep J on quare p to obtain quare p + 1. (p+2) A ~ check on the previou tep, repeat the proce of tep J on the (p+l)t quare, and quare ~ hould reult. The row of the p +1 quare forw the (p 2 +p) block of p treatment. Example 2 1. The tep of Algorithm 2 1 for v = p 2 = 9 are: Square =p+l block block block block The contruction method ha a built in check for either a clerk or a computer program. If it i carried through the (p+2)nd tep, it produce quare 2. A illutrated below, p + 1 reolvable quare have been obtained to produce a BIB deign with parameter v = 9, k = 3, b = 12, r = 4, and A = 1, for which tatitical analye are readily available. If fewer than p + 1 replicate are deired, one may ue quare 1, quare 2,, quare n for n = 2, 3,, p + 1 replicate. If more than p + 1 replicate are deired, multiple of p + 1 quare plu n quare can be obtained for n < p + 1. pecific order to obtain the n quare. It i not neceary to proceed in any Statitical analye for uch deign

7 -7- (quare lattice deign) are readily available (ee Federer (1955), chapter XI, XIII, and Kempthorne (1952), chapter 22, 23). We now conider the cae where v = pk, k < p. From the BIB deign contructed by Algorithm 2.1, conider quare 2 to p+l. In thee quare, "cut out 11 (delete) the treatment number from pk + 1 to p 2 Thi 11 variety cutting 11 delete treatment and reduce the block ize to k < p In Example 2.1, for v = 6 = 3(2), the number 7, 8, and 9 are deleted, reulting in 3 rectangle with 3 incomplete block of ize 2. When p > 3 additional :multiple of p treatment :may be deleted. Thee deign have been called rectangular lattice deign (ee Harhbarger (1947, 1949, 1951); Robinon and Waton (1949)). Both Ke:rnpthorne (1952) and Federer (1955) have ued a different :method of contruction than preented here, but Ke:mpthorne (1952) in chapter 25, eentially give the above deign. Both intra-block and inter-block analye have been preented. A computer program for uing Algorithm 2.1 to contruct the p + 1 quare and the (p+2)nd quare a a check i given in Appendix C of the Cornell Univerity Mater Thei by M. Khare. The treatment number up to 31 2, the block ize k, and the replication number are given in Table II.l of thi thei. The poible treatment number up to v = 150 for p > 31 are alo included in the table For unequal block ize Uing all the p + 1 quare from the p 2 deign and the :method of "variety cutting 11 on thee quare, one obtain p quare for v = pk treatment with p incomplete block of equal ize k a decribed in ubection 2.1. If in addition, quare 1 with k incomplete block of ize p i included, then every pair of treatment will occur together A = 1 ti:rne in the incomplete block. For intance, in Example 2.1 all 4 quare can be ued for v = 3(2) = 6 treatment with quare 1 having 2 incomplete block of ize 3 and other having incomplete block of ize 2.

8 -8- Omitting the firt quare and uing "variety cutting", one can contruct deign having any number of treatment not equal to kp in incomplete block of ize k and k + 1 in the remaining p quare. To illutrate, reolvable block for v = 7 = 3(2) + 1 treatment in block of ize 2 and 3, from Example 2.1 are obtained by deleting (i.e., uing "variety cutting" :method) treatment 8 and 9 in quare 2, 3, and 4. There are (pt) = 3(1) = 3 block of ize k + 1 = 3 and p 2 - pt = 9-3(1) = 6 block of ize k = 2 in the reulting plan Deign for v = n 2 treatment, where n = mp, R... i the mallet prime m in.!! and ~ i any integer, or where n = p, ~ prime power Uing the firt p + 1 quare from the n + 1 quare obtained by uing the "ucceive diagonalizing" method on the quare of n 2 treatment in block of ize n, one can obtain incomplete block deign with zero and one concurrence. The concurrence number increae in group of p, if more quare are included. Therefore, if more quare are deired, one could duplicate the (p + 1) quare or one could ue the additional quare obtained from "ucceive diagonalizing". For example, at leat three quare can be obtained for all the even treatment number (n=2:m), four for n=3m treatment, and ix for n=5m treatment With (0,1) concurrence. ll1 The ame :method can be ued for the n = p, a prime power. For example, for v = (3 2 ) 2 = 9 2 = 81 treatment, 1 to 4 replicate are obtained with a zero or one concurrence, 5 to 7 with a zero, one or two concurrence, and 8 to 10 with a zero, one, two or three concurrence. Thee concurrence can be checked from the (NN')vxv matrice obtained for the normal equation, N' being the tranpoe of the incidence matrix N Additional (0,1) concurrence plan for v=n 2 deign The method of "ucceive diagonalizing" reult in the mallet prime of n, ay p, plu one reolvable zero-one concurrence replicate for v = n 2

9 -9- treatment. Then, by "variety cutting", one obtain :p + 1 replicate of a zero one concurrence deign for v = nk treatment. In order to obtain additional replicate, it i neceary that there be :more than :p - 1 orthogonal latin quare. Denoting a et of t :mutually orthogonal latin quare of order n a OL(n, t), we can contruct t + 2 replicate for v = n 2 treatment, and the reulting deign will be a zero-one concurrence deign. For example, for n = 10, there i an OL(lO, 2) et which :mean that we can obtain t + 2 = 4 replicate for a zeroone concurrence deign. For n = 12, there exit an OL(l2, 5) et which mean that we can obtain = 7 replicate of a zero-one concurrence deign. The number of orthogonal latin quare for variou n f a :prime :power are tabulated in Raghavarao (1971), Table For n = a :prime :power, there i a complete et, t = (n -1),. of orthogonal latin quare. Thu, for n equal to a :prime or :prime :power, n + 1 replicate of a zero-one concurrence deign are available. Incomplete block deign with unequal block ize may alo be obtained a decribed in ubection CONSTRUCTION PROCEDURE FOR A TWO-CONCURRENCE CLASS OF INCOMPLETE BLOCK DESIGNS FOR v = :p 2 k TREATMENTS, :p A PRIME NUMBER AND k ~ :p We firt give an algorithm for contructing reolvable incomplete block deign, making ue of Algorithm 2.1, for v =:p 2 k treatment in :p 2 block of ize k and for which the concurrence of :pair of treatment are either zero or one. Secondly, we :preent a cla of :pairwie balanced, but not variance balanced, incomplete block deign for v = :p 2 k treatment in block of unequal ize :p and k Deign for v = :pk + t, t = 1, 2,, (:p - 1), in unequal block of ize k and k + 1 are alo :preented in thi ection. In the third ubection, deign for non- :prime number ( n), including :prime :power, are :preented for v = n 3 treatment, n an integer. Then, additional :plan for rf deign are given.

10 -10- The procedure involve ~irt contructing a balanced incomplete block (BIB) deign ~or v = p 3 treatment, p a prime n:umber, in block o~ ize p uing the "ucceive diagonalizing" method a decribed ~n Algori tbm 3.1 and then applying the method o~ 'variety cutting" to reduce the treatment number. Thu, the parameter o~ the reolvable BIB deign are v =rf, k =p, r =p 2 +p + 1, b = rp 2 and A = 1. Federer (1955) and Kempthorne (1952) give the analye ~or partially a 4 2 m balanced and balanced deign when v =, k =, and r = 3,,, + + 1, = p a prime power, and for v = n 3, k = n, and r = 3, n an integer greater than one. Uing their procedure, one may obtain tatitical analye for r = t + 2 replicate whenever an OL(n,t) et exit ~or any integer n For equal block ize ~ The procedure for contructing reolvable balanced incomplete block (BIB) deign for v = p 2 treatment with parameter v =p 3, k =p, r = p 2 + p + 1, i formalized in Algorithm 3.1. ALGORITHM 3 1. The tep in contructing BIB deign with parameter v = p3, k =p, r =p 2 +p + 1, b =p 2 (p 2 +p + 1) and A= 1, for p ~prime n:umb.er ~ a follow:: (0) Write the v =p3 treatment conecutively in p 2 block of ize p a decribed in tep _! of Algorithm 2.1. Then aign number (1) through (p2 ) to p 2 block (i.e., ~) and arrange the block number in p + 1 group ~ obtained uing Algorithm 2.1 on p 2 treatment, conidering treatment number 1 through p 2 a block number 1 through p 2 Partition the p 2 block in p et of p block. (1) On the firt grouping of the p 2 block from tep Q, apply Algbritbm 2.1 to the p 2 treatment in each et of p block eparately to obtain p + 1 of the poible p 2 + p + 1 replicate (Rep) for v = p 3 treatment in block of ize p A a check, if the procedure i continued to the (p + 2)nd tep, Rep g will reult.

11 -11- (2) For the econd grouping of' p 2 block f'ro:m tep Q with p 3 treatment written properly in each block, apply Algorithm 2.1 to each et of' p block to obtain p + 1 new arrangement f'or p 3 treatment. The f'irt arrangement mut be o:mi tted a it i identical to Rep _! f'rom tep!, except f'or. :_ permutation of' the p 2 block of' ize p. Thi give. :_ et of' p additional replicate. (3) Repeat tep ~ on the third grouping of' p3 treatment f'ram tep Q to obtain an additional et of' p replicate, omitting the f'irt arrangement. (p+l) Repeat tep ~ on the (p+i)t grouping of' p 3 treatment f'rom tep Q to obtain. :_ new (p+l)th et of' p replicate, omitting the f'irt arrangement. Thu, there are p + 1 replicate f'ram tep 1 and p f'ro:m each of' the additional p tep, reulting in p 2 + p + l replicate to f'orm a BIB deign. If' f'ewer than p 2 + p + 1 replicate are deired, one :may ue replicate 1, 2,, r f'or r = 3, 4,..., p 2 + p + 1. If' :more than p 2 + p + 1 replicate are deired, the additional replicate can be obtained by electing any n arrangement, n $ p 2 +p + 1, f'ro:m the p 2 +p + 1 arrangement. Statitical analye are available uing the procedure in Federer (1955) and Kempthorne (1952). We now conider the cae where v = p 2 k, k < p From the BIB deign contructed by Algorithm 3.1, conider replicate (p + 2) to p 2 + p + 1 f'ro:m tep 2 through tep p + 1. On thee arrangement, ue "variety cutting" and delete treatment number p 2 k + 1 to p 3 Thi "variety cutting" reduce the block ize to k < p. The reulting deign i a zero-one concurrence one For unequal block ize Uing all but the f'irt p + 1 replicate f'ram p 3 deign and the :method of' "variety cutting" on them, one obtain p 2 replicate f'or v = p 2 k treatment, each with p 2 incomplete block of' equal ize k a decribed in ubection 3.1. If',

12 -12- in addition, replicate 1 through p + 1 f'ro:m group 1, each with pk incomplete block of' ize p, are included, then every pair of' treatment will occur together once in the incomplete block. Olni tting replicate 1 through p + 1 f'ro:m Group 1, one can contruct deign having any number of' treatment not equal to p 2 k, in incomplete block of' ize k and k + 1 in the remaining p 2 replicate. The concurrence of' pair of' treatment i either zero or one f'or thi deign Deign f'or v = rf3 treatment, where n = :mp, 12. i the :mallet prime in E: and ~ i any integer, or n = p:m, ~ prime power Obtain the f'irt p + 1 quare with zero-one concurrence f'ram an n 2 deign a decribed in ubection 2.3, f'or the arrangement of' n 2 incomplete block in p + 1 grouping f'or n 3 deign. Then, apply Algorithm 3.1 on thoe p + 1 grouping to obtain p + 1 replicate f'ram the f'irt grouping, p f'ro:m the econd.. grouping omitting the f'irt one, p f'rom the third grouping omitting the f'irt one,..., p f'ro:m (p +l)t grouping omitting the f'irt one. Thu, there are p (p ) + (p + 1) = p 2 + p + 1 replicate f'or v = rf3 of' n with zero-one concurrence. treatment in incomplete block The concurrence of' pair of' treatment increae a the number of' replicate increae beyond p 2 +p + 1. Uing the :method of' "variety cutting", one :may obtain deign f'or v = n 2 k treatment in unequal block of' ize k and k + 1. To illutrate, conider n = 2 2, i.e., v = 4 3 = 64 treatment where there are r = 7 replicate with incomplete block of' ize n = 4 which have zero-one concurrence of' treatment pair. For v = n 2 k = 16(3) = 48 in equal block of' ize 3, omit the 3 replicate f'ram the f'irt grouping and ue the :method of' "variety cutting" on the ret of' the replicate, i.e., delete treatment 49 through 64 in replicate 4, 5, 6 and 7. For v = 16(3) = 48 treatment in block ize of' 4 and 3, include the replicate f'ram the f'irt grouping and delete the block with treatment 49 through 64 and thu, there are 12(3) = 36 incomplete block of' ize 4 and 64 incomplete block of'

13 -13- ize 3. And finally, for v = 16(2) + 7 = 39 treatment, omit the 3 replicate from the firt grc:u:ping and ue the "variety cutting" method on treatment 40 through 64 in replicate 4 through 7, reulting in block of ize 2 and Additional (0,1) concurrence plan for v=n3 deign The method of "ucceive diagonalizing" reult in p 2 + p + 1 reolvable zero-one concurrence replicate for v = rf3 treatment, p being the mallet prime of n. Then by "variety cutting", one obtain :p 2 + p + 1 replicate of a zero-one concurrence deign for v = n 2 k treatment. In order to obtain additional replicate for a zero-one concurrence deign, one may ue a et of t mutually orthogonal latin quare of order n reulting in t + 2 replicate for a zero-one concurrence deign. Uing thee replicate for the t + 2 arrangement of block, a decribed in. tep 0 of Algorithm 3.1, one may obtain the replicate for v = rf3 treatment from thee grouping when n i not equal a prime power. power. There i a complete et of n 2 + n + l replicate available for n =a :prime For example, for n = 2 2 = 4, there exit t = 3 orthogonal latin quare and therefore a complete et of n 2 + n + 1 = l = 21 replicate are available for a zero-one concurrence deign. To obtain n 2 = 16 replicate for v = n 2 k = 16(3) = 48 treatment in block of ize 3, delete treatment number 49 through 64 from replicate in grouping 2 through 5. Incomplete block deign with unequal block ize may alo be obtained a decribed in ubection m EXTENSION TO v p 4, rf5,, p TREATMENTS, :m ANY INTEGER Uing Algorithm 2.1 and 3.1, one may obtain reolvable balanced incomplete deign for v = p4, rp,..., pm treatment in block of ize k = p There are (pm- 1)/(p -l) =pm-l +pm :p 2 + p + l arrangement available for balanced deign, uing appropriate arrangement of p:m-l block in a replicate. Then replicate for v = pm-~ treatment may be obtained in equal and unequal

14 -14- block ize for (0,1) concurrence by uing the method of "variety cutting" on kpm-l + 1 through :prn treatment in the balanced deign. rn-1 rn-2 There are :p + :p + + :p 3 + :p 2 arrangement available for obtaining replication with equal block ize k. m For v :p, m = 1, 2, 3,.. treatment in block of ize k :S:: :p, = 2, 3,..., m-1, one may obtain deign uing the "ucceive diagonalizing" :method and conequently the "variety cutting method", but thee deign will have concurrence more than (0,1). To obtain deign with (0,1,2) concurrence, one may m obtain available deign for v = :p treatment in block of ize k = :p fro:m the literature, and then ue "variety cutting" to reduce the number of treatment. For example, Federer and Robon (1952) give 10 arrangement for v = 25 treatment in block of ize 2 2 and for v = 35 in block of ize 3 2, and the method of "variety cutting" :may be ued to reduce the number of treatment. Uing their :procedure, one could contruct additional :plan for other :p and additional re:pli cate. 5. EFFICIENCY FACTORS FOR THE CONSTRUCTED DESIGNS The efficiency factor for a deign i calculated a the harmonic mean of the non-zero eigenvalue of the matrix C, a decribed e.g., by Jarrett and Hall (1978) and by Raghavarao (1971) p:p The matrix C for an equireplicated and unequal block ize deign i obtained a C vxv = ( ri - NK- ~' ) I r ' ( 5.1) where N' i the tranpoe of the incidence matrix Nand K=Diag(k 1,k 2,...,~); kl' k2,, ~ are the block ize for block 1, 2,, b, re :pecti vely. k. = k, K -1 i replaced by the calar 1 I k The efficiency factor i l When v-1 E = (v-1)1 L: lit.., where the "A.' are the non-zero eigenvalue of the matrix i=l l l c.

15 The :propoed deign with unequal block ize k and k + 1, are a efficient or more o than the a deign given by Patteron and William (1976) Efficiency factor for a (0,1) concurrence deign contructed for v 25 treatment in block of ize k and k + 1 in r = 4 replicate are given in Table 5.1. Table 5.1. Efficiency factor (r=4) v Block ize Efficiency factor , , , , , ,5.8o For 3 replicate deign for any v = n 2, n an even integer, our method :produce zero-one deign wherea that of Patteron and William (1976) :produce a zero-one-two (i.e., a(o,l,2)) deign. Efficiencie for ome of the contructed deign are given in Table 5.2. Our deign are a, or :more, efficient a other in the literature. Table 5.2. Efficiency factor for ome deign (r=3) Efficiency factor v k Our deign* a(o,l,2) deign ~~The efficiencie are the ame a for lattice deign. For :pairwie balanced deign for v = :pk treatment with block of ize :p and k in :p + 1 replicate, the efficiencie are given in Table 5. 3, for ome of the contructed deign.

16 -16- Table 5.3. Efficiency factor (r=6) v Block ize Efficiency factor 10 5, , ,4 7W The BIB a-deign do not exit, wherea our procedure tart with the contruction of BIB deign. It i upected that (0,1) concurrence deign are alway better than (0,1,2) concurrence deign. We know thi i true for variance balanced deign (Shafiq and Federer (1979)). The relative efficiency of baic balanced ternary to binary deign i given by Shafiq and Federer (1979) a E 3 /E2 = l- 2r 2/r(k-l), where r 2 i the number of occurrence of the number 2 in each row of NN' They alo preent reult demontrating that balanced incomplete block deign with a zero-one concurrence will be more efficient than zero-two concurrence deign. 6. INTRA-BLOCK AND INTER-BLOCK ANALYSES FOR INCOMPLETE BLOCK DESIGNS For completene we give the nor.mal equation for intra-block and inter-block analye. A repone equation for a reolvable incomplete block deign i (6.1) where Q.. =l if treatment j occur in hith incomplete block and 0 otherwie, h~j the Eh.. are NIID(O,cr 2 ), the leat quare equation in matrix for.m are ~J E :;v] [fl +: + ~xl] = [:] vxv -vxl -t (6.2) where K i a diagonal matrix of block ize, Nvxb = (~j) i the v X b incidence matrix for the deign, ~b i the b X 1 vector of incomplete block total, and

17 -l7- ~t i the v x l vector of treatment total. The leat q_uare olution with the uual retraint for l.l i :Y ' the overall mean, and of l.l +ph i yh ' the mean of the ~th complete block. The reduced intra-block normal eq_uation for the treatment effect ~ are (ay) Adding i JvXv' where Jvxv i a matrix of one, to the above equation to make them full rank matrice and auming that~-~- " =0, we obtain leat q_uare olu- J J tion a: A -l l -l ~ = ( ri - N K_ N J ) Q - vxv vxb-oxb bxv k vxv _ t (6.4) 2 -L_ l )-l Then, a E ( ri - NK ~ 1 + k J i ued to obtain variance of linear contrat,.._ I among~. J For the inter-block analyi, ubtitute (K+Icr~/a~)-l fork-lin (6.~-3) to obtain inter-block olution T. for the~., J J and then obtain the etimated variance of linear contrat of 1-. fro:m (ri- N(K +Icr 2 /a~)-~ 1 + (k+ a 2 /a~)-ljf\ where B 2 -J E ~ E ~ E and cr~ are ubtituted for a~ and a~, repectively. Here, we conider the ~i to be NIID(O, a~) REFERENCES BOSE, R. C., CLATWORTHY, W. H. and SHRIKHANDE, S. S. (l954). Table of partially balanced deign with two-aociate clae. N. f Agr. Expt. Sta. Tech. Bull. l07. BOSE, R. C. and NAIR, K. R. (l962). Reolvable incomplete block deign with two replication. Sankbya 24, CLATWORTHY, W. H. (l973). Table of Two-Aociate-Cla Partially Balanced Deign. Appl. Math. Ser. 63. Wahington, D.C.: National Bureau of Standard. COCHRAN, w. G. and COX, G. M. (l957). Experimental Deign. Second edition. New York: Wiley.

18 -18- DAVID, H. A. (1967). Reolvable cyclic deign. Sarikhya ~ ~ FEDERER, w. T. (1955). Experimental Deign: Theory and Application. New York: Macmillan. FEDERER, w. T. and BALAAM, L. N. (1972). Bibliography on Experiment and Treatment Deign Pre Edinburgh: Oliver and Boyd. FEDERER, w. T. and LADIPO, (1978). A component of variance due to competition. Autrl. J. Statit. 29(2), FEDERER, w. T. and ROBSON, D. S. (1952). General theory of prime-power lattice deign. VI. Incomplete block deign and analyi for p 5 varietie in block of p 2 plot. Cornell Univ. Agr. ~ Sta. Mem HARSHBARGER, B. (1947). Rectangular lattice. Va. Agr. Expt. Sta. Mem. _!, HARSHBARGER, B. (1949). Triple rectangular lattice. Biometric 2, HARSHBARGER, B. (1951). Nearbalance rectangular lattice. Va. ~ Sci., New Serie, ~' JARRETT, R. G. and HALL, w. B. (1978). Generalized cyclic incomplete block deign. Biometrika 65(2), JOHN, J. A., WOLOCK, F. W. and DAVID, H. A. (1972). Cyclic deign. Appl. Math. Ser. 62. Wahington, D.C.: Bureau of Standard. KEMPTHORNE, 0. (1952). The Deign and Analyi of Experiment. New York: Wiley. PATTERSON, H. D. and WILLIAMS, E. R. (1976). A new cla of reolvable incomplete block deign. Biometrika 63(1), RAGHAVARAO, D. (1971). Contruction and Combinatorial Problem in Deign of Experiment. New York: Wiley. RAO, C. R. (1947). General method of analyi for incomplete block deign. J. Am. Statit. Aoc. 42, ROBINSON, H. F. and WATSON, G. S. (1949). An analyi of imple and triple rectangular lattice deign. N. Q Agr. Sta. Tech. Bull. 88, SHAFIQ, M. and FEDERER, w. T. (1979). Generalized N-ary balanced block deign. Biometrika 66(1), YATES, F. (1936). A new method of arranging variety trial involving a large number of varietie. ~ Agr. Sci. 26, Revied 6/79