On Selection of Best Sensitive Logistic Estimator in the Presence of Collinearity
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1 Amrcan Journal of Appld Mathmatcs and Statstcs, 05, Vol. 3, No., 7- Avalabl onln at Scnc and Educaton Publshng DOI:0.69/ajams-3-- On Slcton of Bst Snstv Logstc Estmator n th Prsnc of Collnarty C. E. ONWUKWE *, I. A. AKI Dpartmnt of Mathmatcs/Statstcs and Computr Scnc Unvrsty of Calabar P. M. B. 5, Cross Rvr Stat, Ngra *Corrspondng author: malchrsonwukw@gmal.com Rcvd Sptmbr 6, 04; Rvsd Dcmbr 6, 04; Accptd January 08, 05 Abstract Collnarty s a major problm n rgrsson modlng. It affcts th prdcton ablty of ordnary last squar stmators. Collnarty s stablshd n logstc rgrsson modls whn th dffrnc btwn th last and hghst gn valu of th nformaton matrx s mor n rlaton to th last gn valu. Ths rsults n nflatd varanc of stmatd rgrsson paramtrs. Consquntly, th rsultng modl s not rlabl and wll rsult n ncorrct conclusons about th rlatonshp among th varabls. To ovrcom th problm of collnarty n logstc rgrsson modl a numbr of stmators wr proposd. Ths artcl compars th prformanc of four stmators - ordnary logstc stmator, logstc rdg stmator, gnralzd logstc rdg stmator and modfd logstc rdg stmator n th prsnc of collnarty, to ascrtan whch s mor ffctv n varanc rducton. To stablsh suprorty among th abov stmators, analyss s carrd out on a cas study n Unvrsty of Calabar Tachng Hosptal, Calabar Cross Rvr Stat, Ngra. Rsult showd that modfd logstc stmator prformd bttr than othr stmator consdrd du to th fact that t had th smallst varanc. Kywords: collnarty, canoncal transformaton, rspons probablty, logstc rdg stmator, logt, nformaton matrx, lnk functon Ct Ths Artcl: C. E. ONWUKWE, and I. A. AKI, On Slcton of Bst Snstv Logstc Estmator n th Prsnc of Collnarty. Amrcan Journal of Appld Mathmatcs and Statstcs, vol. 3, no. (05): 7-. do: 0.69/ajams Introducton Ordnary Last Squars (OLS) stmaton s wdly usd n rgrsson analyss. Logstc rgrsson has provn to b on of th most vrsatl tchnqus n gnralzd lnar modls whch allows for th modlng of catgorcal varabls. Mthod of last squars prforms wll undr som basc assumpton such as whr rror ar ndpndnt and followng normal dstrbuton wth man zro and havng constant varanc (Jadhav and Kashd, 0). In ral lf stuaton, som varabls ar sn to rlat wth ach othr thrby ntroducng multcollnarty n modls. Prsnc of multcollnarty can mak ordnary last squar stmator to b unstabl du to larg varancs whch lad to poor prdcton (Batah t al, 008; Batah, 0; Josh, 0; Nja, 03). To ovrcom ths problm, svral masurs had bn prsntd. Rmds nclud rdg rgrsson mthod by Horl and Knnard, (970) and th tratv prncpal componnt mthod Marx and Smth (990). Snc multcollnarty producs larg varancs n ordnary last squar stmaton, rdg rgrsson attmpts to fnd paramtr stmats that hav smallr varanc and hnc smallr MSE by nlargng th small Egn valus (Nldr and Wddrburn, 97; Hawkn and Yn, 00; Vago and Kmny, 006).. Ordnary Rdg Rgrsson Estmator Consdr a multpl lnar rgrsson modl. Y X () Whr Y s (nx) vctor of obsrvatons, β s a (px) vctor of unknown rgrsson coffcnts, X s a matrx of ordr (nxp) of obsrvatons on p prdctor (rgrssor) varabls x, x, x p and s an (nx) vctor of rrors wth E() = o and var() = σ. Th last squar stmator of β s gvn by ˆ X X X Y. Th lnar modl can b wrttn n canoncal form as Y Z () whr Z = XT, T s th matrx of gn vctors of X'X Z'Z T T'X'XT dag(,,, ) whr λ s th th gn valu of X'X T', T'T TT' I p Th OLS stmator of α s gvn by whr p Z'Z Z'Y J Z'Y (3)
2 8 Amrcan Journal of Appld Mathmatcs and Statstcs whr Z' Z J, TOLS (4) OLS A dag, K, K,, k (5) K p K K K p,k 0 K s a basng constant. K can b gnralzd as k = (K, K, k p ) so that KI dag k, K, k p Th gnralz ordnary Rdg stmator s obtand as whr GOR T GOR T KA (6) GOR A dag K, K,, K p p λ s th th gn valu of (X'X + KI) Ths procdur s xtndd to modl logstc rdg stmator and ts subsqunt modfcaton, th modfd logstc rdg rgrsson stmator. 3. Ordnary Logstc Rgrsson Estmator Th ordnary logstc stmator uss th tratv wghtd last squars mthod. Th ordnary logstc stmat of β s gvn by ˆ X WX X WZ (7) whr, W s a dagonal matrx of wghts Z s a column matrx of adjustd dpndnt varabls. 4. Logstc Rdg Rgrsson Estmator Th gnralzd rdg rgrsson can b xprssd n canoncal form as GLS T GLS T KA (8) GLS K K, K,,K, A dag K P λ s th th gn valu of (X'WX + KI) Th logstc Rdg rgrsson stmator of β s gvn by X'WX KI (9) 5. Modfd Logstc Rdg Rgrsson Estmator Modfd logstc rdg rgrsson stmator was proposd by Nja t al (03). Ths s gvn n canoncal form as follows MLS T KA GLS (0) Whr A dag K λ s th th gn valu of ( X ' W X KI ) 0. Th modfd logstc rdg stmator of β s gvn by ' X W X KI X ' W Z () 6. Mthodology If th probablty of an vnt takng plac s P, thn th odd of that vnt s gvn by: Odd p p That s, odd s th probablty of an vnt takng plac dvdd by th probablty of th vnt not takng plac. Th log of th odds s known as logt gvn as logt P p log p Logstc rgrsson lk othr rgrsson has a dpndnt varabl and ndpndnt varabl(s). In logstc rgrsson th dpndnt varabl s a logt whch s th natural log of th odds, p Log odds logt P log p Logstc rgrsson s a modlng stratgy that rlats th logt to a st of xplanatory varabl wth a lnar modl (Bndr and Grovn, 997; Hosmr and Lmshow, 008; Lamot 0). That s, p In X p β 0 = th constant β = th rgrsson coffcnt X = th prdctor varabl So that p 0X p 7. Th Modl 0 0 (var ( ) X 'WX ), P N 0X 0X W ar modlng th probablty that a prson slctd from a subpopulaton has rspratory nfcton gvn by, whr, β 0 = constant β = sx xp 0 X X 3X3 P xp X X X 0 3 3
3 Amrcan Journal of Appld Mathmatcs and Statstcs 9 β = locaton β 3 = % of xposur Th stmaton of s ar as follows:. Ordnary logstc stmator (X'WX) W s a dagonal matrx of wghts gvn by W m µ µ,,,3,4 m s th sub populaton total µ s th rspons probablty and Z s a column matrx of adjustd dpndnt varat gvn by y Z µ,,,3,4 m η s th lnk functon y s numbr of favourabl outcom. Logstc Rdg Estmator (X'WX+KI) Computaton for Z and W ar th sam as thos of ordnary logstc stmator. KI s dagonal matrx of Tkhonov constants (small postv basng constants). K K K 3 K4. Gnralzd Logstc Rdg Estmator (X'WX+K) Th computaton for Z, W and K ar th sam as thos of logstc rdg xcpt that; K K K3 K4 v. Modfd Logstc Rdg Estmator X ' W X K X ' W Z whr, W Z m y m Th varanc of th paramtr s gvn by Var( ) = σ (X WX) - N N whr s th rror. Th stmaton of paramtrs and calculaton of varancs wr don wth MATLAB tratvly. 8. Data Collcton Th data for ths rsarch wr obtand from th Unvrsty of Calabar Tachng Hosptal, Calabar, n Cross Rvr Stat of Ngra. Ths was facltatd by a wll structurd qustonnar that was admnstrd to patnts attndng th famly mdcn clnc of th hosptal wthn a prod of two wks. A total of 80 qustonnars wr ssud out and 69 wr proprly flld and rturnd whch s prsntd n Tabl. Data ar obtand on locaton of patnts rsdnts, sx and lvls of xposur. Th xplanatory varabls ar sx, locaton and prcntag lvl of xposur of whch th frst two ar dchotomous and th thrd s contnuous. Th rspons varabl s dchotomous. Tabl. Data on Rspratory Infcton Locaton Gndr % lvl of No Dsas xposur Dsas Total Rural Fmal Rural Mal Urban Fmal Urban Mal Rsult of Analyss Tabl. Paramtr stmats st traton Estmators 0 3 Ordnary logstc: Logstc rdg: Gnralzd logstc rdg: Modfd logstc rdg: nd traton Estmators 0 3 Ordnary logstc: Logstc rdg: Gnralzd logstc rdg: Modfd logstc rdg: rd traton Estmators 0 3 Ordnary logstc: Logstc rdg: Gnralzd logstc rdg: Modfd logstc rdg:
4 0 Amrcan Journal of Appld Mathmatcs and Statstcs Tabl 3. Varancs of th dffrnt stmators Estmators Var 0 Var Var Var 3 Ordnary logstc: Logstc rdg: Gnralzd logstc rdg: Modfd logstc rdg: Fttd Modl Th probablts that a prson slctd from a sub group has rspratory nfcton as gvn by th dffrnt stmators ar as follows:. Ordnary logstc stmator X xp 0.755X 0.53X3 P X xp 0.755X 0.53X3. Logstc rdg stmator X xp 0.074X X3 P X xp 0.074X X3 3. Gnralzd logstc rdg stmator X xp 0.78X X3 P X xp 0.78X X3 4. Modfd logstc rdg stmator X xp 0.80X 0.040X3 P X xp 0.80X 0.040X3 From Tabl 3 (varancs of th dffrnt stmators) w can s that modfd logstc rdg stmator has th last varancs of th paramtrs and hnc w tak th modl obtand usng modfd logstc stmator. Th modl gvn by modfd logstc rdg stmator can b xpland as follows:. Th probablty that a fmal lvng n a rural ara wth 0% lvl of xposur s Th probablty that a mal lvng n a rural ara wth 6% lvl of xposur s Th probablty that a fmal lvng n an urban cntr wth 39% lvl of xposur s Th probablty that a mal lvng n an urban cntr wth 4% lvl of xposur s Dscusson of Fndngs Rsult prsntd n Tabl show sgnfcant dffrnc n th paramtr stmats by th dffrnt stmators. It s obsrvd that th stmats obtand by usng ordnary logstc stmator s sgnfcantly dffrnt from thos of th rdg stmators. In Tabl 3 t s sn that thr s sgnfcant dffrnc n th varancs of th paramtr stmats from th dffrnt stmators. Lookng closly at th rsult, modfd logstc rdg stmator s mor snstv and prforms bttr than th othr stmator du to ts ablty to rduc th varanc assocatd wth multcollnarty. Th probablty shows that mals lvng n rural ara wth an xposur lvl of 39% hav a hghr probablty of havng rspratory nfcton.. Concluson Bas on th fndngs of ths study, t can b concludd that modfd logstc rdg stmator s mor supror to othr stmators (ordnary logstc, logstc rdg and gnralzd logstc rdg) on th bass of varancs of th paramtr stmats. Also prsons lvng n rural aras ar sn to b mor pron to havng rspratory nfcton. Acknowldgmnt I acknowldg th fforts of Dr. M. E. Nja who has contrbutd mmnsly to th succss of ths work by puttng m through th computatonal procdurs nvolv. I also apprcat th ffort of Mr. Kayod whch has ld to th actualzaton of th goal of ths work. Rfrncs [] Batah, F. S. M. Ramanathan, T. V., Gor, S. D. (008). Th Effcncy of modfd Jackknf and Rdg Typ Rgrsson Estmators: A comparson Survys n Mathmatcs and ts applcaton 3 -. [] Batah, F. S. (0). A nw Estmator by gnralzd Modfd Jackknf Rgrsson. Estmator: Journal of Basarah Rsarchs (Scncs), 37 (4) [3] Bndr, R. and Groovn, U. (997). Ordnal Logstc Rgrsson n Mdcal Rsarch. Journal of th Royal Collg of Physcan of London. Spt/Oct 997: v 3 (5): [4] Hawkns, D. M. Yn, X. (00). A fastr algorthm for rdg rgrsson. Computatonal statstcs and data analyss. 40, [5] Horl, A. E. and Knnard, R. W. (970). Rdg Rgrsson Basd Estmaton for non-orthogonal Problms. Communcaton s statstcs: Thory and Mthods [6] Hosmr, D. w. and Lmshow, S. (008). Appld Logstc Rgrsson nd Edton. Wly. [7] Josh, H. (0). Multcollnarty Dagnoss n Statstcal Modlng and rmds to dal wth t usng bars. Cytl Statstcal Softwar Srvcs PVT Ltd. Pon Inda. [8] Judhav, N. H. & Kashd, D. N. (0). A jackknf Rdg M. Estmator for Rgrsson modls wth multcollnarty and outlrs. Journal of statstcal thory and practc. 5: 4, [9] Lamot, W. W. (0). Multpl Logstc Rgrsson. Boston. Boston Unvrsty Prss. [0] Marx, B. D. and Smth, E. P. (990). Prncpal componnt stmaton for gnralzd rgrsson. Bomtrka. 77 (): 3-3 (990). [] Nldr, J., Wddrburn, R. W M. (97). Gnralzd Lnar Modls. Journal of th Royal Statstcal socty, A 35,
5 Amrcan Journal of Appld Mathmatcs and Statstcs [] Nja, M. E. (03). A nw Estmaton procdur for Gnralzd Lnar Rgrsson Dsgns wth nar Dpndncs. Accptd for publcaton. Journal of Statstcal; Economtrc Mthods. [3] Nja, M. E., Ogok, U. P. & Nduka, E. C. (03). Th logstc Rgrsson modl wth a modfd wght functon. Journal of statstcal and conomtrc Mthod, Vol. No [4] Vago, E. & Kmny, S. (006). Logstc Rdg Rgrsson for clncal Data Analyss (A cas study). Appld Ecology and Envronmntal Rsarch 4 () 7-79.
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