Chapter 9 Confounding If the number of factors or levels increase in a - PDF document

Analysis of Variance Shalabh, IIT Kanpur CKaper  If Ke uer of facor or OeveO Lcreae L a facorLaO e[perLe Ke Ke uer of reae NoZ Kere arLe Zo queLo. FLrO\ ZKa doe cofoudLJ ea ad ecodO\ KoZ doe L Ke A Babab | CKaper  | CofoudLJ | Suppoe Ke foOOoZLJ OocN arraJee L oped BOocN 1 BOocN 2 TKe OocN effec of OocN 1 ad 2 are ad  repecLveO\ Ke Ke averaJe repoe correpodLJ o reae coLaLo abab 11EyaaEybb yababrepecLveO\. +ere    1yaybyaby ad    1abab  deoe Ke repoe ad reae correpodLJ o  abab ad 1 repecLveO\. IJorLJ Ke facor 1/2 L BAB  1 yaEybEyabEy L e[preLOe a foOOoZ  122111.AababababSo Ke OocN effec L o pree L ad L L o L[ed up ZLK Kee Ze a\ Ka Ke aL effec L o cofouded. SLLOarO\ for Ke aL effec B 122111.BabbaabbaSo Kere L o OocN effec pree L ad Ku L o cofouded. For Ke LeracLo effec 1122121.ABabababab +ere Ke OocN effec are pree L . I fac Ke OocN effec are ad are L[ed up ZLK Ke reae effec ad cao e eparaed LdLvLduaOO\ fro Ke reae effec L . So L[ed up ZLK Ke OocN. a b | CKaper  | CofoudLJ | AOeraLveO\ Lf Ke arraJee of reae L OocN are a foOOoZ BOocN 1 BOocN 2 Ke Ke aL effec L e[preLOe a 1122121Aabababab Oerve Ka Ke OocN effec ad are pree L KL e[preLo. So Ke aL effec L  arraJee of reae. So Ke aL effec KL arraJee of reaeWe oLce Ka L L L our coroO o decLde ZKLcK of Ke effec L o e cofouded. TKe order L ZKLcK reae are ru L a OocN L deerLed radoO\. TKe cKoLce of Ke OocN o e ru fLr L aOo radoO\ decLded. TKe foOOoZLJ oervaLo eerJe fro Ke aOOocaLo of reaFor a given effect, when two treatment combinations with the same signs are assigned to one eatment combinations with the same but opposite signs are assigned to For e[apOe L Ke cae L cofouded Ke ZLK – LJ are aLJed o OocN 2. SLLOarO\ ZKe ZLK  LJ are aLJed o OocN 1 ZKerea TKe reao eKLd KL oervaLo L Ka Lf ever\ OocN Ka reae coLaLo L Ke for of OLear cora Ke effec are eLaOe ad Ku ucofouded. TKL L aOo evLde fro Ke Keor\ of OLear eLaLo Ka a OLear paraerLc fucLo L eLaOe Lf L L L Ke for of a OLear cora. TKe cora ZKLcK are o eLaOe are aLd o e cofouded ZLK Ke dLfferece eZee or OocN effec. TKe cora ZKLcK are eLaOe are aLd o e ucofouded ZLK OocN or free fro OocN effec. a b | CKaper  | CofoudLJ | e BOocN deLJ BIBD veru facNoZ Ze e[pOaL KoZ cofoudLJ ad BIBD copare oJeKer. CoLder a 2 facorLaO e[perLe ZKLcK eed Ke OocN L]e o e . Suppoe Ke raZ aerLaO avaLOaOe o coduc Ke e[perLe L uffLcLe oO\ for a OocN of L]e 4. Oe ca ue BIBD L KL cae ZLK paraeer 144bkr  BIBD Ke effLcLec\ facor L vE  .jjBIBDVarjjCoLder oZ a ucoeced deLJ L ZKLcK  ou of 14 OocN Je reae coLaLo L abcabcad reaLLJ  OocN Je reae coLaLo L OocN 2 a 1abbcacI KL cae aOO Ke effec    BCABBC ad are eLaOe u L o eLaOe ecaue Ke reae coLaLo ZLK aOO  ad aOO – LJ LLOocN1LOocN21111ABCabcabcabcabbcac are coaLed L Ke ae OocN. I KL cae Ke varLace of eLae of ucofouded aL effec ad LeracLo L /.  jjRBDVarjjad Kere are four OLear cora o Ke oaO varLace L 42/ ZKLcK JLve Ke facor / ad ZKLcK L aOOer Ka Ke varLace uder BIBD. We oerve Ka a Ke co of o eLJ aOe o eLae A Ze Kave eer eLae of   BCABBC ad ZLK Ke ae uer of repOLcae a L BIBD. SLce KLJKeroLeracLo are dLffLcuO o Lerpre ad are uuaOO\ o OarJe o L L ucK eer o ue cofoudLJ arraJee ZKLcK provLde eer eLae of Ke LeracLo L ZKLcK Ze are ore Lereed. Noe Ka KL e[apOe L for uderadLJ oO\. A ucK Ke cocep eKLd LcopOee OocN | CKaper  | CofoudLJ | TKe arraJee of reae coLaLo L dLffere OocN ZKere\ oe predeerLed effec eLKer aL or Lerac caOOed a cofoudLJ arraJee. For e[apOe ZKe Ke LeracLo L cofouded L a 2 facorLaO e[perLe Ke Ke cofoudLJ arraJee coL of dLvLdLJ Ke eLJK reae coLaLo Lo Ke foOOoZLJ Zo e abcabc 1abbcacWLK Ke reae of eacK e eLJ aLJed o Ke ae OocN ad eacK of Kee e eLJ repOLcaed Ke ae uer of Le L Ke e[perLe Ze a\ Ka Ze Kave a cofoudLJ arraJee of a 2 facorLaO L Zo OocN. I a\ e oed Ka a\ cofoudLJ arraJee Ka o e ucK Ka oO\ predeerLed LeracLo are cofouded ad Ke eLae of LeracLo TKe LeracLo ZKLcK are cofouded are caOOed Ke defLLJ cora of Ke cofoudLJ arraJee. A cofouded cora ZLOO Kave reae coLaLo ZLK Ke ae LJ L eacK OocN of Ke cofoudLJ arraJee. For e[apOe Lf Ke effec 111ABabc  L o e cofouded Ke pu aOO facor coLaLo ZLK  LJ L.e. 1  abc ad L oe OocN ad aOO oKer facor coLaLo ZLK – LJ L.e. abac ad L aoKer OocN. So Ke OocN L]e reduce o 2 facorLaO e[perLe. Suppoe Lf aOoJ ZLK cofouded Ze Za o cofoud aOo. To oaL ucK OocN coLder Ke OocN ZKere L cofouded ad dLvLde Ke L abcabcL dLvLded Lo foOOoZLJ Zo OocN cabcad Ke OocN 1abbcacL dLvLded Lo foOOoZLJ Zo OocN bcac | CKaper  | CofoudLJ | TKee OocN of 4 reae are dLvLded Lo 2 OocN ZLK eacK KavLJ 2 reae ad Ke\ are oaLed L Ke foOOoZLJ Za\. If oO\ L cofouded Ke Ke OocN ZLK  LJ of reae coLaLo L cacbcabcreae coLaLo L abab111ABCabc  abcabcLL foOOoZLJ OocN ZLK  LJ ZKe 111Cabc  cabacabc2 facorLaO e[perLe. IdeLf\ Ke reae coLaLo KavLJ coo  LJ L Kee Zo OocN L L ad LL. TKee reae coLaLo are ad So aLJ Ke Lo oe OocN. TKe reaLLJ reae coLaLo ou of   SLLOarO\ OooN Lo Ke a foOOoZLJ OocN ZLK – LJ ZKe 1abbcac foOOoZLJ OocN ZLK – LJ ZKe 1abab facorLaO e[perLe. IdeLf\ Ke reae coLaLo KavLJ coo – LJ L Kee Zo OocN L a ad . TKee reae coLaLo are 1 ad ZKLcK Jo Lo oe OocN ad Ke reaLLJ Zo reae coLaLo ad ou of cacbc ad Jo Lo aoKer OocN. So Ke OocN 1adababacbccabcWKLOe aNLJ Kee aLJe of reae coLaLo Lo four OocN eacK of L]e Zo Ze oLce Ka aoKer effec vL]. aOo Je cofouded auoaLcaOO\. TKu Ze ee Ka ZKe Ze cofoud Zo facor a KLrd facor L auoaLcaOO\ JeLJ cofouded. TKL LuaLo L quLe JeeraO. TKe defLLJ cora for a cofoudLJ arraJee cao e cKoe arLrarLO\. If oe defLLJ cora are eOeced Ke oe oKer ZLOO aOo | CKaper  | CofoudLJ | NoZ Ze pree oe defLLLo ZKLcK are uefuO L decrLLJ Ke cofoudLJ arraJee. GeeraOL]ed LeracLo GLve a\ Zo LeracLo Ke JeeraOL]ed LeracLo L oaLed \ uOLpO\LJ Ke facor L OO Ke er ZLK a eve e[poe. For e[apOe Ke JeeraOL]ed LeracLo of Ke facor ad L BCBCDABCDAD ad Ke JeeraOL]ed LeracLo of Ke facor A BBC ad L 232 A BBCABCABCBA e of aL effec ad LeracLo cora L caOOed Ldepede Lf o eer of Ke e ca e oaLed a a JeeraOL]ed LeracLo of Ke oKer eer of For e[apOe Ke e of facor A BBC ad L a Ldepede e u Ke e of facor BBCCD ad L o a Ldepede e ecaue BBCCDABCDAD ZKLcK L TKe reae coLaLo pqrabc… L aLd o e orKoJoaO o Ke LeracLo xyzABC Lf ....pxqyrz L dLvLLOe \ 2. SLce ......pqrxyz are eLKer 0 or 1 o a reae coLaLo L orKoJoaO o a LeracLo Lf Ke\ Kave a eve uer of Oeer L coo. Treae coLaLo 1 L orIf 1112212....ad...pqrpqrabcabc are oK orKoJoaO o ...xyzABC Ke Ke produc 121212ppqqrrabcL aOo orKoJoaO o ...xyz SLLOarO\ Lf Zo LeracLo are orKoJoaO o a reae coLaLo Ke KeLr JeeraOLaO o L. NoZ Ze JLve oe JeeraO reuO for a cofoudLJ arraJee. Suppoe Ze ZLK o Kave a cofoudLJ arraJee L 2 OocN of a 2 facorLaO e[perLe. TKe Ze Kave Ke foOOoZLJ | CKaper  | CofoudLJ | TKe L]e of eacK OocN L TKe uer of eOee L defLLJ cora L 21..21 LeracLo Kave If facor are o e cofouded Ke Ke uer of K order LeracLo ZLK facor L 12.... So Ke oaO uer of facor o e If a\ Zo LeracLo are cofouded Ke KeLr JeeraOL]ed LeracLo are aOo TKe uer of Ldepede cora ou of 21 defLLJ cora L ad re are oaLed a JeeraOL]ed LeracLo. Nuer of effec JeLJ cofouded auoaLcaOO\ L 21. To LOOurae KL coLder a 2 facorLaO 5 ZLK 5 facor vL].  BCD ad . TKe facor are o e cofouded L 2 OocN 3. So Ke L]e of eacK OocN L 24. TKe uer of defLLJ cora L 21. TKe uer of Ldepede cora ZKLcK ca e cKoe arLrarLO\ L 3L.e.  ou of  defLLJ cora. Suppoe Ze cKooe foOOoZLJ CDE ad Ke Ke reaLLJ 4 ou of ed a ACECDEACDEADACEABDEABCDEBCDCDEABDEABCDEABC2223.ACECDEABDEABCDEBE | CKaper  | CofoudLJ | p ACDE A BCDEKe Ke defLLJ cora are oaLed a 222ABCDACDEABCDEBE2222ABCDABCDEABCDEE2222ACDEABCDEABCDEB32332.ABCDACDEABCDEABCDEACDI KL cae Ke aL effec A a ruOe r\ o cofoud a far a poLOe KLJKerorder LeracLo oO\ ecaue Ke\ are Afer eOecLJ Ldepede defLLJ cora dLvLde Ke 2 reae coLaLo Lo 2Jroup of 2 coLaLo eacK ad eacK Jroup JoLJ Lo oe OocN. PrLcLpaO Ne\ OocN TKe Jroup coaLLJ Ke coLaLo 1 L caOOed Ke prLcLpaO OocN or Ne\ OocN. I coaL aOO Ke reae coLaLo ZKLIf Kere are Ldepede defLLJ cora Ke a\ reae coLaLo L Ke prLcLpaO aO OocN ZrLe Ke reae coLacKecN eacK oe of Ke for orKoJoaOL\. If Zo reae coLaLo eOoJ o Ke prLcLpaO OocN KeLr produc aOo eOoJ o WKe a feZ reae coLaLo of Ke prLcLpaO OocN Kave ee deerLed oKer reae coLaLo ca e oaLed \ uOLpOLcaLo ruOe. NoZ Ze LOOurae Kee ep L Ke foOOoZLJ e[apOe. | CKaper  | CofoudLJ | CoLder Ke e up of a 2 facorLaO e[perLe L ZKLcK Ze Za o dLvLde Ke oaO reae effec Lo 2 Jroup \ cofoudLJ Kree effec ad DBEABC. TKe JeeraOL]ed LeracLo L KL cae are DBEBCDACECDEI order o fLd Ke prLcLpaO OocN fLr ZrLe Ke reae coLaLo L a adard order a foOOoZ 1 bcdbceabcecdeacdeabcdePOace a reae coLaLo L Ke prLcLpaO OocN Lf L Ka a eve uer of Oeer L coo ZLK Ke cofoudLJ effec A DBE ad TKe prLcLpaO OocN Ka 1acdbce ad abdeacdbce. OaL oKer OocN of cofoudLJ arraJee fro Ke prLcLpaO OocN \ uOLpO\LJ Ke reae coLaLo of Ke prLcLpaO OocN \ a reae coLaLo o occurrLJ L L or L a\ oKer OocN aOread\ oaLed. I oKer Zord cKooe reae coLaLo o occurrLJ L L ad uOLpO\ ZLK Ke L Ke prLcLpaO OocN. CKooe oO\ dLLc OocN. I KL cae oaL oKer OocN \ uOLpO\LJ ababcacbcabc OLNe a L Ke ArraJee of Ke reae L OocN ZKe A DBE are cofouded PrLcLpaO BOocN 1 BOocN 2 BOocN 3 BOocN 4 BOocN 5 BOocN 6 BOocN  BOocN  bc bceabceaceacdecde For e[apOe OocN 2 L oaLed \ uOLpO\LJ ZLK eacK facor coLaLo L Ke prLcLpaO OocN a 1aaacdaacdcdbceaabceabdeaabdebdeOocN 3 L oaLed \ uOLpO\LJ ZLK 1 acdbce ad ad LLOarO\ oKer OocN are oaLed. If a\ oKer reae coLaLo L cKoe o e uOLpOLed ZLK Ke reae L Ke prLcLpaO OocN Ke Ze Je a OocN ZKLcK ZLOO e oe aoJ Ke OocN 1 o . For e[apOe Lf L uOLpOLed ZLK Ke reae L Ke prLcLpaO OocN Ke Ke OocN oaLed coL of 1aeaeacdaecdebceaeabcabdeaebd ZKLcK L ae a Ke OocN . | CKaper  | CofoudLJ | AOeraLveO\ Lf A CDABCD ad ABCDE are o e cofouded Ke Ldepede defLLJ cora are   CDABCDABCDEacad.cdacadWKe a effec L cofouded L ea Ka L L o eLaOe. TKe foOOoZLJ ep are foOOoZed o rLe ZLK cofouded effec OaL Ke u of quare due o aL ad LeracLo effec L Ke uuaO Za\ a Lf o Drop Ke u of quare correpodLJ o cofoudLJ effec ad reaL oO\ Ke u of FLd Ke oaO u of quare. OaL Ke u of quare due o error ad aocLaed deJree o