Analysis of Covariance of Reinforced Balanced Incomplete Block Designs With a Single Explanatory Variable

Transcription

1 Journal of Statstcal heory and Alcatons, Vol. 3, No. 3 (Seteer 04), Analyss of Coarance of Renforced Balanced Incolete Block Desgns Wth a Sngle xlanatory Varale D. K. Ghosh UGC BSR Faculty Fellow, Deartent of Statstcs, Saurashtra Unersty Rakot, Guarat, Inda ghosh_dkg@redffal.co M. G. Bhatt Deartent of Statstcs, H & H.B. Kotak Insttute of Scence Rakot, Guarat, Inda ahendra.statstcs@gal.co and S. C. Bagu Deartent of Matheatcs and Statstcs, he Unersty of West Florda Pensacola, FL 354, USA sagu@uwf.edu Renforced alanced ncolete lock desgns (BIBDs) are ery useful n statstcal lannng of exerents as they can e constructed for any nuer of treatents for gen nuers of relcatons. Das (958) was frst to ntroduce the statstcal analyss of arance (ANOVA) of these desgns, and n the sae year Gr also deeloed the sae statstcal analyss of arance for renforced artally alanced ncolete lock desgns (PBIBDs). In ths artcle, we focus on the ethod of statstcal analyss of coarance (ANCOVA) of renforced alanced ncolete lock desgn (BIBD) when a sngle exlanatory arale s aalale n the exerent. Keywords: Augented desgns, Renforced desgns, Renforced BIBD, Renforced PBIBD, and Ancllary arale. AMS (03) Matheatcs Suect Classfcaton: 6J, 6k he corresondng author 35

2 D.K. Ghosh, M.G. Bhatt, and S.C. Bagu. Introducton he analyss of coarance (ANCOVA) s tycally used to adust or control for dfferences etween treatent effects ased another contnuous arale called exlanatory (or extraneous, or ancllary) arale (or coarate). In analyss of arance (ANOVA) f one or ore exlanatory arales are susected to affect the outcoe (deendent arale), then gnorng exlanatory arales n ANOVA wll ncrease error su of squares and wll ake t harder to detect any real dfferences n treatent effects. Whereas the analyss coarance n ths case wll e ore recse y reducng error su of squares and ncreasng the statstcal ower of the exerent when a good coarate has een used wthn the ANCOVA. Keeng these n nd we wll deelo ANCOVA for renforced BIBD, Renforced BIBD has seeral adantages oer other desgns artcularly wth resect to usng any nuer of treatents for gen nuers of relcatons. hus t rngs no ltaton on statstcal lannng of exerents n dfferent felds. In factoral exerents when the nuer of treatent conatons are large, the confoundng technques are used to reduce the lock sze. hs rocedure rodes ost accurate estaton of lower order nteractons at the cost of soe of the less ortant hgherorder nteractons whch are confounded wth locks. Slar ethods can e adoted for ncolete lock desgns n unfactoral stuatons such as one descred elow for the urose of ncreasng recson. If the nuer of relcatons of all ars of treatent n a desgn are the sae, t creates an ortant class of desgns known as alanced ncolete lock desgns (BIBD). hs classes of desgns rode equal recsons of all estates of all ars of treatent effects. hs was ntroduced y Yates n 936 for agrcultural exerents. It s known that the alanced ncolete lock desgn s not always sutale for aretal trals snce t requres large nuer of relcatons. In order to ake ncolete lock desgns aalale for all treatents wth saller nuer of relcatons, Das (958) and Gr (958) ntroduced ncolete lock desgns augented y certan nuer of addtonal treatents n each lock. hese new desgns are called renforced BIBD. Suose we hae a BIBD wth treatents, locks of sze k each, and r relcatons. We otan renforced alanced ncolete lock desgns y addng new treatents, say new ones, n each lock to ths BIBD. hen the resultng renforced alanced ncolete lock desgns wll hae ( ) treatents dstruted n locks each of sze ( k ) such that each of newly ntroduced treatents are relcated tes and the orgnal treatents are relcated r tes each. In renforced BIBD t s ossle to ntroduce arorate coarates n order to ncrease the recson n the desgn. In ths artcle, we deelo an exact ethod of analyss of coarance of renforced alanced ncolete lock desgns wth a sngle exlanatory arale. Suary of the results otaned n ths aer are as follows. In secton, we descre the odel and notatons for renforced alanced ncolete lock desgns wth one exlanatory arale. he estates of the odel araeters are roded n secton 3. In secton 4, we consder a odel for y -arate wthout the exlanatory arale and otaned least square (LS) estates for araeters of ths odel. In ths sae secton we 36

3 ANCOVA of Renforced BIBD 3 defne an ANOVA odel for x -arate (the exlanatory arale) only, and otaned LS estates for the araeters. Actually ths facltates the calculatons for the orgnal odel. rror su of squares and adusted treatent su of squares, error su of roducts and adusted treatent su of roducts for the aoe odfed odels are roded n secton 4. and secton 4., resectely. In secton 5 we consder the odel n resence of the exlanatory arale. In ths secton, we roded error su of squares and adusted treatent su of squares, error su of roducts and adusted treatent su of roducts n resence of the exlanatory x -arate. Fnally, arances of the estale treatent contrast are otaned n secton 6. We resent three tyes of arances, e.g., () Var ( t t), () Var ( t t), and () Var ( t t ), where t and t are the estates of orgnal treatent effects; t and t are the estates of the newly ntroduced treatent effects.. Model and Notatons Let n (,,, ;,,, ) denote the nuer of tes the th treatent occurs n th lock and n (,,, ;,,, ) denote the nuer of tes th newly ntroduced treatent occurs n th lock. In fact, f treatent aears n lock n 0 otherwse. In the context of renforced BIBD, we hae the followng relatonshs: ( k ) n n n n., n.. n. ( k ), n. r for,,, n for,,, nn for,,,, nn rfor,,, ;,, nn for, and,,. Suose that y (,,, ;,,, ) are the resonses fro the unt n th lock wth th treatent. he lnear addte ANCOVA odel for renforced alanced ncolete lock desgn s gen y y t x e (.) 37

4 D.K. Ghosh, M.G. Bhatt, and S.C. Bagu where s a constant ean effect; t s the effect of th treatent; s the effect of th lock; x denote the exlanatory arale fro the unt n th lock wth th treatent; e are the error coonents that are assued to e norally dstruted wth a zero ean and a constant arance ; s the regresson coeffcent. 3. staton of Paraeters wth a Sngle xlanatory Varale he estate of the unknown araeters, t,, and are otaned y nzng the error su of squares,, where s gen y ( ).(3.) n y t x he equaton (3.) rodes the followng noral equatons, x (3.) G ( k ) r t t ( k ) G ( ) ( ) ( x) (3.3) B k n t n t k B ( x) r rt n,,,, (3.4) ( x) t n,,, (3.5) n Gx t ( x) t ( x) B ( x) nx (3.6) where G s the grand total; G x s the grand total for the x -arate; B s the th lock total; B ( x) s the th lock total for the x -Varate; and are the treatent totals for orgnal treatents and newly ntroduced treatents, resectely. Below we ge ther atheatcal exressons. G n y, Gx nx ny, ( x) nx,,., B,,,, ; ny, B ( x) nx ; ny and ( x) n x, 38

5 ANCOVA of Renforced BIBD 5 Now alyng the restrctons, 0 and n the aoe noral r t t 0 equatons, the exressons for the estates of and are gen y G G x k ( ) (3.7) ( ) B k nt nt B( x) ( k ) (3.8) where the estate of t and t are otaned n ths secton elow and the estate s roded n secton 5. Now susttutng n (3.5), and after slfcatons we hae r Q t t t Q (3.9) ( x) ( k ) ( k ) where Q s the adusted total for th treatent and Q( x) s the adusted total for th treatent for x -arate. he exressons for ( k ) Q and ( x) Q are gen elow: Q n B and Q( x) ( x) nb( x), ( k ),,. Now the restrcton reduces (3.9) to r t t 0 Q t Q (3.0) ( x) hus, the estate of t ay e exressed as Also, usng (3.) we ay wrte t Q Q ( x ),,,. (3.) t Q Q( x). (3.) Slarly, the estate of th treatent effect t s otaned y susttutng fro (3.8) n (3.4) and alyng (3.). herefore, after ore slfcatons t s gen y t ( k Q ) ( r )( Q Q ) ( k Q ) (3.3) ( x) ( x) z r 39

6 D.K. Ghosh, M.G. Bhatt, and S.C. Bagu where z r( k ) r ; t stand for estate of the orgnal effects,,,, ; t stand for estate of the newly ntroduced treatent effects,,, ; Q and Q ( x) are gen, resectely, y Q n B and Q( x) ( x) nb( x). ( k ) ( k ) 4. staton of Paraeters wthout an xlanatory Varale he odel (.) wthout the exlanatory x -arate can e wrtten as y t e (4.) y ( y) ( y) ( y) where the notatons y, t( y), ( y), and e ( y) hae slar eanng and satsfy slar assutons as dscussed n secton. Now usng least square ethod of estaton as efore, we otan the estates of the araeters as follows. G y k ( ) (4.) ( y) B( k ) y nt( y) nt( y) ( k ) (4.3) t( y) ( k ) Q ( r ) Q [( rk ) r ] r,,,, (4.4) t ( y) Q,,,. (4.5) Now relacng y n (4.) y x, we get the followng new odel. he odel n ters of x -arate ay e wrtten as x t e (4.6) x ( x) ( x) ( x) where x, t( x), ( x), and e ( x) hae slar eanng and satsfy slar assutons as dscussed n secton wth resect to the exlanatory x -arate. Now usng least square ethod of estaton as efore, we otan the estates of the araeters gen (4.6) as Gx x k ( ) (4.7) (x) B( x) ( k ) x nt( x) nt( x) ( k ) (4.8) 40

7 ANCOVA of Renforced BIBD 7 t( x) ( k ) Q( x) ( r ) Q ( x) [( rk ) r ] r,,,, (4.9) t Q ( x) ( x),,,. (4.0) he estates gen n equatons (4.)-(4.5) and (4.7)-(4.0) are necessary to coute the exressons and gen n secton 4. and 4. resectely. In turn these wll hel n calculatng, the regresson coeffcent, gen n secton 5 elow. 4.. rror Su of Squares and Adusted reatent Su of Squares for y -arate wth no xlanatory Varale he error su of squares for the odel (4.) can e wrtten as y ( y) ( y) ( y) ny G t t B Now susttutng y and ( y ) fro (4.) and (4.3), recetely, n the aoe equaton, we otan n y B t( y) Q t( y) Q ( k ) (4..) Under the null hyothess Ho : t( y) 0,,,, and t ( y ) 0,,, the aoe equaton (4..) reduces to ny B ( k ). (4..) hus the adusted treatent su of squares for y -arate on the assuton of no exlanatory arale s gen y t Q t Q ( y) ( y) (4..3) Slarly, usng odel (4.6) the quanttes, the error su of squares for x -arate and, the error su of squares for x -arate under Ho : t( x) 0,,,, and t,,,, and, adusted treatent su of squares for x - ( x) 0 arate, can easly e otaned. In the slar anner the exresson for wll e 4

8 D.K. Ghosh, M.G. Bhatt, and S.C. Bagu t( x) Q( x) t( x) Q( x). 4.. rror Su of Products and Adusted reatent Su of Products he error su of roducts of x and y can e otaned usng odels (4.) and (4.6) as (4..) n x y G t t B x ( x) ( x) ( x) Now susttutng x and ( x ) fro (4.7) and (4.8), resectely, n the aoe equaton (4..), we get nx y BB( x) t( x) Q ( x) t( x) Q( x) ( k ) (4..) Under the hyothess Ho : t( x) 0,,,, and t( x) 0,,, the aoe equaton (4..) reduces to n x y B B (4..3) ( x) ( k ) hus the exresson for adusted treatent su of roducts s gen y t ( ) Q t ( ) Q x x. (4..4) 5. rror Su of Squares and Adusted reatent Su of Squares for y -arate n Presence of an xlanatory x -arate In ths secton, we wll focus on otanng estate of regresson coeffcent, error su of squares, and adusted treatent su of squares for y -arate n resence of ancllary x - arate. Below we state a useful lea. hs lea hels to dere the estates of error su of squares, adusted treatent su of squares, and regresson coeffcent ( ). Lea 5.. For analyss of coarance of renforced BIBD n resence of a sngle exlanatory arale the araeters y, t( y), t( y), and ( y) are estated as y x; t ( y) t t( x) ; t ( y) t t( x) ; t ( ) y ( x ). Proof. Susttutng the alues of x fro (4.6) n (.), we hae y ( ) ( t t ) ( t t ) ( ) ( e e ) (5.) x ( x) ( x) ( x) ( x) 4

9 ANCOVA of Renforced BIBD 9 Now, on coarng the coeffcents of (5.) wth (4.) we otan y x; t ( y) t t( x) ; t ( y) t t( x) ; t ( ) y ( x ). hs coletes the roof. 5.. state of Regresson Coeffcent We get the estate of fro the noral equaton (3.6). Now on susttutng and fro (3.7) and (3.8), resectely, t t( y) t( x), and t t ( y) t( x) n (3.6), we otan. Fnal we hae (5..) Under the hyothess Ho : t( x) 0,,,, and t( x) 0,,, (5..) wll reduce to (5..) 5.. rror su of squares and adusted reatent Su of Squares Fro (.), the error su of squares of y -arate wth an exlanatory arale x can e found as. (5..) n y G t t B n x y On susttutng and fro (3.7) and (3.8), resectely, t t( y) t( x), and t t t n (5..) we get ( y) ( x). (5..) Under the hyothess Ho : t( x) 0,,,, and t( x) 0,,, (5..) wll reduce to. (5..3) hus the adusted treatent su of squares of y -arate gen the exlanatory x -arate s otaned as the dfference ( ). he aoe results are suarzed n the for of 43

10 D.K. Ghosh, M.G. Bhatt, and S.C. Bagu ale and ale. ale shows sus of squares and sus of roducts and ale shows adusted error and treatent sus of squares. ale. Varance-Coarance ale for Analyss of Coarance of Renforced BIBD wth a Sngle xlanatory Varale Source d.f. Block(unad.) reatent(ad.) rror r r otal x B y B B S S S ale. Adusted Sus of Squares Source d.f. x reat(ad.) rror r reat+rror r y Ad. S.S. Now the arorate test statstcs for the hyothess Ho : t( x) 0,,,, and t,,, and s gen y ( x) 0 F ( ) f f (5..4) where F follows an dstruton wth f and f r. he null hyothess H o s reected f F F ; f, f where F ; f, f s an uer crtcal ont of dstruton wth f and f degrees of freedo. 6. Varance of reatent Contrasts hree tyes of treatent contrasts can e consdered for ths tye of desgns. Varances of these contrasts are descred as follows. Case-I. he estated arance of the estale treatent contrast ( t t ),,,, can e found usng (3.3) as 44

11 ANCOVA of Renforced BIBD Var ( ( k ) ( Q x Q x) t t ) rk ( ) r ( ) ( ). (6.) Snce the arance etween two adusted treatent estates s dfferent for each coarson, Fnney (946) and Das and Gr (986) suggested that an aerage alue of ( Q( x) Q ( x) ) can e relaced y ( x) ( ) and hence aerage arance of treatent contrast can e otaned as ( ) ( ) 'aerage' Var ( k x t ) t rk ( ) r ( ) (6.) where s the error ean squares. It can e otaned fro ale as ( r ). Case-II. he estated arance of the estale treatent contrast ( t t ),,,, ;,, can e found fro (3.) and (3.3) as ( k ) r Var( z rz t ) t r Q( x) ( k ) Q( x) Q ( x) z r where z r( k ) r. (6.3) Case-III. he estated arance of the estale treatent contrast ( t ),,, can e found fro (3.) as Q x Q x ( ) ( ) Var( t ) t. (6.4) he estated aerage arance of treatent contrast ( t ) can e found on susttutng the aerage alue of ( Q ) t ( x) Q ( x) y ( x) ( ) t n (6.4) as 45

12 D.K. Ghosh, M.G. Bhatt, and S.C. Bagu 'aerage'var( t ) t ( ) ( x) (6.5) where s otaned as efore. Acknowledgeents he authors are thankful to the dtor-n-chef and an anonyous referee for ther careful readng of the aer. Dr Bagu s research was artly suorted y UWF faculty catalyst ntate award. References [] M.N. Das, Renforced ncolete lock desgns, Journal of the Indan Socety of Agrcultural Statstcs, 0 (958) [] M.N. Das and N.C. Gr, Desgn and Analyss of xerents, nd ed., (Wley astern Ltd., New Delh, 986). [3] D.J. Fnney, Standard errors of yeld adusted for regresson on an ndeendent easureent, Boetrcs, (946) [4] N.C. Gr, On renforced PBIB desgns, Jour. Ind. Agr. Stat., (958) 4-5. [5] F. Yates, Incolete randozed locks, Annals of ugencs, 7() (936)