The Combination of Several RCBDs

Transcription

1 Ausaia Joua of Basic ad Appied Scieces 5(4): ISSN The Comiaio of Sevea RCBDs Musofa Usma Pee Nuho 3 Faiz AM Efai ad 3 Jama I Daoud Depame of Mahemaics Facuy of Sciece Lampug Uivesiy Idoesia Schoo of Saisics ad Acuaia Sciece Uivesiy of Kwazuu- Naa Souh Afica 3 Depame of Scieces Facuy of Egieeig Ieaioa Isamic Uivesiy Maaysia Asac: The appicaio of Expeimea Desig owadays is vey exesive i may eseach aeas especiay i Egieeig Agicuue Educaio ad Life scieces I may expeimea desigs someimes he eseaches wa o compae paamees fom some desig of expeimes This pape wi discuss he appoach o comie sevea RCBDs (Radomized Compee Boc Desigs) fo he fixed effec mode The comied mode is o fu coum a ad has cosai o is paamees The appoach used i his pape is MRM (Mode Reducio Mehod) o asfom he cosaied mode io ucosaied mode The he aaysis of iees wi e ased o he ucosaied mode Key wods: RCBDs MRM(Mode Reducio Mehod) ohogoa Maix Lieihood Raio Tes esig hypohesis INTRODUCTION The expeimea desig as a oo o fid he ifomaio of iees i eseach has ee exesivey used i may aeas of eseach such as i Egieeig Agicuue Medica scieces ad i Educaio The appicaio of he expeimea desig i Egieeig ca e foud fo exampe i Musofa e a (008) Fie (006) Mogomey ad Ruge (994) The aaysis of comied of sevea mode has ecome a ieesig aeas of eseach fo exampe see Peeso (994) has discussed he comiaio of aaysis of sevea expeimea desig appied i he aeas of agicuue The comiaio of some iea egessio mode y usig he geea iea mode ca e foud i Thei (97) The comiaio of some iea egessio mode y usig dummy vaiae aso ca e foud i may egessio oo such as Nee e a (990) Guaai (970) I his pape wi discuss he appoach o aayze of he comiaio of sevea RCBDs mode ude fixed effec mode The comied mode is o fu coum a ad has cosai o is paamees The appoach used i his pape is MRM (Mode Reducio Mehod) give i Hocig (985) o asfom he cosaied mode io ucosaied mode The he aaysis of iees wi e ased o he ucosaied mode To esimae ad es he hypohesis of iees wi e cosideed wo cases of he vaiace covaiace sucue RCBD aaysis mehods: I is we ow ha he RCBD mode is Y i i i whee Y i is he osevaio fom he ih oc ad h eame g i is he eo fom he ih oc ad h coum ad assume ha i is iid (0σ ) Ude he fixed effec mode i is assume ha he paamee has a cosai i 0 ad 0 () i The mode () ad is cosai () ca e wie as () Y X (3) Coespodig Auho: Musofa Usma Depame of Mahemaics Facuy of Sciece Lampug Uivesiy Idoesia 67

2 Suec o Gθ=0 whee Y is x veco of osevaio X I I Aus J Basic & App Sci 5(4): ε has omay disiuio wih mea 0 ad vaiace σ ad 0 0 G 0 0 ad he oece poduc of maix A sx ad B xu deoed y AqB is sxu maix fomed y muipyig each eeme a i y eie maix B (Mose 996 Thei 97) Mode (3) is o fu coum a ad has cosai o is paamees To dea wih his ype of poem hee ae some appoach ca e used fo exampe see (Magus 988; Hocig 985) Hocig (985) poposed of Mode Reducio Mehod (MRM) o asfom he o fu a ad cosaied mode io ucosai fu a mode The idea is as foow: Suppose ha he iea mode Y X (4) Suec o Gθ=g whee Y is -veco of osevaio X is xp desig maix of a #pθ is p-veco of uow paamees ε is -veco of eo wih ε - N(0 I) whee - is ead is disiued as ad N(μ V) deoes he muivaiae oma disiuio wih mea veco µ ad covaiace maix V ad G is qxp maix of a q Assume ha θ ad G ae paiio so ha he cosai ae wie as G G g (5) Whee G is qxq of a q Sovig fo θ yieds G gg G (6) Paiio X as he same way as X = [X X ] ad susiuig io mode (4) we oai Y = X θ +e (7) whee Y = Y XG g X = X XG G ad θ =θ The mode (7) is caed ucosaied mode Hocig (985) I mode (3) y assumpio ha he oc ad eame ae coeced he he a of he desig maix X i (3) is +- As ca e show i he foowig emma (see Musofa 995) Lemma : By assumpio of coecedess he a of X i (3) is +- Poof: Because he fis ad he as coum of desig maix X I I is iea comiaio of he ohe coums he he a of X # +- By assumpio of coecedess β i β i is esimae i i τ - τ is esimae e 68

3 Aus J Basic & App Sci 5(4): : H 3 : whee 0 0 H 0 I I ( ) x( ) ( ) x( ) Ra(H) = +- Sice Hθ is esimae he hee is a maix A such ha H=AX Theefoe +- =a(h) = a (AX) # a(x)#+- So a (X)=+- Appicaio of Mode Reducio Mehods i RCBD: Coside mode (3) X = I I [I E F] whee E I F I To es he hypohesis ha Ho: Hβ=0 agais he aeaive hypohesis Ha: Hβ =0 We woe he mode (3) Y E F (8) Suec o 0 0 i i To asfom he mode (8) he o fu a cosaied mode io ucosaied fu a mode y MRM(Mode Reducio Mehod) fis we asfom he paamee () () pemuaio maix TT is a ohogoa such ha T y Whee ad () () TT I We have io 69

4 Aus J Basic & App Sci 5(4): whee Mode (8) ca e wie as Y E F TT Suec o o Y X i i 0 0 (9) (0) Suec o G 0 Whee X E F T ad G is x veco u Now paiio G ad X as G G G X X X By MRM we have he ucosaied mode Y X () Whee X AD A E F F G D I CT I ( ) ( ) C 0 ad is he maix F wihou he as coum of F he maix () () I ca e show ha D is osigua so he a of he desig maix X is +- which is fu coum a The mode () is saisfied a he assumpio of Gauss Maov mode see Gayi976; Thei 97; Mose 996 Theefoe he esimaio of paamee esimaio of cofidece ieva aio of paamees ad esig hypohesis ca e deived fom his mode () The ideas of he asfomaio mode fom he cosaied ad o fu a mode io ucosaied mode y MRM wi e appied o he comiaio of sevea RCBDs mode The Aaysis of he Comiaio of Sevea RCBDs: Supposed ha hee ae RCBDs mode Y i i i () 70

5 Aus J Basic & App Sci 5(4): Y i i i : : Yi i i whee Y ii is he osevaio fom he ih oc h eame i he h desig µi is he gad mea i he h desig β i is he effec of he ih oc i he h desig τ i is he effec of he h eame i he h desig ad g ii is a eo fom he ih oc h eame ad he h desig i= ; = ; = Mode () ca e wie as Y diag X X X (3) whee X E F as give i mode (8) whee ad 3 Va diag I I I I ad assume ha Va(g )= = ad ad he cosai o is paamees ae 0 0 i i Mode (3) ad is cosai (4) ca e wie as foow: (4) Υ = Г Θ + Ψ (5) Suec o Ω Θ = 0 Whee Y is ()x veco of osevaio ad Г = diag X X X Ω = diag G G G 0 0 G 0 0 = Mode (5) is o fu a mode ad has cosai o is paamees Ude he assumpio of coecedess ad appy he idea of Lemma i ca e show ha he desig maix Г i (5) is (+-) To asfom he o fu a cosaied mode io fu coum a ad ucosaied mode we use he pocedue MRM o he aove mode as foow Defie he asfom maix Λ Λ is a ohogoa maix ad diag T T T whee T i is a ohogoa maix TT diag I I I o I I I ad 7

6 Aus J Basic & App Sci 5(4): Mode (5) ca e wie as Υ = ГΘ + Ψ (6) Suec o Ω Θ = 0 Г = diag E F E F E F diag G G G 0 0 G = 0 0 Now we asfom he mode Υ = Г Λ ΛΘ + Ψ o ad Υ = Г Θ + Ψ (7) Suec o whee 0 diag E F T E F T E F T ad ad () () () () () 3 () 3 : : ( ) 3 () 3 () 3 : : ( ) 3 Nex we fid secod asfomaio maix Λ such ha () () ( ) ( ) 7

7 () () ( ) ( ) So mode (6) ca e wie as Aus J Basic & App Sci 5(4): Y Suec o whee 0 ad I 0 (8) diag Now paiio ad as ad is he fis coum of ad he is he es of he coum of I By appyig mode educio mehods he we have he ucosai mode Y (9) whee Y Y g sice g = 0 he Y Y he we have ad sice I () () () () ( ) ( ) Desig maix Γ has a of size (+-) Theefoe mode (9) is fu coum a mode ad ucosai mode To aayze he mode amey o esimaio ad esig he paamee of he mode (9) we ca use he sadad mehods Esimaio ad Tesig Hypohesis: Ude he assumpio ha he vaiace of each mode i () : he he vaiace ad covaiace maix Va I ae ow ad equa (0) ad assume ha he disiuio of Ψ is muivaiae oma wih mea zeo veco ad vaiace ad covaiace maix saisfied (0) The he esimaio of 73

8 Aus J Basic & App Sci 5(4): ˆ Y ad ˆ Y I Y () () whee he A sad fo geeaized ivese of a maix A ( see Gayi ( )) The esimaio give i () ad () has he opima popey Uifomy Miimum Vaiace Uiased Esimaio (UMVUE) see Gayi (976) Thei (97) Based o mode (9) we ca es some paamees of iees Fo isa we ca es ha he h mode give i () ae equa Ude mode (9) he hypohesis ca e wie as i i i i i Ho: 3 i 3 ad ae ow cosas Which is equivae o es he hypohesis ha Ho: H h (3) Whee H H 0 ( ) x I 0 I 0 I 0 H diag 0 I 0 I 0 I a(h)= (+-) h h is ow veco (+-)x The ieihood aio es is give y H ˆ ˆ h H H H h ( ) Y I Y ( ) Ude he u hypohesis λ has F disiuio wih degees of feedom -(+-) ad (+-) If he vaiaces of each mode () : maix of he comied mode is give y Va diag I I I o Va I ae ow u uequa he he vaiace covaiace 74

9 whee diag Aus J Basic & App Sci 5(4): The appoach o dea wih his ype of poem we ca use geeaized iea mode (Musofa 995; Bhapa976; Gayi 976; Rao 973; Thei 97; Aod 980) The he esimaio of paamee veco Θ is give y ˆ I I Y Wih he covaiace maix Va I Ude his assumpio o es he hypohesis give i (3) he ieihood aio es saisics is ( ) ˆ ˆ ( ) ˆ H ˆ h H I H H h Y ) I Y Ude he u hypohesis λ has a F disiuio wih -(+-) ad (+-) degees of feedom REFERENCES Aod SF 980 The Theoy of Liea Modes ad Muivaiae Aaysis New Yo: Joh Wiey & So BhapaVP 976 Geeaized Leas Squaes Esimaio ad Tesig The Ameica Saisicia 30(): p73-74 Chisese 987 Pae Aswe o Compex Quesios: The Theoy of Liea Mode New Yo: Spige-Veag Das MN ad NC Gii 979 Desig ad Aaysis of Expeimes New Dehi: Wiey Ease Limied Fie TL 006 Poaiiy ad Poaiisic Reasoig: Fo Eecica Egieeig New Jesey: Peaso Peice Ha Gayi FA 969 A Ioducio o Maices wih Appicaios i Saisics Bemo Caifoia: Wadswoh Puishig Co Gayi FA 976 Theoy ad Appicaio of Liea Mode Bemo Caifoia: Wadswoh Puishig Co Guaai D 970 Use of Dummy Vaiae i Tesig fo Equaiy ewee Ses of Coefficies i Liea Regessio: A Geeaizaio The Ameica Saisicia pp: 8- Hocig RR 985 The Aaysis of Liea Mode Moeey Caifoia: Boos/Coe Puishig Compay Magus JR ad H Newdece 988 Maix Diffeeia Cacuus wih Appicaios i Saisics ad Ecoomics New Yo: Joh Wiey & Sos Mogomey DC ad GC Ruge 994 Appied Saisics ad Poaiiy fo Egiees New Yo: Joh Wiey & Sos Mose BK 996 Liea Modes: A mea Mode Appoach New Yo: Academic Pess Musofa U FAM Efai ad JI Daoud 008 Expeimea Desig: Fo Scieiss ad Egiees Kuaa Lumpu: IIUM Pess Musofa U 995 Tesig Hypoheses i a Spi po Desig whe some osevaios ae Missig Upuished Disseaio Kasas Sae Uivesiy Kasas USA Nee J W Wassema ad MH Kue 990 Appied Liea Saisica Modes Boso: Iwi Ic Peeso RG 994 Agicuua Fied Expeimes: Desig ad Aaysis New Yo: Mace Dee Rao CR 973 Liea Saisica Ifeece ad Is Appicaios Secod ediio New Yo: Joh Wiey & Sos Thei H 97 Picipes of Ecoomeics New Yo: Joh Wiey & Sos (4) (5) (6) 75