On Three-Way Unbalance Nested Analysis of Variance

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1 Joural o Mathmatcs ad Statstcs 8 : -4 0 ISS Scc Publcatos O Thr-Wa Ubalac std alss o Varac Smala S. Sa ad ug. Uagbu Dpartmt o Statstcs Facult o Phscal Sccs Uvrst o gra sua gra bstract: Problm statmt: I ths stud w gv a smpl aaltcall tractabl procdur or solvg thr-wa ubalacd std alss o Varac OV. I ma ralstc stuatos ubalacd dsg was uavodabl du to atural costrats ad mssg data. pproach: Hr w prst a comprhsv approach or addrssg solutos o problms arsg rom ubalacd std OV. W cosdr th F-statstcs udr th drt modls. Rsults: Spcal attto was gv to th costructo o approxmat F-tst whr xact F-tst dos ot xst. Psudo-dgrs o rdoms wr drvd usg th Sattrthwat s tp approxmato. ocluso: I all drvatos w assum that th cts act dpdtl ad that th ma squars ar dpdt. umrcal xampl s gv to llustrat th soluto procdur. K words: Ubalacd data ma squar psudo-dgrs o rdom approxmat F-tst std dsg statstcal sotwar last squars mthod atural barrrs lar modl modl paramtrs ITRODUTIO Ths stud s tdd to b a tutoral or thos wshg to orm thmslvs about thr-wa ubalacd std OV. It ocuss o th bacgroud udrstadg o modl paramtrs stmato drvato o sum o squars o cts drvato o xpctd ma squars o cts sttg up o varac compots ad costructo o approxmat F-statstc whr xact F-tst dos ot xst. rtal ths tools ar ot w but sght to how th ar appld to thr-wa ubalac std OV would op w vstas our wa o hadlg ubalacd hrarchcal arragmts. Th stud s mad to b aaltcall ad computatoall accssbl. Radrs d ol som pror owldg o two-wa balacd std dsg; s or xampl Motgomr 008 ad Dowd ad hlo 004 or dscusso o balacd std Dsg. xprmt wth thr actors ad s sad to b thr-wa ubalacd std dsg o actor sa s std wth aothr actor sa ad actor s std wth actor such that ach lvl has b lvls ach lvl has j lvls ad j obsrvatos ar draw rom ach lvl. It s prtt to mto that ths arragmt dos ot prmt tracto btw actors. alss o varac laout s calld ubalacd t has uqual subclass umbrs. I addto aalss o varac modl s sad to b ubalacd th varac o th drc btw a two tratmts s ot a costat but dpds o th tratmts. Ma xprmtal stuatos could lad to std arragmt. Thus ths d o dsg has oud xtsv applcatos dustrs bologcal sccs clcal studs. Th aalss o std dsg s dcult ad th problm s complcatd wh acd wth ubalacd std dsg. Most statstcal sotwar ow corporat commads ad gudls or carrg out computatos o ubalacd std aalss o varac. Kasha ad Salh 00 dscussd thr mthods o stmatg varac compots or mxd-modl. Howvr thr ar svral mthods or stmatg varac compots wh th dsg s ubalacd. ach mthod lucs th corrspodg approxmat F-tst. Sgr 970 gav a mthod or stmatg varac compots ubalacd dsg. H usd uwghtd ma hs stmats ad showd that ths stmats ar ubasd. ush ad drso 963 cosdrd umrcal comparsos btw varacs o compots o varac du to drt mthods. Ttj ad Moor 968 dvlopd a ast procdur or computg approxmat F-tst ubalacd std aalss o varac. Saha ad Ojda 004 dscussd ubalacd std aalss o varac or radom ct modl. Ths stud ams at provdg bacgroud owldg o thr-wa ubalac std aalss o varac a smpl straghtorward sl-cotad orrspodg uthor: Smala S. Sa Dpartmt o Statstcs Facult o Phscal Sccs Uvrst o gra sua gra

2 J. Math. & Stat. 8 : -4 0 accout o th udrlg thor. That s th stud xposs th aaltcal procdurs that ar hdd wh aalss s prormd wth th ad o statstcal sotwar. Ubalacd std dsg could ars rom a umbr o actors whch clud mssg data; los to ollow-up or subjcts gt sc lmtd rsourcs ad atural barrrs. Mssg data ca rsult rom ovrt rrors masurmts patts ot showg up or schduld vsts a clcal tral loss o sampls olto ad o 009. MTRILS D MTHODS Statstcal modl: Th lar modl or thr-wa ubalacd std arragmt s gv b Saha ad Ojda 004 as q. : jl µ + + j a j b + j + jl c j l j Whr: jl Th l th obsrvato wth th th lvl o actor wth jth lvl o actor wth th lvl o actor µ Th ovrall ma Th ct du to th th lvl o actor j Th ct du to th jth lvl o actor std wth th th lvl o actor j Th ct du to th th lvl o actor std wth th jth lvl o actor std wth th th lvl o actor jl Th rsdual rror o th obsrvato jl Th ollowg rstrctos ar mposd o q. : j j j j j j 0 0ad 0 ot that: j j jl l l ad j l jl jl whr s a costat. Rstrcto allows or stmatg th modl paramtrs usg th last squars mthod. stmato o modl paramtrs ad sums o squars: Th Last squars mthod has b wdl usd stmatg modl paramtrs s or xampl aspour t al. 008; Kavtha ad Duraswam 0 ad Rchr ad ruc 008. Th last squars mthod ad rstrcto ar usd to stmat th modl paramtrs. Th stmats ar as ollows: Whr: µ j... j j..... ad.. j.. jl jl j. jl jl jl jl T... jl T... l j j... j j ot that dots o subscrpts dot summg or avragg ovr th subscrpts. S appdx I o how ths stmats ar drvd. Th sums o squars ar gv blow: SS jl T... T T j.. T... j j..... j j..... jl j j SS SS j. j.. j j. j.. jl j T T j. j.. j j j j j. j jl j. jl jl jl j SS T SSTotal jl jl xpctd ma squars: W gv th xpctd ma squars or xd ct modl modl I radom ct modl modl II ad mxd ct modl modl III. W rr th radr to appdx II or th drvato o th xpctd ma squars udr th drt modls. Fxd ct modl Modl I: Factor s xd actor s xd ad actor s xd. ssumptos o th modl: Th assumptos o th modl ar gv q. 3: T T

3 J. Math. & Stat. 8 : j 0 j 0jl 0 σ 3 j j Th xpctd ma squars ar: j j j + σ + σ I I I I Whr: j j j + σ σ 0 0 σ j σ σ 0 ad jl 0 j Th xpctd ma squars or th modl ar: III + σ + σ + σ III σ + σ + σ III σ + σ ad III σ 3 5 a b b a ab c c b j j j ab c c j j j j ad stads or xpctd ma squar du to a gv actor udr a partcular Modl. Radom ct modl Modl II: Factor s radom actor s radom ad actor s radom. ssumptos o th modl q. 4: 0 j 0 0 ad jl 0 σ σ σ σ j Whr: Th xpctd ma squars ar: II σ + σ + σ + σ II σ + σ + σ σ + σ II 3 j j j j j j j j 4 Mxd ct modl Modl III : Factor s radom actor s xd ad actor s radom. ssumptos o th modl q. 6: j 0 σ 0 j σ σ 0 ad jl 0 j Th xpctd ma squars ar gv blow: III σ + σ + σ III j j j + σ + σ III σ + σ ad III σ 3 6 Mxd ct modl Modl III : Factor s radom actor s radom ad actor s xd. ssumptos o th modl q. 7: j 0ad jl 0 0 σ 0 σ j σ 7 j Th xpctd ma squars or th modl ar: III σ + σ + σ III σ + σ j j j + σ III Mxd ct modl Modl III : Factor s xd actor s radom ad actor s radom. ssumptos o th modl q. 5: 3 III σ

4 J. Math. & Stat. 8 : -4 0 Mxd ct modl Modl III : Factor s xd actor s radom ad actor s xd. ssumptos o th modl q. 8: j 0 0 σ j σ 0ad jl 0 j Th xpctd ma squars or th modl ar: III + σ + σ III σ + σ III ad III j j j + σ σ 8 stmato o th varac compots: Frst w cosdr th radom ct modl Modl II ad gv procdur or stmatg th varac compots as ollows: II σ ad st sc II ; whr " " rads stmatd b. Ths procdur s smlar to Hdrso s Mthod I. Saha ad Ojda 004 dscussd Hdrso s mthods or stmatg varac compots or ubalacd data: Whr: II σ + σ ad s c II 3 / II σ + σ + σ ad + + σ sc II σ + σ + σ + σ II ad st σ σ M S M S M S 3 3 M S M S 3 M S M S M S M S M S Whr: 3 3 Th sam procdur s usd to obta th varac compots or th drt modls ad th rsults ar gv blow. Mxd ct modl Modl III : Factor s xd; actor ad actor ar radom. Th varac compots ar: σ 3 Whr: 3 Mxd ct modl Modl III : Factor s xd actor ad actor ar radom. Th varac compots ar: 3 ad υ υ

5 J. Math. & Stat. 8 : -4 0 whr υ 3. Mxd ct modl Modl III : Factor s xd actor ad actor ar radom. Th varac compots ar: ad Mxd ct modl Modl III : Factor s radom actor ad actor ar xd. Th varac compots ar: ad Tst procddurs or th modls: Th xpctd ma squars hlp th dtrmato o approprat tst statstcs or tstg hpothss about th cts. xpctd ma squars dtrm whch hpothss ar tstd b ach ma squar. F-statstc ca ol b ormd wh udr approprat hpothss two xpctd ma squars hav th sam valu Maso t al I ths stud w shall cosdr th tst statstcs udr th drt modls. Modl I Fxd-ct modl: Th hpothss H: 0 s tstd b / H: j 0 s tstd b / ad H: j 0 s tstd b /. Th OV stmat o σ s. It s obsrvd that udr ull hpothss I σ I σ ad I σ. Modl II Radom ct modl: Th hpothss to b tstd ar: H : σ 0 H : σ 0 H : σ 0 W obsrv that II udr th ull hpothss H 0 : σ 0 dos ot hav th sam valu as II bcaus ad. Hc thr s o xact F-tst or tstg H : σ 0. Thror w 0 costruct a w ma squar whch s dpdt o. W mplo th approach proposd. 5 Lt σ + σ + σ. Ths s th xpctd ma squar du to actor udr th ull hpothss H 0 : σ 0. Th varac stmats ar: 3 3 ad 3 Thror: Whr: 3 3 W ow show that: σ + σ + σ + + σ + σ + σ + σ 3 σ + σ + σ Th approxmat F-statstcs: Th hpothss H 0 :σ 0 s tstd b: F whr F s approxmatl F-dstrbutd wth a- ad dgrs o rdom. Whr: + +

6 J. Math. & Stat. 8 : Drvato o psudo-dgrs o rdom : W costruct th dgrs o rdom b applg a approxmato du to Sattrthwat 946. Sattrthwat approxmato s basd o assumg that a varac stmator has a ch-squar dstrbuto ad solvg or th mpld dgrs o rdom usg th mthod o momts Vallat ad Rust 00: + + I w assum that ad ar dpdt th q. 9 ad 0: Var Var + Var + Var Rcall that: SS ad that SS σ χ : Var SS Var 4 σ 9 0 Var SS σ Var χ Var SS sc th varac o ch-squar dstrbuto s. 4 Substtutg th rsult σ hav q. : Var SS 0 w Var σ xtdg our da o 9 w gt: xt w cosdr tstg H 0 :σ 0. Thr s o xact F-tst or tstg ths hpothss bcaus /II udr H : σ 0 dos ot hav th sam 0 valu as c /II sc 3. W drv th approxmat F-tst usg th sam procdur as prstd abov ad th rsult s as ollows: Th hpothss H 0 :σ 0 s tstd b F. F s approxmatl F-dstrbutd wth dgrs o rdom whr: + + ad Fall th hpothss H 0 :σ 0 s tstd b. Modl III Mxd ct modl; whr actor s xd: Th hpothss to b tstd ar: H 0 : H : σ 0 H : σ 0 Th hpothss H 0 :σ 0 s tstd b: F whr F s approxmatl F-dstrbutd wth a- ad dgrs o rdom whr:

7 J. Math. & Stat. 8 : -4 0 Th hpothss H 0 :σ 0 s tstd b F ; F s approxmatl F-dstrbutd wth ad dgrs o rdom. Whr: + ad + Th hpothss H 0 :σ 0 s tstd b. Modl III Mxd ct modl; whr actor s xd: Th hpothss to b tstd ar: 0 H : σ 0 0 j 0 H : 0 H : σ 0 Th hpothss H 0 :σ 0 s tstd b: 0 0 H : σ 0 H : σ 0 0 j H : 0 Th hpothss H 0 :σ 0 s tstd b: F whr F s approxmatl F-dstrbutd wth a- ad F dgrs o rdom whr: + ad + Th hpothss H 0 : j 0 s tstd b: F ad th hpothss H 0 : j 0 s tstd b: F υ F whr F s approxmatl F-dstrbutd wth a- ad υ dgrs o rdom whr: υ + υ ad υ υ υ υ + υ Th hpothss H 0 : j 0 s tstd b: F whr F s approxmatl F-dstrbutd wth ad dgrs o rdom. Th hpothss H 0 :σ 0 s tstd b. Modl III Mxd ct modl; whr actor s xd: Th hpothss to b tstd ar: 7 Modl III Mxd ct modl; whr actor ad actor ar xd: Th hpothss to b tstd ar: H 0 : 0 0 H : σ 0 0 j H : 0 Th hpothss H 0 : σ 0 s tstd b: F whr F s approxmatl F-dstrbutd wth a- ad dgrs o rdom whr: + ad +

8 J. Math. & Stat. 8 : -4 0 Th hpothss hpothss H 0 : j 0 s tstd b H 0 : σ 0 s b F ad th. RSULTS D DISUSSIO umrcal xampl: W gv llustratv xampl wth hpothtcal data o th hardss crushg strgth o a partcular tablt producd b a pharmacutcal compa. Th compa has two producto sts wth a partcular rgo. Two o th machs or producg th tablt ar radoml slctd rom st o ad thr machs rom st two. asd o th producto capact o ach mach two batchs o th producd tablts ar radoml slctd rom ach o th machs at st o. t st two two batchs ar slctd rom mach o thr batchs rom mach two ad o batch rom mach thr. Th masurs o th crushg strgth o th tablts radoml slctd rom ach o th batchs ar rcordd Tabl blow. Th compa wats to vstgat th batch to batch varablt wth machs mach to mach varablt wth sts ad st ct ar sgcat sourcs o varato th crushg strgth o th tablt. Ths s a cas whr th st s xd machs ad batchs ar radom. W ow procd to costruct th j -tabl Tabl 3 to ma th computato asr. Th j -tabl s tabl o couts o th umbr o obsrvatos or th th St j th Mach ad th batch. omputato o sum o squars: Lt:... j j.. j j R T j j. j j jl R T R T R T R jl jl Thror th sums o squars ar: SS R R SS R R st mach j SS R R SS R R 85.7 atch j j rror jl j SS R R Total jl Th rsults o ths computatos ar summarzd Tabl 3 blow. Th valus mard b astrs 8 colum v ar obtad b th approxmat F-tst dvlopd ths stud. o xact F-tst xsts or such tsts. How th dgrs o rdom ar calculatd: b a b b a j b b + b c j 3 j j alculato o th wghtg actors: Th valus o th j s Tabl ar usd or computg th wghtg actors as ollows: j j jj j j j υ

9 J. Math. & Stat. 8 : -4 0 Tabl : Drug hardss St I II Mach atch 3 Hardss g T j T j T T Tabl : j- tabl Tabl 3: OV Tabl or drug hardss Sourc o varato Dgr o rdom Sum o squars Ma squar xpctd ma squar F-tst St / a + 6.9σ + 8.4σ + σ * Mach atch rror Total σ + 8.3σ + σ * 7.σ + σ σ Th valus o th approxmat F-tsts or actors ad ar obtad as ollows: jl jl µ j j F F O th bass o th rsults show Tabl 3 w coclud that thr s o drc Sts thr s o sgcat drc mach-to-mach varablt wth th Sts ad thr s o drc batch-tobatch varablt wth th machs at o prct ad v prct sgcat lvls rspctvl. Drtatg wth rspct to µ ad quatg to zro w hav: jl j j µ 0 3 Solv or µ b summg o j ad l 3 w gt: jl µ 0 jl ppdx I: Procdur or stmatg modl paramtrs: W provd hr procdur or stmatg th paramtrs o. Th last squars mthod ad rstrcto ar usd to stmat th modl paramtrs as ollows q. ad 3: 9 b q.. Thror: jl µ jl T

10 J. Math. & Stat. 8 : -4 0 jl Solv or µ 0 Sc: jl j b summg o j ad l 4: c 0 b : j j j j j µ µ jl jl T Solv or j b summg o ad l 3: jl jµ j j j 0 ; l Sc: l b: 0 j jl l j.. j j..... µ j µ j T Solv or j b summg o l 3: µ jl j l j j j Tj. µ j j j.. j.. Solv or jl b substtutg µ w gt: jl jl j. ad j j ppdx II. Drvato o xpctd ma squars: W drv th xpctd ma squars or xd ct modl modl I radom ct modl modl II ad mxd ct modl modl III. Fxd ct modl Modl I: I... 0 whr I stads or xpctd ma squar du to actor udr Modl I. W obta th quvalts o assumptos o th modl ad as: µ Whr: Thror: I... µ + ; jl jl jl jl σ. j j..... j I... ad usg th W obta th quvalts or... ad j.. usg ad 3 as: µ µ j.. j j.. Thror: I j + + σ j j..... j j j j

11 J. Math. & Stat. 8 : -4 0 j j. j.. j I W obta th quvalts or... ad j.. usg ad 3 as: µ j.. j j.. µ j. j j j. Thror: I j + + σ j j. j.. j j j j I jl j. jl j. σ jl jl Radom ct modl Modl II: II... W obtad th quvalts o... ad usg ad 4 as: µ µ Whr: j j j.... j j j j.. j ad... j j j j Thror: II Whr: a j j j j j j + j j j j j j σ + σ + σ + σ j j j j j j j j. II j j..... j W obtad th quvalts or... ad j.. usg ad 4 as: µ

12 J. Math. & Stat. 8 : -4 0 µ j.. j. j j.. Whr:. j j j Thror: II j j +.. j.. + j..... j j j j j j j j j j j j j j + j j..... j + σ + σ + σ µ j.. j. j j.. Thror: j j. j + j. j.. j II Whr: j j j j j j j + σ + σ j j. j.. j 3. 3 j j j Mxd ct modl Modl III : III... Whr: j j W obtad th quvalts o... ad usg ad 5 as: µ j j j j j j j. j.. j II W obtad th quvalts o j. ad j.. usg ad 4 as: µ j. j j j. Thror: III µ ɺ σ + σ + σ

13 j j..... j III W obtad th quvalts o... ad j.. usg ad 5 as: µ µ j.. j. j j.. Thror: III j + j.. j.. + j..... j σ + σ + σ j j. j.. j III W obtad th quvalts o j. ad j.. usg ad 5 as: III µ j. j j j. µ j.. j. j j.. j j. j + j. j.. j 3 σ + σ jl j. jl j. III σ jl jl Usg th sam procdur as dscrbd abov th xpctd ma squars or othr modls ca b asl drvd. J. Math. & Stat. 8 : OLUSIO W hav attmptd to provd solutos to problms arsg rom thr-wa ubalacd std OV rsultg rom a combato o mssg data ad atural rstrcto o th dsg. prmar goal was to costruct psudo-f-tst whr xact F-tst dos ot xact. Th approxmat dgr o rdom drvd ths stud ot rsult o-tgr valu whch tur lads to trpolato th tabl o Prctag pots o th F-dstrbuto. Th laout or th llustratv xampl has larg dgrs o rdom or th stmato o th rror compot whch compsats or th hgh sum o squars du to rror that s xprmtal rror. Th procdur provdd hr ca b asl mplmtd ad xtdd to m-wa ubalacd std aalss o varac. RFRS olto S. ad. o 009. Pharmacutcal Statstcs: Practcal ad lcal pplcatos. 5th d. Iorma Halthcar US. w Yor IS: pp: 656. ush. ad R.L. drso 963. comparso o thr drt procdurs or stmatg varac compots. Tchomtrcs 5: Dowd S.M. ad D. hlo 004. Statstcs or Rsarch. 3rd d. Wl-Itrscc hchstr w Jrs IS: X pp: 67. Kasha P. ad G. Salh 00. stmato o gtc corrlatos o swt cor brd ls usg SS mxd modl. m. J. grc. ol. Sc. 5: DOI: /ajabssp Kavtha S. ad K. Duraswam 0. daptv urouzz rc sstm approach or th automatc scrg o dabtc rtopath udus mags. J. omput. Sc. 7: DOI: /jcssp Maso R.L. R.F. Gust ad J.L. Hss 003. Statstcal Dsg ad alss o xprmts: Wth pplcatos to grg ad Scc. d d. Joh Wl ad Sos w Jrs IS: pp: 78. Motgomr D Dsg ad alss o xprmts. 7th d. Joh Wl ad Sos US. IS: pp: 656. aspour M. M.H. shar. Hassa ad.r. Froozja 008. ollocato Dscrt Last Squar DLS mthod or lastct problms ad grd rrgulart ct assssmt. m. J. ppld Sc. 5: DOI: /ajassp

14 J. Math. & Stat. 8 : -4 0 Rchr.. ad G.. ruc 008. Lar Modls Statstcs. d d. Joh Wl ad Sos w Jrs IS: pp: 67. Saha H. ad M.M. Ojda 004. alss o Varac or Radom Modls: Ubalacd Data. st d. ruhausr osto IS: pp: 480. Sattrthwat F approxmat dstrbuto o stmats o varac compots. omtrcs : 0-4. PMID: Sgr P mthod o stmatg varac compots ubalacd dsgs. Tchomtrcs : Ttj G.L. ad R.H. Moor 968. O tstg sgcac o compots o varac th ubalacd std aalss o varac. omtrcs 4: PMID: Vallat R. ad K.F. Rust 00. Dgrs o rdom approxmatos ad ruls-o-thumb. J. Ocal Stat. 6: