On the Development of an Exponentiated F Test for One-way ANOVA in the Presence of Outlier(s)

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1 Mthmtcs d ttstcs 4: DOI: 0.389/ms O th Dvlopmt of Epottd Tst fo O-wy ANOA th sc of Outls Adpou K.A * httu O.I huwu A.U Dptmt of ttstcs Uvsty of Id Ng opyght 06 y uthos ll ghts svd. Authos g tht ths tcl ms pmtly op ccss ud th tms of th tv ommos Attuto Lcs 4.0 Ittol Lcs Astct Th clsscl sh-dco tst whch comps svl populto ms dpds o th udld ssumptos whch clud; dpdt of popultos costt vc d sc of outl mog oths.aguly th souc of volto of som of ths ssumptos s th outl whch ld to uqul vcs. Outl lds to qulty th vcs of th popultos whch cosqutly lds to th flu of th clsscl- to t coct dcso tms of th ull hypothss. A ss of oust tsts hv cd out to mlot ths lpss wth som dgs of ccucs d lmttos tms of fltg th typ o d th pow of dfft comto of pmts t vous smpl szs whl stll uss th covtol -tl. Ths study focuss o dvlopg oust -tst clld pottd tst wth th toducto of o shp pmt to th covtol -dstuto cpl of tg dcsos o ANOA tht oust to th stc of outl. Th pfomc of th oust tst ws compd wth th stg -tsts th lttu usg th pow of tst. Rl lf d smultd dt w usd to llustt th pplclty d ffccy of th poposd dstuto ov th stg os. Epmtl dt wth lcd d ulcd dsg w usd wth popultos szs 3 d 5 w smultd wth 0000 plctos d vyg dgs of outls w ctd domly. Th sults otd dct tht th oposd Epottd- tst s ufomly most powful th th covtol- tsts fo lyss of vc th psc of outl d s thfo commdd fo us y schs. Kywods sh-dco Dstuto Epottd- Dstuto O Wy ANOA Outl Mmum Llhood Mthod. Itoducto Th clsscl -tst to comp svl populto ms dpds o th ssumpto of costt vc of th populto d th omlty sh Rold []. Wh ths ssumptos spclly th qulty of vc voltd th clsscl -tst lcs lty to t cocto dcso. Th hv ss of oust tsts th lttu whch clud th Wlch tst [7] cott-mth [4] Bow-osyth [5] Whd.s Glzd tst [6] Kwd Rog tst y Hsy [7] Glzd tst[3] mtc Bootstp tst y Kschmoothy [9] mog oths to ovcom ths polm ut ths tsts stll show som dgs of ccucs tms of fltg th typ o d pows dfft comtos of pmts d vous smpl szs... O-wy ANOA Assumptos osdg tstg th modl gv s Y + τ + ~ N L y sttstcl tst o-wy ANOA hs svl ssumptos. Howv som of ths ssumptos stgt qumts whl oths c wvd..3. Outls Ovvw A outl s dt vlu tht s som wy cosstt wth th st of th dt st. Outls th sduls- osvto fo whch th sdul s mo tm th would pctd fom dom vto lo Th dtfcto of outl my chg wth th poposd modl. Rsos fo Outls olms wth th pmtl pocdu Eos th codg tscpto o dt put pocdus 3 A coct o complt modl spcfcto 4 A gu osvto tht s comptl wth th st

2 Mthmtcs d ttstcs 4: Effcts of movg outls: ould glgl. Mght ffct pmt stmts. Mght cs/dcs RsM. Wht to do? hc ogl soucs d coct oous vlus. Rpt lyss wth th offdg pots movd. Assss ffct o fl cocluso pmt stmts tc. If th sults mdly dfft th o my d to pot oth lyss. Wg! Do ot dscd outls wthout cful cosdto s dt oft psv to ot. A outl mght th most mpott osvto th study. Ths study focuss o dvlopg glzd dstutos tht cpl of hdlg dt tht o-oml. Also ttmpts wll md to dvlop Roust tst clld pottd tst tht s lss sstv to y sous voltos du to outls\..4. Wht Hpps If You olt th Assumpto of Equl cs? o hypothss tsts l ANOA you st sgfcc lvl. Th sgfcc lvl s th polty tht th tst coctly cts th ull hypothss Typ I o. Ths o cuss you to coctly coclud tht th goup ms dfft. If you st th sgfcc lvl to 0.05 th osvd o t s lso Th gt th dffc tw th tgt d ctul o t th mo sstv o-wy ANOA s to voltos of th qul vcs ssumpto..5. ossl Alttvs If You Dt olts O-wy ANOA Assumpto Tsfomtos: coctg o-omlty d uqul vcs y tsfomg ll th dt vlus Nopmtc tsts: dlg wth o-omlty y mployg tst tht dos ot m th omlty ssumpto of th o-wy lyss of vc Th Kusl Wlls tst dos ot ssum tht th dt omlly dstutd; tht s ts g dvtg. If you' usg t to tst whth th mds dfft t dos ssum tht th osvtos ch goup com fom popultos wth th sm shp of dstuto so f dfft goups hv dfft shps o s swd to th ght d oth s swd to th lft fo mpl o thy hv dfft vcs th Kusl Wlls tst my gv ccut sults gld d dv [9]. Htoscdstcty s o wy whch dfft goups c hv dfft shpd dstutos. If th dstutos omlly shpd ut hghly htoscdstc you c us Wlch's t-tst fo two goups o Wlch's ov fo mo th two goups. If th dstutos oth o-oml d hghly htoscdstc I do't ow wht to commd..6. mtc Wy Out Th m d of ppomt tsts s to possly fd coffct sy m d dgs of fdom sy v such tht * m ~ v Th m focus of th wo s to dvlop glzd -ttstc fo tstg th qulty of ms wh th s volto of ssumptos du to outl.. lsscl Tst d Estg Roust Tsts.. -Tst /ANOA Tst of Equlty of Ms osd mutully stochstclly dpdt dom vls hvg oml dstutos wth uow ms... spctvly d uow ut commo vc. Lt X X... X pst dom smpl of sz fom th oml dstuto wth m d vc. J.... I tstg th hypothss H H 0 :... : ot ll 's qul... A llhood to tst s s follows: Th pmt { } d Ω... 3 < < 0 < < 4 {... } <... < ω d Th llhood fuctos fo Ω d ω 0 < < 5 spctvly X L ω 6 π L Ω π X 7

3 64 O th Dvlopmt of Epottd Tst fo O-wy ANOA th sc of Outls Th log-llhood fuctos log L ω log π log L Ω log π X X W th dfftt 8 d 9 wth spct to 8 9 d d st th sultg psso to zo w ot th mmum llhood stmt of d. ˆ.... ˆ X X ˆ ˆ X X Thus th sh dco sttstc s dvd s X X X X / /.. Estg Roust Tsts of Equlty of Ms ud olto of Assumptos Wlch s Tst d th Mw + w 3 s s s s 0 Wh w w * w K w [ > f M w d w d pst um of popultos to compd sugoup vc d sugoup populto sz spctvly. Th cott-mth Tst s Th Bow-osyth Tst B 3 X X If H 0 s tu th dst of Kwd-Rog Tst 3 ~ R X X s K E

4 Mthmtcs d ttstcs 4: Dot E th fo th htoscdstcty ANOA th Kwd-Rog ppomto ts fom d M E M f > mtc Bootstp Tst Ths llows smplg fom dstuto ud H 0 whos pmts th smpl vcs. Gt dpdt ~ 0 Ζ d dom vl w st Y Ζ d... Y Y K K Z 3.3. oposd Epottd -Dstuto d -Tsts Lt X dom vl wth -dstuto ts dsty d dstuto fuctos spctvly dfd s f > > > ; I A 3 Usg th fucto pvously usd y Ndh d Kotz [] clld th pottd l fucto s follows [ ] f c g c 4 If w put d 3 4 w hv th poposd pottd- dstuto s c c I g As fo th poposd pottd tst t s dvd s follows π f 6 Φ Γ γ Φ 7 stly w ot pottd oml dstuto foud usg 6 d 7 4 π c g > > > > c 8 X L π ω Ω L π 9 Tg th log of llhood fuctos 9 d dffttg wth spct to d w hv π c 0 c + 3 ˆ

5 66 O th Dvlopmt of Epottd Tst fo O-wy ANOA th sc of Outls ˆ ω log Th poposd pottd -sttstc s π π c + c 3 + / π 4. Applctos π + / 3 W cosd th lcd d ulcd css fom smll to lg smpl szs fo 3 d 5 fo compg th poposd pottd-f tst d oth Roust tsts th lttu. Outls ctd to oml populto t oth low d hgh t ctg o costt vcs of dfft foms. lcultos wll md of th cto t of ch tstg pocdu d compd wth th oml lvl 0.05 wh th ms ll qul. Th typ I o ts of th WT BB d th poposd Epottd- tsts stmtd y th popoto of tst sttstc tht cd th ctcl vlus clcultd fom th dstutos. multo cosstg of 0000 plctos ws u fo ch of th smpl szs d pmt cofgutos. Th sults show th tls low Tl. multd Typ I Eo Rts Wh K 3 o Equl mpl zs WT B B 007 ROOD EXO NA NA NA NA NA NA NA NA Tl. multd Typ I Eo Rts Wh K 3 o Equl mpl zs WT B B 007 ROOED EXO Tl 3. multd Typ I Eo Rts Wh K 3 o Equl mpl zs WT B B 007 ROOED EXO

6 Mthmtcs d ttstcs 4: Tl 4. multd Typ I Eo Rts Wh K 3 o Equl mpl zs WT B B 007 EXO Tl 5. multd Typ Eo Rts Wh K 5 d Equl mpl zs WT B B 007 EXO NA 0.04 NA NA NA NA NA NA Tl 6. multd Typ Eo Rts Wh K 3 d Uqul mpl zs WT B B 007 ROOED EXO NA NA NA NA NA NA

7 68 O th Dvlopmt of Epottd Tst fo O-wy ANOA th sc of Outls Tl 7. multd Typ Eo Rts Wh K 5 d Uqul mpl zs WT B B 007 ROOEE X NA NA NA Adpou 5. oclusos Th vll tsts fo th o-wy ANOA modl wth htoscdstc o vcs hvsous typ I o polms tht hv ovlood; ths hs potd out y D [7]. I ths pp w hv dvlopd th Epottd- tst s lttv to lsscl tst d compd to som of th stg Roust tsts th lttu whch clud: th Wlch tst [7] th cot-mth tst [4] th Bow-osyth tst [5] th Kwd Rog tst [7] d mtc Bootstppg tst y Kschmoothy t l [9]. o g of chocs of th smpl sz d pmt cofgutos w hv vstgtd th pfomc of th ov tsts usg Mot lo smulto. I tms of cotollg th Typ I o t o c coclud stog tm tht th oposd Epottd- tst s th oly pocdu tht pfoms stsfctoly gdlss of th smpl szs vlus of th o vcs d th um of ms g compd oth th psc d sc of outl. Th Kwd-Rog tst cm dstt scod. REERENE [] Adpou K. A huwu A. U d httu O.I 04 Th ttstcl popts of pottd Ngm Dstuto. Ittol Joul of Mthmtcs d ystm vol 7 04 [] Ast A. d Low008.Th t-rylgh dstuto llty msu. cto o hyscl d Egg ccs ocdgs of th Amc ttstcl Assocto : [3] Ast A. moy. d L. 008 th t-to dstuto ttstcs 46: [4] Bd G. A. 984.ompg th ms of two dpdt smpls. Appld ttstcs [5] Bow M.B. osyth A.B. 974.Th smll smpl hvo of som sttstcs whch tst th qulty of svl ms Tchomtcs [6] huwu A. U. Adpou K. A 0: O th ow Effccy of Atfcl Nul Ntwo ANN d th lsscl Rgsso Modl. d oul of pogss ppld Mthmtcs vol 3 No 0 [7] D A. N. 00. otutos to ttstcl Ifc fo om d d Rdom Modls. h.d. Dsstto sumttd to th Dptmt of Mthmtcs d ttstcs Uvsty of Myld - Bltmo outy. [8] Eug N. L. d moy 00. Bt-oml dstuto d ts pplctos. ommuctos ttstcs-thoy d Mthods 34: [9] gld MW dv L. 009 Th Wlcoo-M-Wh ty tsts ud scuty. tt Md.009; 8: [0] moy. L. d Olugg O 005. Th Bt-Wull dstuto Joul of ttstcl Thoy d Applctos 4:-38. [] sh Rold 98. "tuds op to. I. A mto of th yld of dssd g fom Bodl". Joul of Agcultul cc : [] Hstgs W.K. 979 Mot lo smplg mthods usg Mov chs d th pplctos. Bomt 57: [3] Hmpl. R. Rochtt E. Z. Roussuw. J. d thl W.A Roust ttstcs. Th ppoch sd o fluc fuctos. Nw Yo: Wly. [4] Jms G Th compso of svl goups of osvtos wh th tos of populto vcs uow. Bomt [5] Jos M..004 mls of dstutos sg fom dstutos od sttstcs. ocdd d Estdstc Ivstgco Optv Tst 3:-43. [6] Kozuows T. d Ndh 008.Th t-lplc dstuto. Joul of computtol lyss d Applctos 03: [7] Kwd M. G. Rog J. H. mll mpl Ifc fo d Effcts fom Rstctd Mmum Llhood Bomtcs [8] Kutchoff R. G O-wy fd ffcts lyss of vc wh th o vcs my uqul. Joul of ttstcl omputto d multo

8 Mthmtcs d ttstcs 4: [9] Kschmoothy K. Lu d Mthw T. A pmtc ootstp ppoch fo ANOA wth uqul vcs: d d dom modls omputtol ttstcs & Dt Alyss [0] L B.W. dozo M Dtmto of totl dty f foods o poducts wth lttl o o stch o-zymtc-gvmtc mthod: ollotv study. J. Assoc. Alytcl hmsts Ittol [] Ndh. d Kotz 006.Th t-potl dstuto. Rllty Egg d ystm fty 9: [] Ndh. & Kotz "Th Epottd Typ Dstutos". Act Applcd Mthmtc [3] Rc W.R d Gs.D. 989.O-wy lyss of vc wth uqul vcs. ocdgs of th Ntol Acdmy of ccs [4] cott A. J. v mth T. M.. Itvl Estmts fo L omtos of Ms Appld ttstcs 0: [5] Tsu K. d Whd. 989.Glzd p-vlus sgfcc tstg of hypothss th psc of usc pmts. Joul of th Amc ttstcl Assocto [6] Whd.s. 987.Tstg gsso qulty wth uqul vcs. Ecoomtc [7] Wlch B.L. 95. O th compso of svl m vlus: A lttv ppoch. Bomt