PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS

Transcription

1 PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS V.K. Sharma I.A.S.R.I., Lbrary Avenue, New Delh-00. Introducton Balanced ncomplete block desgns, though have many optmal propertes, do not ft well to many expermental stuatons as these desgns requre a large number of replcatons. Moreover, these desgns are not avalable for all numbers of treatments and block szes. To overcome these dffcultes a class of bnary, equreplcate and proper desgns that are called Partally Balanced Incomplete Block (PBIB) desgns were ntroduced. In these desgns the varance of every estmated elementary contrast among treatment effects s not the same and hence the name PBIB desgns. The defnton of PBIB desgns s based on the assocaton scheme we, therefore, frst gve the concept of assocaton scheme. Assocaton Scheme Gven v treatment symbols,,.., v, a relaton satsfyng the followng condtons s called an m-class assocaton scheme (m ). () Any two symbols are ether st, nd,..., or m th assocates; the relaton of assocaton beng symmetrc,.e., f the symbol α s the th assocate of β, then β s the th assocate of α. () Each symbol α has n th assocates, the number n beng ndependent of α, () If any two symbols α and β are th assocates, then the number of symbols that are j th assocates of α and k th assocate of β s p jk and s ndependent of the par of th assocates α and β. The numbers v, n and p jk (, j, k =,,..., m) are called the parameters of the assocaton scheme. Example: Consder treatment symbols denoted by numbers to. Let us a form 3 group of 4 symbols each as follows: (,,3,4), (5,6,7,8), (9,0,,). We now defne () any two treatment symbols are frst assocates f they belong to the same group, () any two treatment symbols are second assocates f they belong to the dfferent groups. Here v=, n = 3, n = 8. p =, p = 0 p = 0 p = 8 p = 0, p = 3 p = 3 p = 4. It can be verfed that these values of (, j, k=,) reman unchanged for any choce of p jk two frst or second assocates. These parameters are usually wrtten n the form of the followng matrces. P 0 = ( p j ) =,. 0 3 P = ( p ) 0 8 j = 3 4 Gven an assocaton scheme for the v symbols, we now defne a PBIB desgn as follows:

2 PBIB Desgn Gven an assocaton scheme wth m classes (m ) we have a PBIB desgn wth m assocate classes based on the assocaton scheme, f the v treatment symbols can be arranged nto b blocks, such that () Every symbol occurs at most once n a block. () Every symbol occurs n exactly r blocks. () If two symbols are th assocates, then they occur together n λ blocks, the number λ beng ndependent of the partcular par of th assocates α and β. The numbers v,b,r,k,λ (=,,...,m) are called the parameters of the desgn. It can be easly seen that m nλ = r( k ) = Two-class assocaton schemes and the two-assocate PBIB desgns have been extensvely studed n the lterature and are smple to use. We, therefore, deal wth some of these schemes and the desgns based on them n the followng sectons:. Some Two-class Assocaton Schemes Here we brefly defne some of the well known two-assocate class assocaton schemes and gve ther parameters. () Group Dvsble (GD) Assocaton Scheme Let v = mn symbols be arranged nto m groups of n symbols each. A par of symbols belongng to the same group s frst assocates and a par of symbols belongng to dfferent groups s second assocates. Ths defnes a GD assocaton scheme and t has the followng parameters: n = n-, n = n(m-) P n 0 =, p nm 0 n = (.) 0 ( ) n n( m ) A PBIB () desgn based on a GD scheme s called a GD desgn. A GD desgn s called () sngular f r-λ = 0; () sem-regular, f r- λ > 0, and rk-vλ = 0; () regular, f r-λ > 0 and rk-vλ > 0. ()Trangular assocaton scheme Fll n an n x n square array wth v = n(n-)/ symbols n such a way that the postons n the prncpal dagonal (runnng from upper left-hand to lower rght-hand corner) are left blank, the n(n-)/ postons above the prncpal dagonal are flled by so many symbols, and the n(n-)/ postons below the prncpal dagonal are flled so that the array s symmetrcal about the prncpal dagonal. For any symbol, now, the frst assocates are the treatments that occur n the same row (or n the same column) wth n the above arrangement. The 78

3 remanng symbols are defned as second assocates of. The trangular assocaton scheme has the followng parameters: P = n = (n-), n = (n-)(n-3)/, n n 3 ( n 3)( n 4) n, P 3 = 4 n 8 ( n 4)( n 5) n 8 (.) () Latn-Square, L (=,3,...,) - assocaton scheme Let v = s symbols be arranged nto an s x s square array and - mutually orthogonal latn squares (MOLS) be super-mposed on the array. Two symbols are defned to be frst assocates f they occur n the same row, or column of the array or n poston occuped by the same letter n any of the latn squares. Ths defnes L assocaton scheme that has the followng parameters: n = (s-), n = (s-+) (s-), P = ( )( ) + s ( s + )( ) ( s + )( ) ( s + )( s ) P = ( ) s ( ) s ( ) ( s )( s ) + s (.3) 3. Methods of Constructon It may be mentoned that as n the case of BIB desgns, the complementary desgn of a PBIB wth parameters v, b, r, k, λ s also a PBIB desgn havng the same assocaton scheme wth the parameters v * = v, b * = b, r * = b-r, k * = v-k, λ * = b-r+λ. We descrbe below some methods of constructng PBIB desgns based on the above assocaton schemes. 3. Group Dvsble Desgns () Let D be an ncomplete block desgn. The desgn D * obtaned from D by nterchangng the role of treatments and blocks s called the dual desgn of D. For example, f D wth parameters, v = 4, b = 6, r = 3, k = s I, II,3 Blocks III,4 IV,3 V,4 VI 3,4 then ts dual D * wth parameters v * =6, b * =4, r * =, k * =3 s I, II, III I, IV, V 3 II, IV, VI 4 III, V, VI Let D be an affne resolvable BIB desgn wth parameters v, b, r, k, λ. Obvously, then any two blocks of the same set have no common treatment whle any two blocks from dfferent sets have k /v common treatments. Therefore, D *, the dual desgn of D s a GD desgn wth the followng parameters: v * = b, b * =v, r * =k, k * =r, m=r, n=b/r, λ =0, λ =k /v, 79

4 where the frst assocates of the treatments of D * are the blocks belongng to the same group n D. The followng seres of BIB desgn are affne-resolvable: (a) (b) v = s, b = s(s+), r = s+, k = s, λ=; v = s 3, b = s(s +s+), r=s +s+, k=s, λ=s+; s beng a prme or power of a prme n (a) and (b). These yeld the followng seres of GD desgns: ( a ) v * =s(s+), b * =s, r * =s, k * =s+, m=s+, n=s, λ =0, λ =; ( b ) v * =s(s +s+), b * =s 3, r * =s, k * =s +s+, m=s +s+, n=s, λ =0, λ =s; s beng a prme or power of a prme n ( a)' and( b)'. () Let D be a BIB desgn wth parameters v = m,b,r,k, λ. Obtan a desgn D * from D by replacng the th tretment (=,,...,v) n D by n new treatment symbols,,..., n. Evdently, D * s a group dvsble desgn wth the followng parameters: v * = mn, b * = b, r * = r, k * = nk, m, n, λ =r, λ =λ. (3) By omttng the blocks n whch a partcular treatment, say θ, occurs from a BIB desgn wth the parameters v,b,r,k,λ =, we obtan a GD desgn consstng of the remanng blocks wth the parameters: v * = v-, b * = b-r, r * = r-, k * = k, m = r, n = k-, λ = 0, λ =, Here a par of treatments s frst assocates f they occur wth θ n the orgnal desgn, and second assocate, other wse. In the r blocks n whch θ occurs, we fnd that on omttng θ they become dsjont and the remanng v- = r(k-) treatment symbols form r groups each of k- symbols. Ths defnes the GD assocaton scheme on whch the GD desgn s based. 3. Trangular Desgns () An obvous method of constructon of a trangular desgn s to take the rows (or columns) of the assocaton scheme as blocks of the desgn. Such a desgn wll have the followng parameters: v = n(n-)/, b = n, r =, k = n-, λ =, λ = 0. It may be noted that ths trangular desgn can also be obtaned by dualsng rreducble BIB desgn wth parameters gven below: v = n, b = n( n )/, r = n, k =, λ =. () If there exsts a BIB desgn wth the parameters v = (n-)(n-)/, b = n(n-)/, r = n, k = n-, λ =, then by dualzng t a trangular desgn wth the parameters v* = n(n-)/, b* = (n-)(n-)/, r* = n-, k* = n, λ =, λ = can be constructed. 80

5 3.3 L desgns 3.3. Methods of Constructng L desgns () Let v be a squared number,.e. v = s. We wrte the s treatment symbols n the followng form: 3... s s+ s+ s+3... s A = (s-)s+ (s-)s+ (s-)s+3... s An ncomplete block desgn wth rows of A and columns of A as blocks s called a smple lattce desgn. A smple lattce has v = s, b = s, r =, k = s. It s easy to see that a smple lattce s an L desgn wth λ =, λ =0. () If s s a prme or a prme power, we can construct a seres of L desgns as follows: we supermpose each latn square of the complete set of (s-) mutually orthogonal latn squares on A defned above and form blocks wth treatments whch fall under the same letter of a latn square. Ths gves us an L desgns wth parameters: v = s, b = s(s-), r = s-, k = s, λ = 0, λ = Method of Constructng L (>) desgns A smple lattce s also called a square lattce. These desgns have v=s. Let there exst -(- <s-) mutually orthogonal latn squares of orders s, and let us supermpose each of these squares on A as defned above n Sec. 3.3.(). Treatng the rows of A, columns of A, symbols of A fallng under same letter of st, nd,..., (-)-th latn square as blocks, we get s blocks each of sze s. These s blocks consttute an L desgn havng the parameters: v = s, b = s, r =, k = s, λ =, λ = 0 A large number of two-assocate desgns can be found n Bose, Clatworthy and Shrkhande (954) and Clatworthy (973). 4. Analyss of PBIB () desgns Let D be a PBIB desgn wth two assocate classes havng the parameters v, b, r, k, λ, n,,,j,k =, (4.) p jk Consder the fxed effect model Observaton (y) = General mean (µ) + Treatment effect (τ) + Block effect (β) +Random error (4.) where random errors are assumed to be dentcally ndependently normally dstrbuted wth mean zero and constant varance σ. Mnmzaton of the resdual sum of squares wth respect to the constants ncluded n the model yelds a set of normal equatons whch n vew of the restrctons Στ =0 and Σβ j =0 can be solved to gve 8

6 [ ] τ$ = kbq AS ( Q ) /( AB A B ), =,,..., v (4.3) wth A = r (k-) + λ, A = λ - λ, B = (λ - λ ), p B = r (k-) + λ + (λ - λ ) ( p p) and S (Q ) s the sum of the adjusted totals of those treatments whch are the frst assocates of treatment. The adjusted treatment sum of squares s v Σ $τ Q (= SSTA, say) = ka ( Var ( τ$ τ$ + B) σ m) = = v, say ( AB AB ) f and m are the frst assocates kb σ σ = AB AB = v, say f and m are the second assocates. (4.4) (4.5) The average varance of all estmated elementary treatment contrasts s gven by A.V. = (n v + n v ) / (n + n ) (4.6) Detals of the analyss are llustrated wth the help of followng example. EXAMPLE A varetal tral on wheat crop was conducted usng a two-assocate class P.B.I.B. desgn. The parameters of the desgn are v = b = 9, r = k = 3, λ =, λ = 0, n = 6, n =, P = ( ), P = ( ). 0 0 The data alongwth the block contents are gven below: Blocks Block contents/yelds per plot I (3) 59 (8) 56 (4) 53 II () 35 (7) 33 (4) 40 III () 48 (7) 4 (5) 4 IV (7) 46 (8) 56 (9) 5 V (4) 6 (5) 6 (6) 55 VI (3) 5 (9) 53 (5) 48 VII () 54 (8) 58 (6) 6 VIII () 45 (9) 46 (6) 47 IX () 3 () 7 (3) 35 Carry out the analyss. Analyss: Compute Grand Total (G) = = 96 No. of observatons (n) = 7 8

7 Grand Mean ( y ) = G/n = 96/7 = 48 No. of replcatons = 3 Block sze (k) = 3 CF = G /n = (96) /7 = 608 () () (3) (4) (5) (6) (7) (8) (9) Treat/ T B j Block No s n B j B j / Q τˆ Block whch treat j() j() ()- no. (6) Adj. treat. mean occurs k ,7, / ,8, / ,6, / ,, / ,5, / ,7, / ,3, / ,4, / ,6, / Note : T = =33, etc.; B = = 68, etc. Total of Blocks, n whch treatment occurs, B j = = 399, etc. j() Assocates of Dfferent Treatments (,3,5,6,7,8) Frst assocate (4,9) Second assocate (,3,4,6,7,9) Frst assocate (5,8) Second assocate 3 (,,4,5,8,9) Frst assocate (6,7) Second assocate (,3,5,6,7,8) Frst assocate (,9) Second assocate (,3,4,6,7,9) Frst assocate (,8) Second assocate (,,4,5,8,9) Frst assocate 83

8 (3,7) Second assocate 7 (,,4,5,8,9) Frst assocate (3,6) Second assocate 8 (,3,4,6,7,9) Frst assocate (,5) Second assocate 9 { B Q A S ( Q ) } (,3,5,6,7,8) Frst assocate (,4) Second assocate k ˆ τ =, = ( AB AB) A = r( k ) + λ= 3(3-) + 0 = 6 A = λ - λ = 0 - = - = p p B = (λ - λ ) p = 0, as p 0 B = r(k-) + λ + (λ - λ ) ( ) = 3 x (-) (3-6) = = 9,,..., v Now A B - A B = 6 x 9-0 = 54. Therefore, τˆ = 3 [ 9Q + S (Q )]/54 = Q / + S (Q )/8, =,,...,9 Now, S (Q ) = Sum of Q s for those treatments whch are frst assocates of treatment, = Q + Q 3 + Q 5 + Q 6 + Q 7 + Q 8 = = -5, S (Q ) = 4, S (Q 3 ) =, S (Q 4 ) = -5, S (Q 5 ) = 4, S (Q 6 ) =, S (Q 7 ) =, S (Q 8 ) = 4, S (Q 9 ) = -5. Total S.S.(TSS) = Σ(observaton) - CF = = 490 Treatment S.S. unadjusted (SST U ) = (Σ T )/r - CF = ( )/3-608 = Block S.S. unadjusted (SSB U ) = (Σ B j )/k - CF = ( )/3-608 =

9 Treatment S.S. adjusted(sst A ) = Στ Q = 0x(-5/8) + (-6) x (-50/8) x (3/8) =.67. Block S.S. adjusted (SSB A ) = SST A + SSB U - SST U = = 95. Error S.S. (SSE) = TSS -SSB U - SST A = = The analyss of varance Table s gven below: ANOVA Source d.f. S.S. M.S. F Blocks(unadj.) 8 68 Treatments(adj.) Blocks(adj.) Treatments (unadj) Error Total Table value of F (8,0) = 3.07 (at 5% level of sgnfcance) Treatment effects are not sgnfcant. SE() = Standard error of ( ˆ τ ˆ τ ) = [{ x 3x( + 9) MSE}/ 54] m = ( 8 / 9) x =.99, f the treatments are frst assocates SE() = Standard error of( ˆ τ ˆ τ )= x 3x9MSE / 54 m = = 3.7, f the treatments are second assocates CD = t (0.05,6) x SE()=.056x.99 = 6.4, f two treatments are frst assocates CD = t (0.05,6) x SE()=.056x 3.7 = 6.5, f two treatments are second assocates An estmate of average varance of elementary treatment contrast s A.V. = [6x8.96+x0.030]/8 = 9.95 Average SE = 3.03 An unbased estmate of the dfference between two treatment effects ( ˆ τ ˆ τ ) s: m 85

10 Treatment m Treatment Bold fgures ndcate sgnfcant at 5% level. A comparson of ( ˆ τ ˆ τ m ) wth CD f two treatments are frst assocates and CD when treatments are second assocates, ndcates that treatments and 3, 3 and 7, 4 and 7, 5 and 7, and 7 and 8 are sgnfcantly dfferent. For further detals refer to Dey (986). The same analyss can be carred out usng SAS. The steps and analyss are gven below: data pbb; nput blk trt yld; cards; ; 86

11 proc prnt; proc glm; class blk trt; model yld=blk trt/ss ss; LSMEANS trt/stderr pdff; run; General Lnear Models Procedure Class Level Informaton Class Levels Values BLK TRT Number of observatons n data set = 7 General Lnear Models Procedure Dependent Varable: YLD Source DF Sum of Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square C.V. Root MSE YLD Mean Source DF Type I SS Mean Square F Value Pr > F BLK TRT Source DF Type II SS Mean Square F Value Pr > F BLK TRT Least Squares Means TRT YLD Std Err Pr > T LSMEAN LSMEAN LSMEAN H0:LSMEAN=0 Number

12 Pr > T H0: LSMEAN()=LSMEAN(j) /j NOTE: To ensure overall protecton level, only probabltes assocated wth pre-planned comparsons should be used. References Bose, R.C. Clatworthy, W.H. and Shrkhande, S.S. (954). Tables of partally balanced desgns wth two assocate classes. North Carolna Agrcultural Experment Staton. Tech. Bull. No. 07. Clatworthy, W.H. (973). Tables of two-assocate partally balanced desgns. Natonal Bureau of Sandards, Appled Maths. Seres No. 63, Washngton D.C. Dey, A. (986). Theory of block desgns. Wley Eastern Ltd., New Delh. 88