An Appropriate F -Test for Two-Way Balanced Interactive Model

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1 n pproprat F -st for wo-wa alancd Intractv Modl F.C. Ez 1, F.O dmon 1, C.P. Nnanwa M.I. Ezan 3 1 Dpartmnt of Statstcs, Nnamd-zkw Unvrst, wka, Ngra. Dpartmnt of Mathmatcs, Nnamd-zkw Unvrst, wka, Ngra. 3 Dpartmnt of Computr Scnc, Nnamd-zkw Unvrst, wka, Ngra. bstract h prsnc of ntracton n a st of data/ modl n a two-wa ntractv modl ma lad to a basd rsult whn tstng for th man ffcts. h nusanc paramtr whch s th ntracton was rmovd from th data wthout dstortng th assumpton of homognt condton of analss of varanc. hs s don b a lnar combnaton such that th dffrncs btwn th corrspondng ld row-ws as wll as column-ws dffrnc s a constant and t th total sum of th ld rmans unchangd. Kwords: Nusanc paramtr, Mxd ffct modl, Last squars mthod. 1. Introducton maor st back n dsgn and analss of xprmnts s th prsnc of ntracton. two-wa ntracton ma b dfnd as th chang n rspons du to on at dffrnt lvls of th or two ndpndnt varabls ntract f th ffcts of on of th varabls dffr dpndng on th lvl of th othr varabl. h prsnc of ntracton ma obscur th rsult for th tst of sgnfcanc for th man ffcts accordng to Moor, t al [9]. On ma b tmptd to vhmntl mantan that w should not vn tst for man ffcts onc w know that ntractons ar prsnt. Prsnc of ntracton btwn th two s mans both ffcts ar not ndpndnt. In analss of varanc, a larg F -valu provds vdnc aganst th null hpothss. Howvr, th ntracton tst should b xamnd frst. h rason for ths s that, thr s lttl pont n tstng th null hpothss H of th altrnatv H f H : no ntracton ffct s rctd, snc th dffrnc btwn an two lvls of a man ffct also ncluds an avrag ntracton ffct Cabrra and McDougall [] argud.ovrton [1] gav a quck chck for ntracton usng an xampl on two bnar s and as llustratd blow. smpl sttng n whch ntractons can ars s n a two- xprmnt analzd usng nalss of Varanc (NOV) can b shown n abl 1. Suppos w hav two bnar s and. For xampl, ths s mght ndcat whthr thr of two tratmnts was admnstrd to a patnt, wth th tratmnts appld thr sngl, or n combnaton. W can thn consdr th avrag tratmnt rspons (.g. th smptom lvls followng tratmnt) for ach patnt, as a functon of th tratmnt combnaton that was admnstrd Ovrton [10] argud abl 1: wo-wa classfcaton wthout ntracton In abl 1, thr s no ntracton btwn th two tratmnts. hr ffcts ar addtv. h rason for ths s that th dffrnc n man rspons btwn thos subcts rcvng tratmnt and thos not rcvng tratmnt s - rgardlss of whthr tratmnt s admnstrd (4-6 = -) or not (5-7 = -). It automatcall follows that th dffrnc n man rspons btwn thos subcts rcvng tratmnt and thos not rcvng tratmnt s th sam rgardlss of whthr tratmnt s admnstrd (7-6 = 5-4) abl : wo-wa classfcaton wth ntracton Issn (onln) Fbruar 013 Pag 11

2 In contrast, n th abl thr s an ntracton btwn th tratmnts and hnc thr ffcts ar not addtv. Ez t al [3] dvlopd a common F-tst dnomnator for two-wa ntractv balancd dsgn. In thr work, th rmovd th ntracton from th data and consquntl dvdd th orgnal data b th nvrs of th squar root of th standard rror.muhammad t al [11] argud that th ffct of a mdcaton s th sum of ts drug ffct placbo ffct (manng rspons) and thr possbl ntracton. ccordng to thm, currnt ntrprtaton of clncal trals rsults assums no ntracton. Dmonstratng such an ntracton has bn dffcult du to lack of an approprat dsgn. Howvr, ths papr s st to rsolv such problm.park t al [13] carrd out a rsarch on th prsnc of ntracton btwn drct and carr-ovr tratmnt ffcts b usng a modl n whch th rsdual ffct from a tratmnt dpnds upon th tratmnt appld n th succdng prod. hs mans a modl whch ncluds ntracton btwn th tratmnt drct and rsdual ffcts. h assum that th rsdual ffct do not prsst furthr than on succdng prod.in th prsnc of hghr ordr ntracton, Khrad and Sara [7] dmonstratd an xact prmutaton stratg applcabl to fxd ffct analss of varanc whch can b usd to tst an.jams [5] prsntd a papr smlar to Ovrton [1]. In hs papr, h dmonstratd som common rrors n ntrprtng ntracton ffcts and th approprat stratgs for achvng post hoc undrstandng of th orgn of dtctd ntracton ffcts. ccordng to hm, a lack of ntracton s oftn sgnfd b paralll lns n a plot of cll mans. Convrsl, f th lns ar not paralll, t sgnfs th prsnc of ntracton.in NOV, a larg F-valu provds vdnc aganst th null hpothss. Howvr, th ntracton tst should b xamnd frst. h rason for ths s that, thr s lttl pont n tstng th null hpothss H or H f H : no ntracton ffct s rctd, snc th dffrnc btwn an two lvls of a man ffct also ncluds an avrag ntracton Cabrra and MacDougall [] argud.moor t al [9] argud that thr ar thr hpothss n a twowa NOV wth an F-tst for ach. W can tst for sgnfcanc of th man ffcts, th man ffct and ntracton. It s gnrall a good practc to xamn th tst ntracton frst, snc th prsnc of strong ntracton ma nflunc th ntrprtaton of th man ffcts.som statstcal softwars such as XLS, SPSS, Mntab tc can prform th NOV tst but dos not consdr th mplcatons of th prsnc of ntractons.. Mthodolog Gvn th modl for wo-wa balancd ntractv modl 1,,..., p k k 1,,..., q (1) k 1,,..., r whr k s th kth obsrvaton n th th cll, s a constant, s th avrag ffcts of, s th avrag ffcts of, s th ntracton ffct that xsts btwn and and s th rror assocatd wth k. k h last squar stmats of th paramtrs can b shown to b ˆ..... ˆ..... ˆ From abl 1, th last squar stmats of th cll obsrvatons hav bn calculatd and prsntd n abl Issn (onln) Fbruar 013 Pag 1

3 abl 3: Last squar stmats of wo-wa classfcaton wthout ntracton From abl 3, th ntracton ffcts ar zro. In contrast, thr ar prsncs of ntracton ffcts n abl as shown n abl abl 4: Last squar stmats of wo-wa classfcaton wth ntracton.1 Expctd man squars In ths papr, rut forc mthod s usd n drvng th xpctd man squars. rut forc s a tral and rror mthod usd b applcaton program to dcod ncrptd data rathr than mplong ntllctual stratgs (rnstn [1]).Usng th rut forc mthod th xpctd man squars for Equaton 1 has bn drvd and prsntd n abl 5. S.V d.f SS MS ll ffcts fxd ll ffcts p-1 SS MS qr p 1 q-1 SS MS pr q 1 ntracto n (p-1)(q- 1) Error pq(r-1) SS SS MS r ( p1)( q1) MS qr r pr r r fxd qr p 1 r pr r otal pqr-1 SS abl 5: Complt NOV abl fxd r pr q 1 r r From abl 5, thr s no obvous dnomnator for tstng for th man ffcts whn th modl/data ar fxd, or mxd. For nstanc, f th data ar fxd, th common dnomnator for tstng for th man ffcts s MS. Smlarl, f th data wr, th common dnomnator for tstng for th man ffcts s MS undr th null hpothss H 0. Whn th data ar mxd, th dnomnator of th F -rato vars. h rason for ths s th prsnc of ntracton. If th ntracton s rmovd from th data, abl 5 rducs to abl 6. S.V d.f SS MS ll ffcts fxd p-1 SS MS qr p 1 ll ffcts qr fxd qr p 1 fxd r q-1 SS MS pr q 1 Error (p-1)(q- 1) SS MS pr pr otal pqr-1 SS pr q 1 Issn (onln) Fbruar 013 Pag 13

4 abl 6: Rducd NOV abl From abl 6, th common dnomnator for tstng for th man ffcts s MS undr H 0 and th modl quaton consquntl rducs to Equaton. 1,,..., p k k 1,,..., q () k 1,,..., r h paramtrs hav th sam manng as t s n Equaton 1.. Mthod of rmovng th ntracton h ntractons can b rmovd from th data/modl as follows wthout dstortng th assumptons of analss of varanc.n lnar combnaton such that th dffrncs btwn th corrspondng ld row-ws as wll as columnws dffrncs s a constant and t th total sum of th ld rmans unchangd lmnats th ntracton (Wsstn[15]).Lt x11, x1,..., xpq b th lds or valus n wo-wa crossd ntractv modl wth on obsrvaton pr cll. h data format s shown n abl , q q q.. 3q p p1 p p3 pq p q.. abl 7: Data laout of wo-wa classfcaton wth on obsrvaton pr cll. From abl 7 k k k k k k k 3k k k 4k k k ( pq 1) k (3) ' ' ' ' 11 ' ' 3. Illustratv Exampl n ngnr s dsgnng a battr for us n a dvc that wll b subctd to som xtrm varaton n tmpratur. h onl dsgn paramtr that h can slct s th plat matral for th battr, and h has thr possbl chocs.h ngnr dcds to tst all th thr matrals at thr tmpratur lvls- 15 F,70 F, and 15 F -as ths tmpratur lvls ar consstnt wth th product nd-us nvronmnt. Four battrs ar tstd at ach combnaton of plat matral and tmpratur; all 36 tsts ar run n a. h xprmnt and th rsultng obsrvd battr lf data ar gvn n abl 8. Issn (onln) Fbruar 013 Pag 14

5 Matral tp 1 130, 155, 74, , 188, 159, , 110, 168, 160 mpratur ( 0 F) , 40, 80, , 1, 106, , 10, 150, 139 0, 70, 8, 58 5, 70, 58, 45 96, 104, 8, 60..k Sourc abl 8: Sourc: Lf Data for battr dsgn from [10] p.07 Usng SPSS, th analss of varanc was prformd and prsntd n abl 9. p 111 sum of squars df Man Squar Corrctd modl Intrcpt Matral mpratur Matral*mpratur Error otal Corrctd otal abl 9: NOV abl From th NOV abl 9, th man ffcts and th ntracton ar sgnfcant. Usng th xprsson drvd from abl 7 and Equaton (3), th data n abl 8 ar now transformd as follows: For th corrspondng ntrs for 11 w hav 36k k 7.4 h valus of k for th corrspondng ntrs for 11, 113, and 114 ar , 8.14, and rspctvl. h transformd data ar shown n abl 10. F Sg Matral tp mpratur ( 0 F) , 155, 74, , 10.3, 98.4, , 85.64, 1.84, , , 8.14, , , , , 74.08, , , , 90.8, , 97.0, 114.7, , 6.5, 139.1, k Issn (onln) Fbruar 013 Pag 15

6 abl 10: ransformd data of th lf Data for battr dsgn h NOV tst for th transformd data s shown n abl 11 Sourc p 111 sum of squars df Man Squar F Sg Corrctd modl Intrcpt Matral mpratur Matral*mpratur Error otal Corrctd otal abl 11: NOV abl From abl 11, th ntracton btwn th matral and tmpratur s hghl non-sgnfcant showng th succssful rmoval of th ntracton from th data. Howvr, th matral ffcts ar sgnfcant whl th tmpratur ffcts ar non-sgnfcant.snc th ntractons ffcts ar zro, w thrfor prform th NOV tst n absnc of th ntracton as shown n abl 1. Sourc p 111 sum of squars df Man Squar F Sg Corrctd modl Intrcpt Matral mpratur Error otal Corrctd otal abl 1: Rducd NOV abl h rsults obtand n abl 1 ar th sam as th rsult obtand n abl 11. If w ar to us th mthod of 3.0 w shall hav th sam transformd data n abl 10 and NOV tst n abl Summar nd Concluson W hav succssfull drvd an xprsson that would nabl us rmov th ntracton n our data/modl. From th llustratv xampl gvn, s possbl to commt an rror whn ntracton s prsnt n our data. For xampl, th tmpratur ffcts wr sgnfcant whn ntracton s prsnt as shown n abl 9. Whn th ntracton ffcts wr rmovd from th data, th tmpratur ffcts bcam non-sgnfcant. W thrfor rcommnd that analss of varanc tst should b don whn ntracton s hghl non-sgnfcant or zro. Rfrncs [1] rnstn, D.J. (005), Undrstandng rut Forc, Dpartmnt of Mathmatcs, Statstcsand Computr Scnc, h Unvrst of Illnos, Chcago. [] Cabrra, J., and MacDougall,. (00), Statstcal Consultng. Sprngr. [3] Ez, F.C., Ehwaro, J.C., and Ogum G.E.O (009), Common F-tst Dnomnator for wo- Wa Intractv alancd Dsgn, Natural and ppld Scncs Journal, Vol.10 N0.. [4] Hnkman, K and Kmpthron, O (008), Dsgn and nalss of Exprmnts, Vol. 1Scond Edton, Wll. [5] Jams, M.G. (000), Intracton Effct: hr Natur and Som Post Hoc Exploraton stratgs, xas and M Unvrst [6] Kovach Computng Srvcs (011), How Can I Us XLS to Run a wo-wa Unbalancd Dsgn wth Intracton, ngls, Wals. Issn (onln) Fbruar 013 Pag 16

7 7] KHERD PJOUH, Sara, RENUD, Olvr (010), n Exact Prmutaton Mthod for stng an Effct n alancd and Unbalancd Fxd Effct NOV, Journal of Computatonal Statstcs and Data nalss, Vol.54, pp [8] Mchal, W.. (001), n Introducton to Statstcal Infrnc and Data nalss., Dpartmnt of Mathmatcs, Collg of Wllam and Mar, P.O. ox 8795, Wllamsburg, V [9] Moor, D.S., McCabl, G, Duckworth, W., and Sclov, S. (004), Practc of busnss Statstcs, part v pp Palgrav Macmllan. [10] Montgomr, D.C. (001), Dsgn and analss of xprmnts, John Wll and sons. [11] Muhammad, M.H., Eman,., Sd,., and Muhammad,.H. (010), Intracton btwn Drug and Placbo Effcts: a cross-ovr balancd placbo dsgn tral, Cntr for Clncal Studs and Emprcal Ethcs, Kng Fasal Spcalst Hosptal and Rsarch Cntr, Radh, Saud raba. [1] Ovrton, R.C. (001), Modratd multpl rgrsson for ntracton nvolvng catgorcal varabls- a Statstcal control for htrognous varabl across two groups. Pschol. Mthods, 6(3), pp [13] Park, D.K., os, M., Notz, W.I. and Dan,. (011), Effcnt Cross-ovr Dsgns n th Prsnc of Intractons btwn Drct and Carr-Ovr ratmnt Effcts, Journal of Statstcal Plannng and Infrnc, 141, (), [14] Sabn, L., and ran, S.E. (004), Handbook of Statstcal nalss Usng SPSS, oca Raton, London. [15] Wsstn, E. W. (01), Lnar Combnaton, from Mathworld- wolfran Wb Rsourc. Issn (onln) Fbruar 013 Pag 17

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