Chapter Partial Confounding The objective of c onfou - PDF document

Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur CKapter 10 PartLaO ConfoundLnJ TKe obMectLve of confoundLnJ L to mL[ tKe Oe Lmportant treatment combLnatLon ZLtK tKe bOocN effect 1    M A B Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur ZKere 1 yabab denote tKe vector of totaO repone Ln tKe repOLcatLon and eacK treatment L repOLcated tLme 1.2.... If no factor L confounded tKen tKe factorLaO effect are etLmated uLnJ aOO tKe repOLcate a rrr A A B y ZKere tKe vector of contrat ABAB are JLven b\ 1111 1111 1111 .We Kave Ln tKL cae AABBABABTKe um of quare due to and BAB can be accordLnJO\ modLfLed and e[preed a 11abababbaand 1ABiABABababrepectLveO\. NoZ conLder a LtuatLon ZLtK 3 repOLcate ZLtK eacK conLtLnJ of 2 LncompOete bOocN a Ln tKe foOOoZLnJ fLJure: Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur TKere are tKree factor and. BAB In cae of totaO confoundLnJ a factor L confounded Ln aOO trepOLcate. We conLder Kere tKe LtuatLon of partLaO confoundLnJ Ln ZKLcK a factor L not confounded Ln aOO tKe repOLcate. RatKer tKe factor L confounded Ln repOLcate 1 tKe factor L confounded Ln repOLcate 2 and tKe LnteractLon L confounded Ln repOLcate 3. Suppoe eacK of tKe tKree repOLcate L repeated tLme. So tKe obervatLon are noZ avaLOabOe on repetLtLon of eacK of tKe bOocN Ln tKe tKree repOLcate. TKepartLtLon of repOLcatLon tKe bOocN ZLtKLn repOLcate and pOot ZLtKLn bOocN beLnJ randomL]ed. NoZ from tKe etup of fLJure tKe factor can be etLmated from repOLcate 2 and 3 a Lt L confounded Ln repOLcate 1. tKe factor can be etLmated from repOLcate 1 and 3 a Lt L confounded Ln repOLcate 2 and tKe LnteractLon can be etLmated from repOLcate 1 and 2 a Lt L confounded Ln repOLcate 3. L etLmated from tKe repOLcate 2 onO\ tKen Lt etLmate L JLven b\ 2 2 A ireprepand ZKen L etLmated from tKe repOLcate 3 onO\ tKen Lt etLmate L JLven b\ 3 3 A ireprepZKere A and A are tKe uLtabOe vector of 1 and 1 for beLnJ tKe OLnear functLon to be contrat under repOLcate 2 and 3 repectLveO\. Note tKat eacK L a vectorKavLnJ 4 eOement Ln Lt. NoZ Lnce L etLmated from botK tKe repOLcate 2 and 3 o to combLne tKem and to obtaLn a LnJOe etLmator of  Ze conLder tKe arLtKmetLc mean of andreprep a an etLmator of JLven b\ RepOLcate 1  confounded RepOLcate 2  confounded RepOLcate 3  confounded BOocN 1 BOocN 2 1 BOocN 1 BOocN 2 1BOocN 1 BOocN 2 1 Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur 2 13 1reprepreprep§·§·¨¸¨¸©¹©¹ZKere tKe vector AAAKa  eOement Ln Lt and ubcrLpt Ln A denote tKe etLmate of under ³partLaO confoundLnJ´ . TKe um of quare under partLaO confoundLnJ Ln tKL cae L obtaLned a 2 2 AiAiAumLnJ tKat are Lndependent and Vary for aOO and  tKe varLance of A L JLven b\ 2 23 3pcAiAirepAirepVarAVaryVaryyNoZ uppoe L not confounded Ln an\ of tKe bOocN Ln tKe tKree repOLcate Ln tKL e[ampOe. TKen can be etLmated from aOO tKe tKree repOLcate eacK repeated tLme. Under ucK a condLtLon an etLmate of can be obtaLned uLnJ tKe ame approacK a tKe arLtKmetLc mean of tKe etLmate obtaLned from eacK of tKe tKree repOLcate a 123 1 12 23 3111 reprepreprrr irepAirepAirepiiiAAAyyy¦¦¦ Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur ZKere 121tKe vector 123AAAAKa 12 eOement Ln Lt. TKe varLance of aumLnJ tKat are Lndependent and Vary for aOO and LntKLcaeLobtaLneda 1 2 3 111123 2 2 2rrrpcAiAiAiiiirepreprepVarAVaryyy§·§·§·¨¸¨¸¨¸©¹©¹©¹¦¦¦If Ze compare tKL etLmator ZLtK tKe earOLer etLmator for tKe LtuatLon ZKere L unconfounded Ln aOO repOLcate and Za etLmated b\ r A ii y and Ln tKe preent LtuatLon of partLaO confoundLnJ tKe correpondLnJ etLmator of L JLven b\ 123 A repreprep AAABotK tKe etLmator vL]. and A are ame becaue L baed on repOLcatLon ZKerea p c A L baed on repOLcatLon. If Ze denote tKen p c A become ame a . TKe e[preLon of varLance of and p c A tKen aOo are ame Lf Ze ue 3. ComparLnJ tKem Ze ee tKat tKe LnformatLon on Ln tKe partLaOO\ confounded cKeme reOatLve to tKat Ln unconfounded cKeme L 2 2 222/23/3 If 22  tKen tKe LnformatLon Ln partLaOO\ confounded deLJn L more tKan tKe LnformatLon Ln unconfounded deLJn. Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur In totaO confoundLnJ cae tKe confounded effect L compOeteO\ Oot but Ln tKe cae of partLaO confoundLnJ ome LnformatLon about tKe confounded effect can be recovered. For e[ampOe tZo tKLrd of tKe totaO LnformatLon can be recovered .KL cae for SLmLOarO\ ZKen L etLmated from tKe repOLcate 1 and 3 eparateO\ tKen tKe LndLvLduaO etLmate of are JLven b\ andreprepreprepBotK tKe etLmator are combLned a arLtKmetLc mean and tKe etLmator of baed on partLaO confoundLnJ L 1 3 reprepBiBireprep§·§·¨¸¨¸©¹©¹ ZKere tKe vector BBBKa  eOement. TKe um of quare due to L obtaLned a BiBi§·§·¨¸¨¸©¹©¹AumLnJ tKat are Lndependent and Vary  tKe varLance of B r p cBiVarBVary Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur L etLmated from tKe repOLcate 1 and 2 eparateO\ tKen Lt etLmator baed on tKe obervatLon avaLOabOe from repOLcate 1 and 2 are 2 1ABirepreprepreprepectLveO\. BotK tKe etLmator are combLned a arLtKmetLc mean and tKe etLmator of L obtaLned 1 2 reprepABiABireprepABiABAB§·§·¨¸¨¸©¹©¹WKere tKe vector ABABABconLt of  eOement. TKe um of quare due to A L ABiABABABiand tKe varLance of A under tKe aumptLon tKat are Lndependent and Vary L JLven b\ r p cABiVarABVary Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur BOocN um of quare: Note tKat L cae of partLaO confoundLnJ tKe bOocN um of quare ZLOO Kave tZo component – due to repOLcate and ZLtKLn repOLcate. So tKe uuaO um of quare due to bOocN need to be dLvLded Lnto tZo component baed on tKee tZo varLant.. NoZ Ze LOOutrate KoZ tKe um of quare due to bOocN are adMuted under partLaO confoundLnJ. We conLder tKe etup a Ln tKe earOLer e[ampOe. TKere are 6 bOocN 2 bOocN under eacK repOLcate 1 2 and 3 eacK repeated tLme. So tKere are totaO 6 deJree of freedom aocLated ZLtK tKe um of quare due to bOocN. TKe um of quare due to bOocN L dLvLded Lnto tZo part tKe um of quare due to repOLcate ZLtK 31 deJree of freedom and tKe um of quare due to ZLtKLn repOLcate ZLtK deJree of freedom. NoZ denotLnJ B to be tKe totaO of bOocN and R to be tKe totaO due to repOLcate tKe um of quare due to bOocN L TotaOnumberofbOocNL1222222TotaOnumberoftreatment12212212221222212BlockpciiiiiiiSSBBNrBRRBRRZKere denote tKe totaO of bOocN Ln repOLcate 12. TKe um of quare due to bOocN ZLtKLn repOLcatLon  L BOocNwriSSRTKe um of quare due to repOLcatLon L BOocN212SSR Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur So Ze Kave Ln cae of partLaO confoundLnJ BOocNBOocNBOocNwrrSSSSSSTKe totaO um of quare remaLn ame a uuaO and L JLven b\ TotaO12.pcijkijkSSyNrTKe anaO\L of varLance tabOe Ln tKL cae of partLaO confoundLnJ L JLven Ln tKe foOOoZLnJ tabOe. TKe tet of K\potKeL can be carrLed out Ln a uuaO Za\ a Ln tKe cae of factorLaO e[perLment. SourceSum of quareDeJree of freedomMean quare RepOLcateBOocN ZLtKLn repOLcateFactor Factor A ErrorBOocNrBOocNSSwr p c A p cB p c A B\ ubtractLon3311 6323 BOocN BOocNZr Bpc TotaO 12141 E[ampOe 2: ConLder tKe etup of factorLaO e[perLment. TKe bOocN L]e L and 4 repOLcatLon are made a Ln tKe foOOoZLnJ fLJure. RepOLcate 1 A confounded RepOLcate 2 A confounded RepOLcate 3 confounded RepOLcate 4 A confounded BOocN 1 BOocN 2 1 BOocN 1 BOocN 2 1abc BOocN 1 BOocN 2 1 BOocN 1 BOocN 2 1 bc Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur TKe arranJement of tKe treatment Ln dLfferent bOocN Ln varLou repOLcate L baed on tKe fact tKat dLfferent LnteractLon effect are confounded Ln tKe dLfferent repOLcate. TKe LnteractLon effect confounded Ln repOLcate 1 L confounded Ln repOLcate 2 L confounded Ln repOLcate 3 and confounded Ln repOLcate 4. TKen tKe repOLcatLon of eacK bOocN are obtaLned. TKere are totaO eLJKfactor LnvoOved Ln tKL cae LncOudLnJ 1. Out of tKem tKree factor vL]. and are unconfounded ZKerea and BBCACABC are partLaOO\ confounded. Our obMectLve L to etLmate aOO tKee factor. TKe unconfounded factor can be etLmated from aOO tKe four repOLcate ZKerea partLaOO\ confounded factor can be etLmated from tKe foOOoZLnJ repOLcate: A from tKe repOLcate 2 3 and 4 A from tKe repOLcate 1 3 and 4 from tKe repOLcate 1 2 and 4 and A from tKe repOLcate 1 2 and 3. We fLrt conLder tKe etLmatLon of unconfounded factor A and ZKLcK are etLmated from aOO tKe four repOLcate. TKe etLmatLon of factor from repOLcate 1234 L a foOOoZ: 111416Ajirepj A jrepjAjijji A ¦¦¦ZKere 321 tKe vector 1234AAAAAKa 32 eOement and eacK 1234 L KavLnJ  eOement Ln Lt. TKe um of quare due to L noZ baed on 32 eOement a AiAi§·§·¨¸¨¸©¹©¹ Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur AumLnJ tKat are Lndependent and Vary and  tKe varLance of L obtaLned a r A VarAVarySLmLOarO\ tKe factor L etLmated a an arLtKmetLc mean of tKe etLmate of from eacK repOLcate a B ZKere 321tKe vector 1234BBBBBconLt of 32 eOement. TKe um of quare due to L obtaLned on tKe LmLOar OLne a Ln cae of TKe varLance of L obtaLned on tKe LmLOar OLne a Ln tKe cae of a VarB TKe unconfounded factor L aOo etLmated a tKe averaJe of etLmate of from aOO tKe repOLcate a ZKere 321tKe vector 1234CCCCCconLt of 32 eOement.TKe um of quare due to Cy Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur TKe varLance of L obtaLned a .VarC Ne[t Ze conLder tKe etLmatLon of tKe confounded factor.TKLfactor BAB can be etLmated from eacK of tKe repOLcate 2 3 and 4 and tKe fLnaO etLmate of can be obtaLned a tKe arLtKmetLc mean of tKoe tKree etLmate a 2 23 33 4111reprepreprrrABirepABirepABirepiiiABiABABAByyy§·§·§·¨¸¨¸¨¸©¹©¹©¹¦¦¦ZKere 241 tKe vector 234ABABABABconLt of 24 eOement and eacK of tKe vectorABAB and L KavLnJ  eOement Ln Lt. TKe um of quare due to A L tKen baed on 24 eOement JLven a ABiABiABAB§·§·¨¸¨¸©¹©¹TKe varLance A Ln tKL cae L obtaLned under tKe aumptLon tKat are Lndependent and eacK Ka varLance a 2 3 4 111234pcABirrrABiABiABiiiirepreprepVarABVaryVaryyyrrr§·§·§·¨¸¨¸¨¸©¹©¹©¹¦¦¦ Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur TKe confounded effect L obtaLned a tKe averaJe of etLmate of obtaLned from tKe repOLcate 1 3 and 4 a 134repreprepACiACACACZKere 241 tKe vector 134ABACACconLt of 24 eOement TKe um of quare due to Ln tKL cae L JLven b\ ACiTKe varLance of Ln tKL cae under tKe aumptLon tKat are Lndependent and eacK Ka varLance L JLven b\ .VarAC SLmLOarO\ tKe confounded effect BC L etLmated a tKe averaJe of tKe etLmate of obtaLned from tKe repOLcate 1 2 and 4 a 124repreprepBCiBCBCBCZKere tKe vector 134  BCBCBCBCconLt of 24 eOement. TKe um of quare due to Ln tKL cae L baed on 24 eOement and L JLven a BCi Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur TKe varLance of Ln tKL cae L obtaLned under tKe aumptLon tKat are Lndependent and eacK Ka varLance a VarBC /atO\ tKe confounded effect can be etLmated fLrt from tKe repOLcate 12 and 3 and tKen tKe etLmate of obtaLned a an averaJe of tKee tKree LndLvLduaO etLmate a 123repreprepABCABCABCZKere 241 tKe vector 123ABCABCABCABCconLt of 24 eOement. TKe um of quare due to Ln tKL cae L baed on eOement and L JLven b\ ABCiABCTKe varLance of Ln tKL cae aumLnJ tKat are Lndependent and eacK Ka varLance L JLven .VarABC If an unconfounded deLJn ZLtK repOLcatLon Za ued tKen tKe varLance of eacK of tKe factor  BCABBCAC and L ZKere L tKe error varLance on bOocN of L]e . So tKe reOatLve effLcLenc\ of a confounded effect Ln tKe partLaOO\ confounded deLJn ZLtK repect to tKat of an unconfounded one Ln a comparabOe deLJn L 2 2 226/3/4 So tKe LnformatLon on a partLaOO\ confounded effect reOatLve to an unconfounded effect L . If 224/3 tKen partLaOO\ confounded deLJn JLve more LnformatLon tKan tKe unconfounded deLJn. Analysis of Variance | CKapter 10 | PartLaO ConfoundLnJ | Shalabh, IIT Kanpur FurtKer tKe um of quare due to bOocN can be dLvLded Lnto tZo component – ZLtKLn repOLcate and due to repOLcatLon. So Ze can ZrLte BOocNBOocNZrBOocNrSSSSSSZKere tKe um of quare due to bOocN ZLtKLn repOLcatLon  BOocNZrSSRZKLcK carrLe deJree of freedom and tKe um of quare due to repOLcatLon L BOocNr232SSRZKLcK carrLe 41 deJree of freedom. TKe totaO um of quare L TotaOpcijkijkSSyTKe anaO\L of varLance tabOe Ln tKL cae of factorLaO under partLaO confoundLnJ L JLven a Source Sum of quare DeJree of freedom Mean quare RepOLcate BOocN ZLtKLn repOLcate Factor Factor Factor A ABC Error lockr lockwrAB(pc) AC(pc) BC(pc) ABC(pc) b\ ubtractLon   lockr lockwrAC(pc) BC(pc) ABC(pc) TotaO – 1 Tet of K\potKeL can be carrLed out Ln tKe uuaO Za\ a Ln tKe cae of factorLaO e[perLment.