Analyss of Varance and Desgn of Experments-II MODULE - III LECTURE - 8 PARTIALLY BALANCED INCOMPLETE BLOCK DESIGN (PBIBD) Dr Shalah Department of Mathematcs & Statstcs Indan Insttute of Technology Kanpur
Intralock analyss of PBIBD wth two assocates Consder a PBIBD under two assocates scheme The parameters of ths scheme are,, r, k, λ, λ, n, n, p, p, p, p, p p and y µ + β + τ + ε ;,,,,,,,, The lnear model nolng lock and treatment effects s µ s the general mean effect; β τ ε m s the fxed addte th lock effect satsfyng s the fxed addte th treatment effect satsfyng s the d random error wth ε N σ m ~ (0, ) β 0; r τ 0 and The PBIBD s a nary, proper and equreplcate desgn So n ths case, the alues of the parameters ecome n 0 or, k k for all,,, and r r for all,,, There can e two types of null hypothess whch can e consdered one for the equalty of treatment effects and another for the equalty of lock effects We are consderng here the ntralock analyss, so we consder the null hypothess related to the treatment effects only As done earler n the case of BIBD, the lock effects can e consdered under the nterlock analyss of PBIBD and the recoery of nterlock nformaton The null hypothess of nterest s H0: τ τ τ aganst alternate hypothess H : at least one par of τ s dfferent The null hypothess related to the lock effects s of not much practcal releance and can e treated smlarly
3 Ths was llustrated earler n the case of BIBD In order to otan the least squares estmates of the sum of squares due to resduals ( y µ β τ ) wth respect to µβ, and τ Ths results n the three normal equatons, whch can e unfed usng the matrx notatons The set of reduced normal equatons n matrx notaton after elmnatng the lock effects are expressed as Q Cτ C R N K N, ', Q V N K B R ri K ki ', Then the dagonal elements of C are otaned as n rk ( ) c r, (,,, ), k k the off-dagonal elements of C are otaned as µβ, and τ, we mnmze c λ f treatment and ' are the frst assocates k nn f treatment and ' are the second assocates ( ',,, ) k ' ' k λ
4 and the th alue n Q s [ th Q V Sum of lock totals n whch treatment occurs ] k rk ( ) τ nn ' τ k '( ') Next, we attempt to smplfy the expresson of Q S S Let e the sum of all treatments whch are the frst assocates of th treatment and e the sum of all treatments whch are the second assocates of th treatment Then τ + S + S τ Thus the equaton n Q usng ths relatonshp for,,, ecomes [ rk ] kq r( k ) τ ( λs + λs ) Imposng the sde condton rk ( ) τ λs λ τ τ S ( ) + λ τ + ( λ λ ) S λ τ a rk ( ) + λ and a λ λ τ 0, we hae kq r( k ) τ + λ τ + ( λ λ ) S a τ + S,,,,
These equatons are used to otan the adusted treatment sum of squares Let Q denote the adusted sum of Q ' soer the set of those treatment,s whch are the frst assocate of th treatment We note that when we add the terms for all, then occurs n tmes n the sum, eery frst assocate of occurs tmes n the sum and eery second assocate of occurs tmes n the sum wth p + p n Then usng K ki and we hae [ ] S S r( k ) + λ S + ( λ λ ) nτ + p S + p S rk ( ) + λ + ( λ λ )( p p S + ( λ λ ) pτ p kq k Q Sum of Q 's whch are the frst assocates of treatment a S + a τ ( λ λ ) p rk ( ) + λ + ( λ λ )( p p ) 5 p τ 0, Now we hae followng two equatons n kq and kq : kq a τ + S kq a τ + S Now solng these two equatons, the estmate of τˆ s otaned as k [ Q Q ] ˆ τ, (,,, ) a a
6 We see that Q Q 0, so ˆ τ 0 Thus τˆ s a soluton of reduced normal equaton The analyss of arance can e carred out y otanng the unadusted lock sum of squares as Block ( unad) B G k k, the adusted sum of squares due to treatment as ˆ τ Q Treat( ad) G y The sum of squares due to error as Error () t Total Block ( unad) Treat( ad) Total G y k
7 A test for H : τ τ τ 0 s then ased on the statstc F Tr Treat( ad) /( ) /( k + ) Error( unad) F Tr > F α ;, k + 0 If then H s reected The ntralock analyss of arance for testng the sgnfcance of treatment effects s gen n the followng tale: Source Sum of Degrees of Mean squares F squares freedom Between treatments (adusted) Treat( ad) ˆ τ Q dftreat MS Treat Treat( ad) df Treat MS Treat MSE Between locks (unadusted) Block ( unad) B k G k dfblock Intralock error Total Error () t ( By sustracton) Total G y k df ET k + dft k MSE df Error ET
8 Note that n the ntralock analyss of PBIBD analyss, at the step we otaned and elmnated to otan the followng equaton [ + ] + kq r( k ) λ τ ( λ λ ) S λ τ, S S another posslty s to elmnate S nstead of S If we elmnate S nstead of (as we approached), then the S soluton has less work noled n the summng of f < Q n n >, S Q Q If n n then elmnatng wll nole less work n otanng n otanng 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, the followng estmate of the treatment effect s otaned : ˆ τ k Q a Q a a rk ( ) + λ a λ λ ( λ λ ) p rk ( ) + λ + ( λ λ )( p p ) The analyss of arance s then ased on τˆ and can e carred out smlarly
9 The arance of the elementary contrasts of estmates of treatments (n case of n < n ) ( kq kq ) ( kq kq ) ˆ τ ˆ τ ' ' ' a a s k ( + ) a Var( ˆ τ ˆ τ ' ) k a a a f treatment and ' are the frst assocates f treatment and ' are the second assocates We osere 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) alanced 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 e partally alanced n ths sense