Chapter 7 Partially Balanced Incomplete Block Design (PBIBD)

Transcription

1 Chapter 7 Partally Balanced Incomplete Block Desgn (PBIBD) The balanced ncomplete block desgns hae seeral adantages. They are connected desgns as well as the block szes are also equal. A restrcton on usng the BIBD s that they are not aalable for all parameter combnatons. They exst only for certan parameters. Sometmes, they requre large number of replcatons also. Ths hampers the utlty of the BIBDs. For example, f there are = 8 treatments and block sze s k = 3 (.e., 3 plots s each block) then the total number of requred blocks are 8 b = = 56 and so usng the relatonshp bk = r, the total number of requred replcates s 3 bk r = =. Another mportant property of the BIBD s that t s effcency balanced. Ths means that all the treatment dfferences are estmated wth same accuracy. The partally balanced ncomplete block desgns (PBIBD) compromse on ths property upto some extent and help n reducng the number of replcatons. In smple words, the pars of treatments can be arranged n dfferent sets such that the dfference between the treatment effects of a par, for all pars n a set, s estmated wth the same accuracy. The partally balanced ncomplete block desgns reman connected lke BIBD but no more balanced. Rather they are partally balanced n the sense that some pars of treatments hae same effcency whereas some other pars of treatments hae same effcency but dfferent from the effcency of earler pars of treatments. Ths wll be llustrated more clearly n the further dscusson. Before descrbng the set up of PBIBD, frst we need to understand the concept of Assocaton Scheme. Instead of explanng the theory related to the assocaton schemes, we consder here some examples and then understand the concept of assocaton scheme. Let there be a set of treatments. These treatments are denoted by the symbols,,,. Partally Balanced Assocaton Schemes A relatonshp satsfyng the followng three condtons s called a partally balanced assocaton scheme wth m-assocate classes. () Any two symbols are ether frst, second,, or m th assocates and the relaton of assocatons s symmetrcal,.e., f the treatment A s the th assocate of treatment B, then B s also the th assocate of treatment A. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur

2 () Each treatment A n the set has exactly n treatments n the set whch are the th and the number n ( =,,..., m) does not depend on the treatment A. assocate () If any two treatments A and B are the th assocates, then the number of treatments whch are both th assocate A and k th assocate of B s p k and s ndependent of the par of th assocates A and B. The numbers n,, n,..., n, p (, k, =,,..., m) are called the parameters of m -assocate partally balanced scheme. m k We consder now the examples based on rectangular and trangular assocaton schemes to understand the condtons stated n the partally balanced assocaton scheme. Rectangular Assocaton Scheme Consder an example of m = 3 assocate classes. Let there be sx treatments denoted as,, 3, 4, 5 and 6. Suppose these treatments are arranged as follows: Under ths arrangement, wth respect to each symbol, the two other symbols n the same row are the frst assocates. One another symbol n the same column s the second assocate and remanng two symbols are n the other row are the thrd assocates. For example, wth respect to treatment, treatments and 3 occur n the same row, so they are the frst assocates of treatment, treatment 4 occurs n the same column, so t s the second assocate of treatment and the remanng treatments 5 and 6 are the thrd assocates of treatment as they occur n the other (second) row. Smlarly, for treatment 5, treatments 4 and 6 occur n the same row, so they are the frst assocates of treatment 5, treatment occurs n the same column, so t s the second assocate of treatment 5 and remanng treatments and 3 are n the other (second) row, so they are the thrd assocates of treatment 5. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur

3 The table below descrbes the frst, second and thrd assocates of all the sx treatments. Treatment number Frst assocates Second assocates Thrd assocates , 3, 3, 5, 6 4, 6 4, , 6 4, 6 4, 5, 3, 3, Further, we obsere that for the treatment, the o number of frst assocates ( n ) =, o number of second assocates ( n ) =, and o number of thrd assocates ( n 3) =. The same alues of n, n and n 3 hold true for other treatments also. Now we understand the condton () of defnton of partally balanced assocaton schemes related to p. k Consder the treatments and. They are the frst assocates (whch means = ),.e. treatments and are the frst assocate of each other; treatment 6 s the thrd assocate (whch means = 3) of treatment and also the thrd assocate (whch means k = 3) of treatment. Thus the number of treatments whch are both,.e., the th ( = 3) assocate of treatment A (here A ) and k th assocate of treatment B(here B ) are th (.e., = ) assocate s pk = p = 33. Smlarly, consder the treatments and 3 whch are the frst assocate ( whch means = ); treatment 4 s the thrd (whch means = 3) assocate of treatment and treatment 4 s also the thrd (whch means k = 3) assocate of treatment 3. Thus p 33 =. Other alues of p (, k=,,, 3) can also be obtaned smlarly. k Remark: Ths method can be used to generate 3-class assocaton scheme n general for m n treatments (symbols) by arrangng them n m -row and n -columns. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 3

4 Trangular Assocaton Scheme The trangular assocaton scheme ges rse to a -class assocaton scheme. Let there be a set of treatments whch are denoted as,,,. The treatments n ths scheme are arranged n q rows and q columns where. q qq ( ) = =. These symbols are arranged as follows: (a) Postons n the leadng dagonals are left blank (or crossed). (b) The q postons are flled up n the postons aboe the prncpal dagonal by the treatment numbers,,.., correspondng to the symbols. (c) Fll the postons below the prncpal dagonal symmetrcally. Ths assgnment s shown n the followng table: Rows 3 4 q - q Columns 3 q - q - q q + q - q - 3 q 4 3 q + q - q - q - q(q - )/ q q - q - qq ( ) / Now based on ths arrangement, we defne the frst and second assocates of the treatments as follows. The symbols enterng n the same column ( =,,..., q) are the frst assocates of and rest are the second assocates. Thus two treatments n the same row or n the same column are the frst assocates of treatment. Two treatments whch do not occur n the same row or n the same column are the second assocates of treatment. Now we llustrate ths arrangement by the followng example: Let there be 0 treatments. Then q = 5 as arranged under the trangular assocaton scheme as follows: 5 = = 0. The ten treatments denoted as,,, 0 are Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 4

5 Rows Columns For example, for treatment, o the treatments, 3 and 4 occur n the same row (or same column) and o treatments 5, 6 and 7 occur n the same column (or same row). So the treatments, 3, 4, 5, 6 and 7 are the frst assocates of treatment. Then rest of the treatments 8, 9 and 0 are the second assocates of treatment. The frst and second assocates of the other treatments are stated n the followng table. Treatment number Frst assocates Second assocates, 3, 4 5, 6, 7 8, 9, 0, 3, 4 5, 8, 9 6, 7, 0 3,, 4 6, 8, 0 5, 7, 9 4,, 3 7, 9, 0 5, 6, 8 5, 6, 7, 8, 9 3, 4, 0 6, 5, 7 3, 8, 0, 4, 9 7, 5, 6 4, 9, 0, 3, 8 8, 5, 9 3, 6, 0, 4, 7 9, 5, 8 4, 7, 0, 3, 6 0 3, 6, 8 4, 7, 9,, 5 We obsere from ths table that the number of frst and second assocates of each of the 0 treatments ( = 0) s same wth n = 6, n = 3 and n+ n = 9 =. For example, the treatment n the column of frst assocates occurs sx tmes, z., n frst, thrd, fourth, ffth, eghth and nnth rows. Smlarly, the treatment n the column of second assocates occurs three tmes, z., n the sxth, seenth and tenth rows. Smlar conclusons can be erfed for other treatments. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 5

6 There are sx parameters, z., p, p, p (or p ), p, p and p (or p ) whch can be arranged n symmetrc matrces P and P as follows: p p, p p. = P = p p p p P [Note: We would lke to cauton the reader not to read superscrpt.] 3 4 P =, P = 0. p as squares of p but n p s only a In order to learn how to wrte these matrces P and P, we consder the treatments, and 8. Note that the treatment 8 s the second assocate of treatment. Consder only the rows correspondng to treatments, and 8 and obtan the elements of P and P as follows: p Treatments and are the frst assocates of each other. There are three common treatments : (z., 3, 4 and 5) between the frst assocates of treatment and the frst assocates of treatment. So p = 3. Treatment st assocates Treatment st assocates of treatment, 6, , 8, 9 st assocates of treatment Those three treatments whch are the st assocates of treatments and. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 6

7 p and p : Treatments and are the frst assocates of each other. There are two treatments (z., 6 and 7) whch are common between the frst assocates of treatment and the second assocates of treatment. So p = = p. Treatment st assocates Treatment st assocates of treatment, 3, 4, nd assocates of treatment Those two treatments whch are the st assocates of treatments and the nd assocate of treatment. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 7

8 p : Treatments and are the frst assocates of each other. There s only one treatment (z., treatment 0) whch s common between the second assocates of treatment and the second assocates of treatment. So p =. Treatment st assocates Treatment nd assocates of treatment , 7 nd assocates of treatment The treatment whch s the st assocates of treatments and Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 8

9 p : Treatments and 8 are the second assocates of each other. There are four treatment (z.,,3,5 and 6) whch are common between the frst assocates of treatment and the frst assocates of treatment 8. So p = 4. Treatment nd assocates Treatment 8 st assocates of treatment 4, , 0 st assocates of treatment 8 Those four treatments whch are the nd assocates of treatments Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 9

10 p and p : Treatments and 8 are the second assocates of each other. There are two treatments (z., 4 and 7) whch are common between the frst assocates of treatment and the second assocates of treatment 8. So p = = p. Treatment nd assocates Treatment 8 st assocates of treatment, 3, 5, st assocates of treatment 8 Those two treatments whch are the st assocates of treatments Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 0

11 p : Treatments and 8 are the second assocates of each other. There s no treatment whch s common between the second assocates of treatment and the second assocates of treatment 8. So p = 0. Treatment nd assocates Treatment 8 nd assocates of treatment 8, 9, 0, 4, 7 nd assocates of treatment 8 There are no treatments whch are the nd assocates of treatments and 8 In general, f q rows and q columns of a square are used, then for q > 3 q qq ( ) = =, n = q 4, ( q )( q 3) n =, q q 3 P = ( q 3)( q 4) q 3 4 q 8 P = ( q 4)( q 5). q 8 For q = 3, there are no second assocates. Ths s a degenerate case where second assocates do not exst and hence P cannot be defned. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur

12 The graph theory technques can be used for countng p. Further, t s easy to see that all the parameters n P, P, etc. are not ndependent. Constructon of Blocks of PBIBD under Trangular Assocaton Scheme The blocks of a PBIBD can be obtaned n dfferent ways through an assocaton scheme. Een dfferent block structure can be obtaned usng the same assocaton scheme. We now llustrate ths by obtanng dfferent block structures usng the same trangular scheme. Consder the earler llustraton q where = = 0 were obtaned. k wth q = 5 was consdered and the frst, second and thrd assocates of treatments Approach : One way to obtan the treatments n a block s to consder the treatments n each row. Ths consttutes the set of treatments to be assgned n a block. When q = 5, the blocks of PBIBD are constructed by consderng the rows of the followng table. Rows Columns From ths arrangement, the treatment are assgned n dfferent blocks and followng blocks are obtaned. Blocks Treatments Block,, 3, 4 Block, 5, 6, 7 Block 3, 5, 8, 9 Block 4 3, 6, 8, 0 Block 5 4, 7, 9, 0 The parameters of such a desgn are b= 5, = 0, r =, k = 4, λ = and λ = 0. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur

13 Approach : Another approach to obtan the blocks of PBIBD from a trangular assocaton scheme s as follows: Consder the par-wse columns of the trangular scheme. Then delete the common treatments between the chosen columns and retan others. The retaned treatments wll consttute the blocks. Consder, e.g., the trangular assocaton scheme for q = 5. Then the frst block under ths approach s obtaned by deletng the common treatments between columns and whch results n a block contanng the treatments, 3, 4, 5, 6 and 7. Smlarly, consderng the other pars of columns, the other blocks can be obtaned whch are presented n the followng table: Blocks Columns of assocaton scheme Treatments Block (, ), 3, 4, 5, 6, 7 Block (, 3), 3, 4, 5, 8, 9 Block 3 (, 4),, 4, 6, 8, 0 Block 4 (, 5),, 3, 7, 9, 0 Block 5 (, 3),, 6, 7, 8, 9 Block 6 (, 4), 3, 5, 7, 8, 0 Block 7 (, 5), 4, 5, 6, 9, 0 Block 8 (3, 4), 3, 5, 6, 9, 0 Block 9 (3, 5), 4, 5, 7, 8, 0 Block 0 (4, 5) 3, 4, 6, 7, 8, 9 The parameters of the PBIBD are b= 0, = 0, r = 6, k = 6, λ = 3 and λ = 4. Snce both these PBIBDs are arsng from same assocaton scheme, so the alues of n, n, P and P reman the same for both the desgns. In ths case, we hae n = 6, n = 3, 3 4 P =, P =. 0 Approach 3: Another approach to dere the blocks of PBIBD s to consder all the frst assocates of a gen treatment n a block. For example, n case of q = 5, the frst assocates of treatment are the treatments, 3, 4, 5, 6 and 7. So these treatments consttute one block. Smlarly other blocks can also be found. Ths results n the same arrangement of treatments n blocks as consdered by deletng the common treatments between the par of columns. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 3

14 The PBIBD wth two assocate classes are popular n practcal applcatons and can be classfed nto followng types dependng on the assocaton scheme, (Reference: Classfcaton and analyss of partally balanced ncomplete block desgns wth two assocate classes, R. Bose and Shmamoto, 95, Journal of Amercan Statstcal Assocaton, 47, pp. 5-84).. Trangular. Group dsble 3. Latn square wth constrants ( L ) 4. Cyclc and 5. Sngly lnked blocks. The trangular assocaton scheme has already been dscussed. We now brefly present other types of assocaton schemes. Group dsble assocaton scheme: Let there be treatments whch can be represented as Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur = pq. Now dde the treatments nto p groups wth each group hang q treatments such that any two treatments n the same group are the frst assocates and two treatments n the dfferent groups are the second assocates. Ths s called the group dsble type scheme. The scheme smply amounts to arrange the = pq treatments n a ( p q) rectangle and then the assocaton scheme can be exhbted. The columns n the ( p q) rectangle wll form the groups. Under ths assocaton scheme, hence n = q n = q( p ), ( q ) λ + qp ( ) λ = rk ( ) and the parameters of second knd are unquely determned by p and q. In ths case q 0 0 q P =, P =, 0 q( p ) q q( p ) For eery group dsble desgn, r λ, rk λ 0. 4

15 If r = λ, then the group dsble desgn s sad to be sngular. Such sngular group dsble desgn can always be dered from a correspondng BIBD. To obtan ths, ust replace each treatment by a group of q treatments. In general, f a BIBD has followng parameters b*, *, r*, k*, λ *, then a group dsble desgn s obtaned from ths BIBD whch has parameters b = b*, = q*, r = r*, k = qk*, λ = r, λ = λ*, n = p, n = q. A group dsble desgn s nonsngular f r λ. Nonsngular group dsble desgns can be dded nto two classes sem-regular and regular. A group dsble desgn s sad to be sem-regular f r > λ and rk λ = 0. For ths desgn b p+. Also, each block contans the same number of treatments from each group so that k must be dsble by p. A group dsble desgn s sad to be regular f r > λ and rk λ> 0. For ths desgn b. Latn Square Type Assocaton Scheme Let L denotes the Latn square type PBIBD wth constrants. In ths PBIBD, the number of treatments are expressble as = q. The treatments may be arranged n a ( q q) Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur square. For the case of =, two treatments are the frst assocates f they occur n the same row or the same column, and second assocates otherwse. For general case, we take a set of ( = ) mutually orthogonal Latn squares, proded t exsts. Then two treatments are the frst assocates f they occur n the same row or the same column, or correspondng to the same letter of one of the Latn squares. Otherwse they are second assocates. Under ths assocaton scheme, = q, n = q ( ), n = ( q )( q + ) ( )( ) + q ( q + )( ) P =, ( q + )( ) ( q + )( q ) ( ) q ( ) P =. ( q ) ( q )( q ) + q 5

16 Cyclc Type Assocaton Scheme Let there be treatments and they are denoted by ntegers,,...,. In a cyclc type PBIBD, the frst assocates of treatment are where the + d, + d,..., + d (mod ), n d' s satsfy the followng condtons: () the d' s are all dfferent and 0 < d < ( =,,..., n ); () among the n( n ) dfferences d d ', (, ' =,,..., n, ') reduced (mod ), each of numbers d, d,..., d n occurs α tmes, whereas each of the numbers e, e,..., e n occurs β tmes, where d, d,..., dn, e, e,..., e n are all the dfferent numbers,,,. [Note : To reduce an nteger mod, we hae to subtract from t a sutable multple of, so that the reduced nteger les between and. For example, 7 when reduced mod 3 s 4.] For ths scheme, nα + n β = n ( n ), n α = n α n n+ α + P. = n β n n+ β P β α n β, Sngly Lnked Block Assocaton Scheme Consder a BIBD D wth parameters b**, **, r**, k**, λ ** = and b** > **. Let the block numbers of ths desgn be treated as treatments,.e., = b**. The frst and second assocates n the sngly lnked block assocaton scheme wth two classes are determned as follows: Defne two block numbers of D to be the frst assocates f they hae exactly one treatment n common and second assocates otherwse. Under ths assocaton scheme, = b**, n = k**( r** ), n = b** n, r** + ( k** ) n r** ( k** ) + P =, n r** ( k** ) + n n+ r** + ( k** ) ** ** k n k P =. ** ** n k n n+ k Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 6

17 General Theory of PBIBD A PBIBD wth m assocate classes s defned as follows. Let there be treatments. Let there are b blocks and sze of each block s k,.e., there are k plots n each block. Then the treatments are arranged n b blocks accordng to an m -assocate partally balanced assocaton scheme such that (a) (b) (c) eery treatment occurs at most once n a block, eery treatment occurs exactly n r blocks and f two treatments are the th assocates of each other then they occur together exactly n λ ( =,,..., m) blocks. The number λ s ndependent of the partcular par of th assocate chosen. It s not necessary that λ should all be dfferent and some of the λ ' s may be zero. If treatments can be arranged n such a scheme then we hae a PBIBD. Note that here two treatments whch are the th assocates, occur together n λ blocks. The parameters brk,,,, λ, λ,..., λ, n, n,..., n are termed as the parameters of frst knd and m m p k are termed as the parameters of second knd. It may be noted that n, n,..., n m and all p k of the desgn are obtaned from the assocaton scheme under consderaton. Only λ, λ,..., λ, occur n the defnton of PBIBD. m If λ = λ for all =,,,m then PBIBD reduces to BIBD. So BIBD s essentally a PBIBD wth one assocate class. Condtons for PBIBD The parameters of a PBIBD are chosen such that they satsfy the followng relatons: ( ) bk = r m ( ) n = = m ( ) n λ = r( k ) = ( ) n p = n p = n p ( ) k k k k m pk = k = n f = n f. It follows from these condtons that there are only mm ( ) / 6 ndependent parameters of the second knd. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 7

18 Interpretatons of Condtons of PBIBD The nterpretatons of condtons ()-() are as follows. () bk = r Ths condton s related to the total number of plots n an experment. In our settngs, there are k plots n each block and there are b blocks. So total number of plots are bk. Further, there are treatments and each treatment s replcated r tmes such that each treatment occurs atmost n one block. So total number of plots contanng all the treatments s r. Snce both the statements counts the total number of blocks, hence bk = r. () m = n = Ths condton nterprets as follows: Snce wth respect to each treatment, the remanng ( ) treatments are classfed as frst, second,, or m th assocates and each treatment has n assocates, so the sum of all n ' s s the same as the total number of treatments except the treatment under consderaton. () m = nλ = rk ( ) Consder r blocks n whch a partcular treatment A occurs. In these r blocks, rk ( ) pars of treatments can be found, each hang A as one of ts members. Among these pars, the th assocate of A must occur λ tmes and there are n assocates, so nλ = rk ( ). () Let np = n p = n p k k k k G be the set of th assocates, =,,..., m of a treatment A. For, each treatment n G has exactly p k numbers of k th assocates n G. Thus the number of pars of k th assocates that can be obtaned by takng one treatment from G and another treatment from G on the one hand s np k and np k, respectely. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 8

19 () m m k k k= k= p = n f = and p = n f. Let the treatments A and B be th assocates. The k th assocate of Ak ( =,,..., m) should contan all the n number of th assocates of B( ). When =, A tself wll be one of the th assocate of B. Hence k th assocate of A, ( k =,,..., m) should contan all the ( n ) numbers of th assocate of B. Thus the condton holds. Intrablock Analyss of PBIBD Wth Two Assocates Consder a PBIBD under two assocates scheme. The parameters of ths scheme are b,, r, k, λ, λ, n, n, p, p, p, p, p and p, The lnear model nolng block and treatment effects s y = µ + β + τ + ε ; =,,..., b, =,,...,, where µ s the general mean effect; β s the fxed addte th block effect satsfyng β = 0; τ s the fxed addte th treatment effect satsfyng r = τ = 0 and ε m s the..d. random error wth ε N σ m ~ (0, ). The PBIBD s a bnary, proper and equreplcate desgn. So n ths case, the alues of the parameters become n = 0 or, k = k for all =,,..., b and r = r for all =,,...,. There can be two types of null hypothess whch can be consdered one for the equalty of treatment effects and another for the equalty of block effects. We are consderng here the ntrablock analyss, so we consder the null hypothess related to the treatment effects only. As done earler n the case of BIBD, the block effects can be consdered under the nterblock analyss of PBIBD and the recoery of nterblock nformaton. The null hypothess of nterest s H0 : τ = τ =... = τ aganst alternate hypothess H : at least one par of τ s dfferent. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 9

20 The null hypothess related to the block effects s of not much practcal releance and can be treated smlarly. Ths was llustrated earler n case of BIBD. In order to obtan the least squares estmates of µβ, and ρ, we mnmze of sum of squares due to resduals b = = ( y µ β τ ) wth respect to µβ, and τ Ths results n the three normal equatons whch can be unfed usng the matrx notatons. The set of reduced normal equatons n matrx notaton after elmnatng the block effects are expressed as where Q = Cτ C = R N K N Q= V N' K B, R = ri, K = ki. b ', Then the dagonal elements of C are b n = rk ( ) c = r =, ( =,,..., ), k k the off-dagonal elements of C are obtaned as c λ b f treatment and ' are the frst assocates = k nn = f treatment and ' are the second assocates ( ' =,,..., ) k ' ' k = λ and the th alue n Q s [Sum of block totals n whch th Q = V treatment occurs] k = rk ( ) τ nn ' τ. k '( ') Next, we attempt to smplfy the expresson of Q. Let S be the sum of all treatments whch are the frst assocates of th treatment and S be the sum of all treatments whch are the second assocates of th treatment. Then τ + S + S = τ. = Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 0

21 Thus the equaton n Q usng ths relatonshp for =,,..,, becomes kq = r( k ) τ ( λs + λs ) = rk ( ) τ λs λ τ τ S [ rk ] = = ( ) + λ τ + ( λ λ ) S λ τ. = Imposng the sde condton τ = 0, we hae = where kq = r( k ) τ + λ τ + ( λ λ ) S = a τ + b S, =,,..., * * a = rk ( ) + λ and a = λ λ. * * These equatons are used to obtan the adusted treatment sum of squares. Let Q denotes the adusted sum of Q ' s oer the set of those treatments whch are the frst assocate of th treatment. We note that when we add the terms S for all, then occurs n tmes n the sum, eery frst assocate of occurs p tmes n the sum and eery second assocate of occurs p tmes n the sum wth S [ ] * * p + p = n. Then usng K = kib and kq = k Q = Sum of Q 's whch are the frst assocates of treatment = r( k ) + λ S + ( λ λ ) nτ + p S + p S = rk ( ) + λ + ( λ λ )( p p S + ( λ λ ) pτ = b S + a τ = τ = 0, we hae where a = ( λ λ ) p * b = rk ( ) + λ + ( λ λ )( p p ). * Now we hae followng two equatons n kq and kq : kq = a τ + b S * * kq = a τ + b S * * Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur

22 Now solng these two equatons, the estmate of τ s obtaned as We see that so kb [ Q b Q ] ˆ τ =, ( =,..., ). a b = = * * * * * * ab Q = Q = 0, = ˆ τ = 0. Thus ˆ τ s a soluton of reduced normal equaton. The analyss of arance can be carred out by obtanng the unadusted block sum of squares as SS Block ( unad) b B G =, k bk = the adusted sum of squares due to treatment as SS = ˆ τ Q Treat( ad) = where b G = y. The sum of squares due to error as = SS = SS SS SS Error () t Total Block ( unad) Treat( ad) where SS Total b G = y. bk = = A test for H0 : τ = τ =... = τ s then based on the statstc F Tr SSTreat( ad) /( ) =. SS /( bk b + ) Error( unad) If FTr F α ;, bk b + > then H 0 s reected. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur

23 The ntrablock analyss of arance for testng the sgnfcance of treatment effects s gen n the followng table: Source Sum of squares Degrees of Mean squares F freedom Between treatments (adusted) SS = Treat( ad) ˆ τ Q = dftreat = MS Treat SS = df Treat(ad) Treat MS Treat MSE Between blocks (unadusted) SS b = Block ( unad) B k G bk dfblock = b Intrablock Error SS Error () t (By substracton) df ET = bk b + SS MSE = df Error ET Total SS b Total = = = y G bk dft = bk Note that n the ntrablock analyss of PBIBD analyss, at the step where we obtaned S and elmnated S to obtan the followng equaton ( ) λ τ ( λ λ ) λ τ, = [ + ] + kq r k S = another possblty s to elmnate S nstead of S. If we elmnate S nstead of S (as we approached), then the soluton has less work noled n the summng of Q f n< n. If n > n, then elmnatng S wll nole less work n obtanng Q where Q denotes the adusted sum of Q ' s s oer the set of those treatments whch are the second assocate of th treatment. When we do so, obtan the followng estmate of the treatment effect s obtaned as: where k bq bq * ˆ τ = ab ab * * * * * * Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 3

24 a b = rk ( ) + λ * * a = λ λ = ( λ λ ) p * b = rk ( ) + λ + ( λ λ )( p p ). * The analyss of arance s then based on * ˆ τ and can be carred out smlarly. The arance of the elementary contrasts of estmates of treatments (n case of n < n ) s b ( kq kq ) b ( kq kq ) ˆ τ ˆ τ = * * ' ' ' * * * * ab ab kb ( + b ) f treatment and ' are the frst assocates * * * * * * ab ab ' * kb * * * * ab ab Var( ˆ τ ˆ τ ) = f treatment and ' are the second assocates. We obsere that the arance of ˆ τ ˆ τ depends on the nature of and ' n the sense that whether ' they are the frst or second assocates. So desgn s not (arance) balanced. But arance of any elementary contrast are equal under a gen order of assocaton, z., frst or second. That s why the desgn s sad to be partally balanced n ths sense. Analyss of Varance Chapter 7 Partal. BIBD Shalabh, IIT Kanpur 4