A General Construction Method of Simultaneous Confounding in

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1 hk Uiv J Sci : - Jly A Geerl Costrctio ethod of Simlteos Cofodig i - Fctoril Eerimet A Jlil ertmet of Sttistics iosttistics d formtics Uiversity of hk hk gldesh E-mil: mjlil@ivdhked Received o Acceted for Pblictio o Abstrct A geerl costrctio method of simlteos cofodig i is rime fctoril eerimet is roosed The cocet of mtri method i the costrctio of fctoril eerimet with sigle fctoril effect cofoded is sed to develo the method The rocedre my be eteded for the costrctio of simlteos cofodig of fctoril eerimet with three or more fctoril effects cofoded trodctio The reserchers workig with fctoril eerimets eeriece difficlty esecilly whe the mber of fctors s well s the mber of levels of ech fctor is lrge t becomes more difficlt if we hve o reqired mber of homogeeos lots i rctice sch sittios we re bod to se limited mber of homogeeos lots to lyze the fctoril effects As reslt some fctoril effects or iterctios will be mied with block effect ie cofoded Sice there is o wy to void this the higher order iterctio effects re slly cosidered to be cofoded ose d Kish ose described the costrctio of fctoril desigs sig fiite geometries The tretmets re rereseted by -tles where i re elemets of GF The method is vilble oly whe is rime or rime ower A system of simlteos cofodig i fctoril eerimet hs bee described where itrblock sbgro is costrcted with the commo elemets tke from the fctoril effects of two icomlete blocks ech cofoded with sigle fctoril effect Kemthore s described eqivlet method of ose i which some of the tretmet fctors re desigted s bsic fctors d the others s dded fctors Levels of dded fctors re derived by combitio of the levels of the bsic fctors over GF White d Hltqist eteded the field method to desig with mber of levels of tretmet fctors Joh d e described the costrctio of rticlr clss of sigle relicte block desigs which they cll geerlized cyclic desigs The essetil fetre of the method is tht the -tles givig the tretmets of rticlr block costitte Abeli gro the itrblock sbgro Ptterso described geerl comter lgorithm clled SGN i which levels of tretmet fctors re derived by lier combitios of levels of lot d block fctors The method rovides fiite-field geerlized cyclic d other desigs llick S A & develoed two systems of desigig fctoril effects with simlteos cofodig of two effects oe for d the other for - fctoril eerimets these systems of simlteos cofodig the combitio of levels ws bsed o some miltig mer Jlil et l develoed mtri method of desigig sigle fctoril effect cofoded i P - fctoril eerimet where the level combitios re obtied by mtri oertios of the levels Costrctio method of simlteos cofodig hs bee develoed ideedetly for d fctoril eerimets Jlil d llick The reset work is geerl method of costrctio for simlteos cofodig i Nottio d efiitio - is rime fctoril eerimet The formle give below hve bee sed i determiig the mber of icomlete blocks or itrblocks b d the mber of homogeeos lots i ech icomlete block or itrblock k for sigle or simlteosly cofoded fctoril eerimet symmetricl Let b be the mber of icomlete blocks or itrblocks i cofoded eerimet; d k be the mber of lots i ech icomlete block or itrblock The b d k ; where r is the mber of fctoril r r effects to be cofoded For sigle fctoril effect cofodig i fctoril eerimet the mber of icomlete blocks is comted s b d the mber of combitios of levels i ech icomlete block is give by k For simlteos cofodig two fctoril effects i fctoril eerimet the mber of itrblock sbgros is comted s b d the mber of level

2 combitios i ech itrblock sbgro is give by k The geerl costrctio mtri with its itrblock sbgros of simlteos cofodig i fctoril eerimet c be rereseted s ; d i rticlr the costrctio mtrices for d fctoril eerimets re give resectively by Col d Col Col Col Col Costrctio ethod of Simlteos Cofodig i Fctoril Eerimets the costrctio of is rime fctoril eerimet with sigle fctoril effect cofoded we c write the level combitios by the mtri method described below Jlil et l ; where icomlete blocks ; is give by: { { { with i i i{ i i colm vector of dimesio i ech is j j { times reetitios of the elemets of i ' s i scedig ordered level; ; i ; j ; with the restrictio tht i j ; m : sm vector of dimesio m ; d is clled the djstmet vector A Jlil From the eqtio is clled the key icomlete block of sigle fctoril effect cofoded i fctoril eerimet For l of simlteos cofodig of two fctoril effects i - fctoril eerimet we re to erform the followig stes Ste Fid the ideedet key icomlete blocks for the fctoril effects to be cofoded simlteosly sig Eqtio Let these key icomlete blocks be deoted by which reresets the level combitios of the key icomlete block for the first cofoded fctoril effect d reresets the level combitios of the key icomlete block for the secod cofoded fctoril effect Ste Fid the commo elemets of level combitios row vectors of these two key icomlete blocks d form mtri c be deoted by is clled the key itrblock sbgro of level combitios of two fctoril effects cofoded simlteosly t c be see tht the key itrblock sbgro cotis the combitio of the lowest levels for ll the fctors Ste To fid ll other itrblock sbgros of the costrctio mtri we will follow the comttios rocedres described below The itrblock sbgros below the key block re obtied by ddig the vectors c ; c with ech of the elemets row vectors of the key itrblock sbgro s described below is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of re the itrblock sbgros lced i the first colm of the costrctio mtri After hvig the itrblock sbgros i the first colm we will get the itrblock sbgros of the secod colm which c be comted s show below is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of ;

3 A Geerl Costrctio ethod of Simlteos Cofodig i - Fctoril Eerimet is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of The itrblock sbgros lced i the third colm c be comted s show below is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of Proceedig i this wy we will get the itrblock sbgros lced i the lst colm of the costrctio mtri which c be obtied s: is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of ; is obtied by dditio of the vector with ech of the vector elemets of ; Ths i the costrctio mtri we get itrblock sbgros i colms The method is illstrted with two emles described i the sectio below llstrtios Emle Sose we like to costrct the lyot of - fctoril eerimet where the fctoril effects AC d AC re cofoded simlteosly The l is give by mtri ; ; { { { ; with { ; { d { The djstmet vector cold be obtied by solvig the symbolic eqtio corresodig to the fctoril effect to be cofoded Ste Fid the mtrices d which re the key icomlete blocks cofoded with AC d C A resectively i - fctoril eerimet Ste Selectig the commo vector elemets from d we will get the key itrblock sbgro which is give by Ste Add the vector to ech of the vector elemets of the key itrblock sbgro we get

4 A Jlil To obti the third itrblock sbgro dd the vector to ech of the vector elemets of the key itrblock sbgro Ths The itrblock sbgros of secod colm d of the third lst colm c be comted s show below is obtied by ddig the vector to ech of the row vectors of ; is obtied by ddig the vector to ech of the row vectors of ; is obtied by ddig the vector to ech of the row vectors of ; d is obtied by ddig the vector to ech of the vectors of ; is obtied by ddig the vector to ech of the vectors of ; is obtied by ddig the vector to ech of the vectors of ; Ths we hve the comlete lyot of ie itrblock sbgros s show below t is esy to verify tht s s ; cofods the st effect AC ; colm comriso s s ; cofods the d effect AC ; row comriso s s ; cofods the st geerlized effect AC AC AC comrig - totls; d s s ; cofods the d geerlized effect AC AC comrig J-totls Emle Sose we re to costrct - fctoril eerimet where the fctoril effects AC d AC re cofoded simlteosly Soltio First we fid the two icomlete blocks d / corresodig cofoded effects AC d AC The key icomlete block cofoded with AC is give by { { with { ; { d the djstmet vector c be obtied by solvig the symbolic eqtio mod tkig first two

5 A Geerl Costrctio ethod of Simlteos Cofodig i - Fctoril Eerimet vles of d from the vectors d resectively for AC Similrly the key icomlete block / cofoded with AC is give by / { { with { ; { d the djstmet vector c be obtied by solvig the symbolic eqtio mod tkig first two vles of d from the vectors d resectively for AC Ste Select the commo elemets vectors of d to fid the key itrblock sbgro s show below Ste We get the secod itrblock sbgro by dditio of the vector to ech of the vectors of the key itrblock sbgro; is lced jst below colm side the key block Similrly we get the third forth d fifth itrblocks d by dditio the vectors d to ech of the vectors of the key itrblock sbgros resectively show below Ste After gettig the itrblock sbgros d we get the itrblock sbgros d by dditio of the vector to ech of the vector elemets row vectors of d resectively Similrly dditio of the vector to ech of the vector elemets of d will rodce the itrblock sbgros d A dditio of the vector to ech of the vector elemets of d will rodce the itrblock sbgros d ; d A dditio of the vector to ech of the vector elemets of d will rodce the itrblock sbgros d which comletes the l show below

6 Coclsio this rticle geerl costrctio method hs bee develoed for simlteos cofodig i is rime fctoril eerimet The costrctio of simlteos cofodig i fctoril eerimet becomes esier d rewrdig The method is restricted to fctoril eerimet whe is rime d it c be eteded for simlteos cofodig of three or more fctoril effects Fisher R A ; The Arrgemet of Field Eerimets Jorl of the iistry of Agricltre of Gret riti; - Fisher R A ; The esig of Eerimets Oliver & oyd Edibrgh Fisher R A ; The theory of cofodig i Fctoril eerimets i reltio to the theory of Gros A Ege - ose R C d K Kish ; O the roblem of cofodig i geerl symmetricl desig Skhy - ose R C ; themticl theory of the symmetricl fctoril desig Skhy - Kemthore O ; A Simle Aroch to Cofodig d Frctiol Relictio i fctoril eerimets iometrik - Kemthore O ; The desig d Alysis of Eerimets Joh Wiley & Sos c New York A Jlil s N ; A somewht ltertive roch for costrctio of symmetricl fctoril desigs d obtiig mimm mbers of fctors Clctt Sttisticl Associtio lleti - e A d J A Joh ; Sigle relicte fctoril eerimets i geerlized cyclic desigs: Symmetricl rrgemets JRSS - Ptterso H ; Geertio of fctoril desigs JRSS - s N d N C Giri ; esig d Alysis of Eerimet hil blictio Jlil A; S A llick d S Rhm S Jly ; tri method of esigig Lyot i - Fctoril Eerimets hk Uiversity Stdies - Jlil A d S A llick ; tri method of esigig Asymmetricl Fctoril Eerimet hk Uiv J Sci ;- Jlil A d S A llick ; tri method of esigig Simlteos Cofodig of Two Fctoril Effects i Fctoril Eerimet hk Uiv J Sci ; - Jlil A d S A llick ; tri method of esigig Simlteos cofodig i - Fctoril Eerimet hk Uiv J Sci ; Jlil A; ; Lier Eqtio ethod of esigig Fctoril Eerimet hk Uiv J Sci ;

Chapter 7 Infinite Series

Chapter 7 Infinite Series MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2