Analysis of Variance| Chapter 7 Partial.IBDShalabh, IIT Kanpur - PDF document

| CKp 7 | PLO. ID | CKp 7  OocN DLJn PID TK Oncd LncopO OocN dLJn Kv vO dvnJ. TKy  conncd dLJn  ZOO  K OocN L]  Oo quO. A LcLon on uLnJ K  n nd OocN L] L K oO nu of quLd pOLc L AnoK Lpon popy of K ID L K L L ffLcLncy Oncd. TKL n K OO K L K ocL of n B Kn L Oo K ocL of n A Analysis of Variance | CKp 7 | PLO. ID | EcK n Ln K  K [cOy n Ln K  ZKLcK  K ocL 12... do no dpnd on K n A If ny Zo n nd  K ocL Kn K nu of n ZKLcK  oK ocL nd ocL of L j nd L Lndpndn of K pL of nd TK nu ...12...mjkvnnnpijkm  cOOd K ocL pLOOy W conLd noZ K [pO d on cnJuO nd LnJuO ocLLon cK o undnd . RcnJuO AocLLon ScK ConLd n [pO of ocL cO. L K  L[ n dnod  1 2 3 4 5 nd 6. Suppo K nUnd KL nJn ZLK pc o cK yoO K Zo oK yoO Ln K  oZ  K fL ocL. On noK yoO Ln K  coOun L K cond ocL LnLnJ Zo yoO  Ln K oK oZ  K KLd ocL. Fo [pO ZLK pc o n 1 n 2 nd 3 occu Ln K  oZ o Ky  K fL ocL of n 1 n 4 occu Ln K ocL of n 1 nd K LnLnJ n 5 nd 6  K KLd ocL of n 1  Ky occu Ln K oK SLLOOy fo n 5 n 4 nd 6 occu Ln K  oZ o Ky  K fL ocL of n 5 n 2 occu Ln K cL of n 5 nd LnLnJ n 1 nd 3  Ln K oK cond oZ o Ky  K KLd ocL of n 5. 1 2 4 5 6 Analysis of Variance | CKp 7 | PLO. ID |  of OO K L[ n. Tn nu ocL ocL ocL fo n 1 K nu of fL ocL 2 nu of cond ocL 1 K nu of KLd ocL 2 TK  vOu of 123ndnnn KoOd u fo oK n Oo. NoZ Z undnd K condLLon LLL of K dfLnLLon of pLOOy Oncd ocLLon cK Od o j ConLd n 1 nd 2. TKy  K fL ocL ZKLcK n  1 L.. n 1 nd 2  K fL ocLn 6 L K KLd ocL ZKLcK n of n 1 nd Oo K KLd ocL ZKLcK n of n 2. TKu K nu of n ZKLcK  oK L.. K  3 ocL of n K1 nd ocL of n K2th  1 ocL L SLLOOy conLd K n 2 nd 3 ZKLcK  K fL ocL  ZKLcK n 1 n 4 L K KLd ZKLcK n ocL of n 2 nd n 4 L Oo K KLd ZKLcK n ocL of n 3. TKu 123pijk cn Oo  oLnd LLOOy. : TKL Kod cn  ud o Jn  3cO ocLLon cK Ln JnO fo n yoO y nJLnJ K Ln oZ nd coOun. Analysis of Variance | CKp 7 | PLO. ID | TK LnJuO ocLLon cK JLv L o  2cO ocLLon cK. L K    of n ZKLcK  dnod  12… TK n Ln KL cK  nJd Ln oZ nd coOun ZK. 1TK yoO  nJd  foOOoZ: TK poLLon  fLOOd up Ln K poLLon ov K pLncLpO dLJonO y K n nu 1 2.. copondLnJ o K yoO. FLOO K poLLon OoZ K pLncLpO dLJonO yLcOOy. TKL LJnn L KoZn 1 2 3 4 CoOun 1 2 3 2 1 2 2 3 2 4 3 2 2 qq - 1 21/2 NoZ d on KL nJn Z dfLn K fL nd cond ocL of K n  foOOoZ. TK yoO nLnJ Ln K  coOun 12...iiq  K fL ocL of nd   K cond ocL. TKu Zo n Ln K  oZ o Ln K  coOun  K fL ocL of n . TZo n ZKLcK do no occu Ln K  oZ o Ln K  coOun  K cond ocL of n NoZ Z LOOu KL nJn y K foOOoZLnJ [pO: L K  10 n. TKn  . TK n n dnod  1 2… 10  nJd und K LnJuO  Analysis of Variance | CKp 7 | PLO. ID | 1 2 3 4 5 CoOun 1 2 3 4 2 1 5 6 7 3 2 5   4 3 6  5 4 7  10 Fo [pO fo n 1 K n 2 3 nd 4 occu Ln K  oZ o  coOunn 5 6 nd 7 occu Ln K  coOun o  oZ. So K n 2 3 4 5 6 nd 7  K fL ocL of n. TKn  of K n  of n 1. Tn nu FL ocL Scond ocL 1 2 3 4 5 6 7   10 2 1 3 4 5   6 7 10 3 1 2 4 6  10 5 7  4 1 2 3 7  10 5 6  5 1 6 7 2   3 4 10 6 1 5 7 3  10 2 4  7 1 5 6 4  10 2 3   2 5  3 6 10 1 4 7  2 5  4 7 10 1 3 6 10 3 6  4 7  1 2 5 W ov fo KL O K K nu of fL nd cond ocL of cK of K 10 n 10 L  ZLK 121263nd1.nnnnv Fo [pO K n 2 Ln K coOun of fL ocL occu L[ L vL]. Ln fL KLd fouK fLfK LJKK nd nLnK oZ. SLLOOy K n 2 Ln K coOun of cond ocL occu K L vL]. Ln K L[K vnK nd nK oZ. SLLO concOuLon cn  vLfLd fo oK n. Analysis of Variance | CKp 7 | PLO. ID | TK  L[ p vL]. 111122221122122111221221ondo ppppppp ZKLcK cn  nJd Ln yLc Lc nd  foOOoZ: 111111121112112221222122ppppppppªºªº…»…»…»…»¬¼¬¼W ZouOd OLN o cuLon K d no o d  qu of u 2 Ln L onOy  upcLp.32422120ªºªº…»…»¬¼¬¼In od o On KoZ o ZL K Lc Z conLd n 1 2 nd . No K K n  L K cond ocL of n 1. ConLd onOy K oZ copondLnJ o n 1 2 nd  nd oLn K On of nd  foOOoZ: p Tn 1 nd 2  K fL ocL of cK oK. TK  K coon n vL]. 3 4 nd 5 Zn K fL ocL of n 1 nd K fL ocL of n Tn 2 Tn 1 of n 1 n 2 TKo K n ZKLcK  K 1of n 1 nd 2. Analysis of Variance | CKp 7 | PLO. ID | 1221nd: Tn 1 nd 2  K fL ocL of cK oK. TK  Zo n vL]. 6 nd 7 ZKLcK  coon Znn 1 nd K cond ocL of n 2. So 1221 p p Tn 2 Tn 1 of n 1 ocL of n 2 TKo Zo n ZKLcK  K 1of n 1 nd K 2ocL of n 2. Analysis of Variance | CKp 7 | PLO. ID | p Tn 1 nd 2  K fL ocL of cK oK. TK L onOy on n vL]. n 10 ZKLcK L coon ocL of n 2. So Tn 2 Tn 1 ocL of n 1 ocL of n 2 TK n ZKLcK ocL of n 1 nd 2 10 Analysis of Variance | CKp 7 | PLO. ID | p Tn 1 nd   K c  fou n vL]. 235 K fL ocL of n 1 nd K fL ocL of n . So Tn  Tn 1 of n 1 n  TKo fou n ZKLcK  K 2ocL of n ocL Analysis of Variance | CKp 7 | PLO. ID | 1221nd: Tn 1 nd   K K  Zo n Zn K fL ocL of n 1 nd K cond ocL of n . So 1221 p p Tn  Tn 1 of n 1 n  TKo Zo n ZKLcK  K 1ocL of n ocL Analysis of Variance | CKp 7 | PLO. ID | p Tn 1 nd   K c L no n ZKLcK L ond ocL of n . So coOun of  qu  ud Kn fo 12423344245 K  no cond ocL. TKL L  dJn c ZK cond cnno  dfLnd. Tn  Tn 1 ocL of n 1 ocL of n  TK  no n ZKLcK  ocL of n 1 nd  ocL Analysis of Variance | CKp 7 | PLO. ID | i j FuK L L y o  K OO K p Ln  c.  no Lndpndn. ConucLon of OocN of PITK OocN of  PID cn  oLnd Ln dLffn Zy KouJK n ocLLon cK. Evn dLffn OocN ucu cn  oLnd uLnJ K  ocLLon cK. W noZ LOOu KL y oLnLnJ dLffn OocN ucu uLnJ K  LnJuO cK. ConLd K OL LOOuLon ZK 10 ZLK Z conLdd nd K fL cond nd KLd ocL of n AppocK 1: On Zy o oLn K n Ln  OocN L o conLd K n Ln cK oZ. TKL conLu K  of n o  LJnd Ln  OocN. WKn K OocN of PID  oZ of K foOOoZLnJ O. Fo KL nJn K n  LJnd Ln dLffn OocN nd foOOoZLnJ OocN  TK p of ucK  dLJn  510241nd0.bvrk 1 2 3 4 5 CoOun 1 2 3 4 2 1 5 6 7 3 2 5   4 3 6  5 4 7  10 OocN Tn OocN 1 1 2 3 4 OocN 2 1 5 6 7 OocN 3 2 5   OocN 4 3 6  10 OocN 5 4 7  10 Analysis of Variance | CKp 7 | PLO. ID | AppocK 2: AnoK ppocK o oLn K OocN of PID fo  LnJuO ocLLon cK L  foOOoZ: ConLd K pLZL coOun of K LnJuO cK. TKn dO K coon nd Ln oK. TK Lnd n ZLOO conLu K OocN. ConLd .J. K LnJuO ocLLon cK fo  5. TKn K fL OocN und KL ppocK L oLnd y dOLnJ K coon n Zn coOun 1 nd 2 ZKLcK uO Ln  OocN conLnLnJ K n 2 3 4 5 6 nd 7. SLLOOy conLdLnJ K oK pL of coOun K cK  pnd Ln K foOOoZOocN CoOun of n ocLLon cK Tn OocN 1 1 2 2 3 4 5 6 7 OocN 2 1 3 1 3 4 5   OocN 3 1 4 1 2 4 6  10 OocN 4 1 5 1 2 3 7  10 OocN 5 2 3 1 2 6 7   OocN 6 2 4 1 3 5 7  10 OocN 7 2 5 1 4 5 6  10 OocN  3 4 2 3 5 6  10 OocN  3 5 2 4 5 7  10 OocN 10 4 5 3 4 6 7   TK p of K PID  1010663nd4.bvrk SLnc oK K PID  LLnJ fo K  ocLLon cK o K vOu of 1212ndnnPP Ln K  fo oK K dLJn. In KL c Z Kv 633242 .2120 ºªº »…» ¼¬¼AppocK 3: AnoK ppocK o dLv K OocN of PID L o conLd OO K fL ocL of  JLvn n Ln  OocN. Fo [pO Ln K c of  K fL ocL of n 1  K n 2 3 4 5 6 nd 7. So K n conLu on OocN. SLLOOy oK OocN cn Oo  found. TKL uO Ln K  nJn of n Ln OocN  conLdd y dOLnJ K coon n Z Analysis of Variance | CKp 7 | PLO. ID | TK PID ZLK Zo ocL cO  popuO Ln pcLcO ppOLcLon nd cn  cOLfLd Lno K foOOoZLnJ yp dpndLnJ on K ocLLon cK Reference: Classification and analysis of partially balanced incomplete block designs with two associate classes, R. Bose and Shimamoto, 1952, Journal of American Statistical Association, 47, pp. . LLn qu ZLK CycOLc nd TK LnJuO ocLLon cK K Ody n dLcud. W noZ LfOy pn oK yp of ocLLon cK. L K  n ZKLcK cn  pnd  vpq . NoZ dLvLd K n Lno Joup ZLK cK Joup KvLnJ n ucK K ny Zo n Ln K  Joup  fL ocL nd Zo n Ln K dLffn Joup  K cond ocL. TKL L cOOd K Joup dLvLLO yp cK. TK cK LpOy oun o nJ K vpq n Ln  cnJO nd Kn K ocLLon cK cn  [KLLd. TK coOun Ln K cnJO ZLOO fo K Joup. Und KL ocLLon cK 1nqp111qqprk nd K p of K cond NLnd  unLquOy dLnd 2001 0112qpqqp§·§·¨¸¨¸©¹©¹rkv Analysis of Variance | CKp 7 | PLO. ID | r  Kn KJoup dLvLLO dLJn L Ld o  SucK  LnJuO Joup dLvLLO dLJn cn OZy  dLvd fo  copondLnJ ID. To oLn KL Mu pOc cK n y  Joup of n. In JnO Lf  ID K K foOOoZLnJ p     bvrk  Kn  d fo KL ID ZKLcK K p 1212        .bbvqvrrkqkrnpnq A Joup dLvLLO dLJn L Lf . NonLnJuO Joup dLvLLO dLJn cn  dLvLdd Lno Zo cO – LJuO nd JuO. LJuOnd0rrkv . Fo KL dLJn bvpAOo cK OocN conLn K  nu of n fo cK Joup o K u  dLvLLO nd0.rrkv L dno K LLn qu yp PID ZLK conLn. In KL PID K nu of n  [pLO  . TK n y  nJd Ln  qu. Fo K c of Zo n  K fL ocL Lf Ky occu Ln K  oZ o K  coOun nd cond ocL oKZL. Fo K JnO c Z N   of 2 uuOOy oKoJonO LLn qu povLdd L [L. TKn Zo n  K fL ocL Lf Ky occu Ln K  oZ o K  coOun o copondLnJ o K  O of on of K LLn qu. OKZL Ky  cond ocL. Und KL ocLLon cK 11112211111112niqnqqiiiqqiiqiiqiqiiiiqiiqiqiqiq Analysis of Variance | CKp 7 | PLO. ID | CycOLc Typ AocLLon ScK L K  n nd Ky  dnod y LnJ 12.... In  cycOLc yp PID K fL ocL of n 121...odidididv  OO dLffn nd 012...dvjn onJ K 1 dLffnc  12... ddjjnjj ducd  cK of nu ...ddd occu L ZK cK of K nu 122...eee occu L ZK 121122......dddeee  OO K dLffn nu 12… : To duc n LnJ od  Z Kv o uc fo L  uLO uOLpO of o K K . Fo [pO 17 ZKn ducd Fo KL cK 12111211211nnnnnnnnnn SLnJOy LLnNd OocN AocLLon ScK ZLK p     1nd .bvrkbv L K OocN nu of KL dLJn  d  n L.. . TK fL nd cond ocL Ln K LnJOy OLnNd OocN ocLLon cK ZLK Zo cO  dLnd  foOOoZ: DfLn Zo OocN nu of o  K fL ocL Lf Ky Kv [cOy on n Ln Und KL ocLLon cK 121 2 2 2 2121   1 1 2 1  11  11  11nkrnbnrknrknrknnrkknknknnk Analysis of Variance | CKp 7 | PLO. ID | GnO TKoy of PID A PID ZLK ocL cO L dfLnd  foOOoZ. L K  n. L K  OocN nd K L] of cK OocN L L.. K  pOo Ln cK OocN. TKn K n  OocN ccodLnJ o n ocL pLOOy Oncd ocLLon cK ucK K vy n occu  o onc Ln  OocN vy n occu [cOy Ln OocN nd Lf Zo n  K ocL of cK oK Kn Ky occu oJK [cOy Ln 12... OocN. TK nu L Lndpndn of K pLcuO pL of ocL cKon. I L no ncy K KouOd OO  dLffn nd o of K y  ]o. If n cn  nJd Ln ucK  cK Kn Z Kv  PID. No K K Zo n ZKLcK  K ocL occu oJK Ln TK p 1212......bvrknnn  d  K p of K fL NLnd nd j  d  K p of K cond NLnd. I y  nod K ...nnn nd OO j of K dLJn L oLnd fo K ocLLon cK und conLd... occu Ln K dfLnLLon of PID. If fo OO  12… Kn PID duc o ID. So ID L nLOOy  PID ZLon ocL cO. TK p of  PID  cKon ucK K Ky Lfy K  11ibkvriinviiinrk 1Lf nLf.kijkijijkjkiivnpnpnpnijI foOOoZ fo K condLLon K K  onOy 1/6 Lndpndn p of K cond | CKp 7 | PLO. ID | InpLon of CondLLon of PID TK LnpLon of condLLon   TKL condLLon L Od o K oO nu of pOo Ln n [pLn. In ou LnJ K  pOo Ln cK OocN nd K  OocN. So K oO nu of pOo  . FuK K  n nd cK n L pOLcd L ucK K cK n occu o Ln on OocN. So K oO nu of pOo conLnLnJ OO K n L . SLnc oK K n coun K oO nu of OocN Knc bkvr SLnc ZLK pc o cK n K LnLnJ   n  cOLfLd  fL cond… ocL nd cK n K ocL o K u of OO L K   K oO nu of n [cp K n und conLdLon. 1nrk OocN Ln ZKLcK  pLcuO n occu. In K OocN 1 pL of n cn  found cK KvLnJ  on of L . AonJ K pL K ocL of u occu L nd K  1.nrk ijkijkjkikijnpnpnpL  K  of ocL 12... of  n . Fo  cK n Ln j nu of ocL Ln . TKu K nu of pL of oLnd y NLnJ on n fo nd noK n fo on K on Knd L ijkpcLvOy.jjk | CKp 7 | PLO. ID | 1LfndLfjkjjkj nijpnijzL K n nd  ocL. TK ocL of 12... KouOd conLn OO K nu of ocL of . WKn j LOf ZLOO  on of K ocL of B 12... KouOd conLn OO K 1 nu of B D WLK TZo AocL ConLd  PID und Zo ocL cK. TK p of KL cK  1112221212112212112212            ndbvrknnpppppp TK OLn odO LnvoOvLnJ OocN nd n ffc L 12...12...ijijijyibjv  L K JnO n ffc ii j n ffc LfyLnJ 0ndijm L K L.L.d. ndo o ZLK ~0. TK PID L  Lny pop nd quLpOLc dLJn. So Ln KL c K vOu of K p co 0o1 fo OO 12... nd fo OO 12... . TK cn  Zo yp of nuOO KypoKL ZKLcK cn  conLdd – on fo K quOLy of n ffc nd noK fo K quOLy of OocN ffc. W  conLdLnJ K K LnOocN nOyL o Z conLd K nuOO KypoKL Od o K n ffc onOy. A don OL Ln K c of ID K OocN ffc cn  conLdd und K LnOocN nOyL of PID nd K covy of LnOocN LnfoLon. 012:...  H  O on pL of L dLffn. | CKp 7 | PLO. ID | TK nuOO KypoKL Od o K OocN ffc L of no ucK pcLcO Ovnc nd cn  d LLOOy. TKL Z LOOud OL Ln c of ID. In od o oLn K Oqu L of nd  Z LnLL] of K u of qu du o LduO ijij ZLK pc o nd TKL uO Ln K K noO quLon ZKLcK cn  unLfLd uLnJ K L[ noLon. TK  of ducd noO quLon Ln L[ noLon f OLLnLnJ K OocN CRNKNQVNKB R KkITKn K dLJonO On of 112...  K offdLJonO On of  oLnd  Lfnnd KfLocLLfnnd KcondocL  12...jjijijcnn jjjj z  SuofOocNoOLnZKLcKnoccu1.jijijjijjjQVjrknnN[ Z p o LpOLfy K [pLon of L j  K u of OO n ZKLcK  K fL ocL of n nd j  K u of OO n ZKLcK  K cond ocL of n. TKn jjjj  | CKp 7 | PLO. ID | 12..  co 11221121221121 1 1.jjjjjjjjjjjjkQrkSSrkSSrkSOOOO OOO IpoLnJ K Ld condLLon 221112121112... jjjkQrkSabSjvOOO12212211nd.arka TK quLon  ud o oLn K dMud n u oL j dno K dMud u of ov K  of Ko n ZKLcK  K fL ocL of n. W no K ZKn Z dd K  j fo OO  Kn occu L Ln K u vy fL ocL of j occu L Ln K u nd vy cond ocL of L Ln K u ZLK 11121 K 2121111111212222111111211222122 Su of  ZKLcK  K fL ocL of n 11jjjjjjjkQkQQjrkSnpSpSrkppSpbSaOOOOOOOO222112 122222111111.brkppOOOnd kQkQ12121122221 kQabSkQabS | CKp 7 | PLO. ID | NoZ oOvLnJ K Zo quLon K L of 22121 121222121....kbQbQ ababW  K j L  oOuLon of K ducd noO quLon. TK nOyL of vLnc cn  cLd ou y oLnLnJ K undMud OocN u of qu  BlockunadjkbkK dMud u of qu du o n  TreatadjjjSSQ. TK u of qu du o o   rrortTotalBlockunadjTreatadjSSSSSSSS TotalijSSy012:...  L Kn d on K LLc /1/1TreatadjErrorunadjSSvSSbkbv111Trvbkvb H L Mcd. | CKp 7 | PLO. ID | TK LnOocN nOyL of vLnc fo LnJ K LJnLfLcnc of n ffc L JLvn Ln K foOOoZLnJ O: Souc Su of quDJ of n InOocN Treatadj Blockunadj kbkyucLonErrortTreatdfv Blockdfb bkbv TdMTT ErrorMSE T Total dfbk No K Ln K LnOocN nOyL of PID nOyL  K p ZK Z oLnd j nd OLLnd j 221121jjjjkQrkS OOO noK poLLOLy L o OLLn Lndof . If Z OLLn j Lnd of j  Z ppocKd Kn K oOuLon K O ZoN LnvoOvd Ln K uLnJ of 112Qnn. If Kn OLLnLnJ j ZLOO LnvoOv O ZoN Ln oLnLnJ j ZK j dno K dMud u of L ov K  of Ko n ZKLcK  K cond ocL of n. WKn Z do o oLn K foOOoZLnJ L of K n ffc L o 21112 11212111kbQbQabab | CKp 7 | PLO. ID | 1111112211212 2121112222211.arkbrkppOOO j nd cn  cLd ou LLOOy. TK vLnc of K Ony con of L of n Ln c of 22 121 1 12222212jjjjbkQkQbkQkQababL 2212 12222212 122222122Lfnnd KfLocLLfnnd KcondocL.kbbababababW ov K K vLnc of dpnd on K nu of nd Ln K n K ZKK Ky  K fL o cond ocL. So dLJn L no vLnc Oncd. u vLnc of ny Ony con  quO und  JLvn od of ocLLon vL]. fL o cond. TK L ZKy K