Application of Local Influence Diagnostics to the Linear Logistic Regression Models

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1 Dhaka Unv. J. Sc., 5(): (July) Applcaton of Local Influnc Dagnostcs to th Lnar Logstc Rgrsson Modls Monzur Hossan * and M. Ataharul Islam Dpartmnt of Statstcs, Unvrsty of Dhaka Rcvd on Accptd for Publcaton on Abstract Ths papr focuss th dvlopmnt of th dagnostcs for th prturbatons of caswghts and xplanatory varabls (on or mor) n a lnar logstc rgrsson modl. Th ffct of spcfc prturbaton schm on th stmaton of paramtrs s also assssd. In addton, th ntrprtaton of th valu of curvatur dagnostcs s hghlghtd n ths papr. Ths papr also dmonstrats and xtnds th utlty of th dagnostcs for dchotomous outcom varabls. For llustraton, a subst of th Framngham Hart Study data st s usd. Ky words and phrass: Local nflunc, dagnostcs, logstc rgrsson modl, and curvatur dagnostcs.. Introducton Th lnar logstc rgrsson modl s consdrd as on of th most mportant and wdly applcabl tchnqus n analyzng catgorcal outcom varabls. To assss th ft of a modl, t s ncssary to dntfy th nfluntal lmnts. Cook (986) dvlopd som local nflunc dagnostcs procdurs for lnar rgrsson modls. Extnson of th Cook s approach ar mad for dffrnt modls (s Wssfld, 990; Wssfld and Scndr, 990 (990a); Escobar and Mkr, 99; othrs). Howvr, no attmpt has bn mad so far n ordr to provd a dtald and clar undrstandng about th xtnson of Cook s approach for th lnar logstc rgrsson modls. Th applcaton of th dagnostcs s wdly dscussd by rsarchrs for th lnar rgrsson, th Wbull and othr paramtrc rgrsson modls. In ths papr, an attmpt s mad to xtnd th procdurs of dagnostcs for th lnar logstc rgrsson modls. Although a brf dscusson on th us of th dagnostcs s provdd for th gnralzd lnar modls by Thomas And Cook (989), a dtald xtnson of th dagnostcs for th logstc rgrsson modls s ncssary to undrstand th xtnt and pattrn of nflunc on th stmats of ntrst. Ths papr dmonstrats and xtnds th utlty of th dagnostcs for dchotomous outcom varabls. Prgbon (98) dmonstratd an approach to dntfy th outlrs and thr ffct on th maxmum lklhood ft of a logstc rgrsson modl. * Assstant Drctor, Bangladsh Bank, Mothl, Dhaka000, Bangladsh. Emal: monzur_h@yahoo.com

2 Dhaka Unv. J. Sc., 5(): (July) In ths papr t s shown that th local nflunc tchnqus hav a clar and concs advantag and flxblty ovr th global nflunc. Th casdlton approach s an xampl of global nflunc snc t causs maor prturbatons of a modl. It s notworthy that th nflunc of a group of cass or varabls cannot b dtctd by a global nflunc mthod. Th global nflunc mthod also fals to dntfy th natur of th nfluntal lmnts. Ths papr focuss th dvlopmnt of th dagnostcs for th prturbatons of caswghts, xplanatory varabls (on or mor) n a lnar logstc rgrsson modl. Th ffct of prturbatons on th stmats of paramtrs s also assssd. In addton, th ntrprtaton of th valu of curvatur dagnostcs s hghlghtd n ths papr.. Th Logstc Rgrsson Modl W consdr th logstc rgrsson modl of th dchotomous outcom varabl Y takng valus and 0 wth probablts π and π rspctvly (s Hosmr and Lmhow;989) as X β π Pr( Y X X ) + X β and π Pr(Y 0 XX ) (.) Xβ + whr X s th th row of X and β b th vctor of paramtrs. Th unprturbd loglklhood functon of th logstc rgrsson modl s dfnd by n X β L ( β) β + Y X log( ). (.) Introducng dffrnt typs of prturbaton schms to (.) w can assss th local nflunc on th paramtr stmats as wll as nfluntal lmnts can b dtctd whch s dscussd n scton 3. Th maxmum lklhood stmats can b obtand by maxmzng loglklhood for whch th scor vctor s U ( βˆ ) L( β) 0 β L( β) n (y πˆ ) x 0 ; 0,,...,p β L( β) / and th nformaton matrx s dfnd by I( β ) X QX ; whr Q dag π / (π ). Th β β soluton to th lklhood quatons s obtand usng a numrcal tratv mthod such as Nwton Raphson mthod. Th followng st of tratv quatons s solvd for stmatng β: (.3)

3 Dhaka Unv. J. Sc., 5(): (July) β t+ β t +I(β ) U(β ), t0,,,... (.4) 3. Local Influnc Tchnqu Lt L(β ω) b th loglklhood for th prturbd data whr ω b a n vctor of small prturbatons and lt β ω b th maxmum lklhood stmat from th prturbd data. Lt ω 0 rprsnts null prturbaton so that L(β ω 0 ) L(β). For a unt drcton vctor u, Cook(986) dfnd curvatur dagnostc as C(u) u / Hu (3.) whr u s th gn vctor componnt of H and (,)th lmnt of H s dfnd ( ) L βˆ ω by H ω ω n n. Th matrx H can b mor asly computd by usng th rlaton H / I valuatd at ω 0 and β whr I I(β ) s th varanccovaranc matrx and matrx of ordr (p+) n s dfnd by L β ω β ω ( ) (3.) valuatd at ω 0 and β. Th maxmum curvatur dagnostc C max asssss local nflunc on th paramtr stmats and t s obtand from (3.) by consdrng gn vctor u max of th nflunc matrx H corrspondng to th largst gn valu. Th actual ffct of th locally nfluntal lmnts can b dtrmnd by prturbng th data n th drcton ndcatd by u max. It also gvs th masur of local chang n th stmats of rgrsson coffcnts as masurd by th lklhood dsplacmnt. Th lklhood dsplacmnt for assssng th nflunc of ω s dfnd by D(ω){L(β )L(β ω)} whch compars β ω and β wth rspct to th unprturbd loglklhood. Escobar and Mkr (99) showd that th lklhood dsplacmnt could b approxmatd by takng half of th dagonal 3

4 Dhaka Unv. J. Sc., 5(): (July) D ω H. lmnts of H for prturbng a sngl cas.. ( ) (3.3) By plottng H aganst cas numbr, w can also dtct th most nfluntal cas(s) along wth othr nfluntal cass. Th local nflunc dagnostc procdurs partcularly dpnd on th rang of scal of th prturbaton of ntrst. Thr s no standard rul for th rang of ω. Usually t may rang btwn 0 to (Cook, 986) wth a null prturbaton ω 0 for addtv prturbaton and ω for multplcatv prturbaton. Th prturbaton rang dpnds on th natur of th slctd varabls and th undrlyng modls of ntrst. It s notworthy that succssful dagnostcs dpnd on th choc of ω. 4. Local Influnc Dagnostc Procdurs Th dagnostc procdurs for th prturbaton of caswghts, all xplanatory varabls wth spcal cas of ndvdual and mor than on xplanatory varabls, and ndvdual coffcnt of th logstc rgrsson modl ar proposd n ths scton. Th dagnostcs for th mnor prturbatons of th abov mntond lmnts ar usd to obsrv th changs on th stmats and to dtct nfluntal cass. 4. Caswghts Lt ω / (ω,...,ω n ) b a vctor of wghts whl ω 0 / (,...,) rprsnts n vctor of null prturbaton. To assss th nflunc for th caswght prturbatons, th prturbd loglklhood s n Xβ dfnd by L( β ω) ω [ Y X β log( + )]. (4.) Lt w consdr that th ntrcpt s ncludd n th modl, so th scorvctor U( βˆ ω) s solvd as x βˆ n L( β ω) ω (Y ˆ )X 0 ; whr ˆ π π ˆ ; 0,,...,p (4.) x β β + Th (,)th lmnts of matrx of ordr (p+) n s gvn by ( β ω) L β ω yx x β + x β x ( y π ) x ; 0,,...,p ;,,..., n (4.3) valuatd at ω 0 and β. Th nflunc matrx H of ordr n n s obtand by followng th rlaton 4

5 Dhaka Unv. J. Sc., 5(): (July) H / I undr th prturbaton schm. Thus th maxmum curvatur C max can b asly computd as dfnd n (3.). 4. Explanatory Varabls In ths scton, w consdr a gnral mthod for prturbng th whol dsgn matrx X.. for th modfcaton of all xplanatory varabls. Lt β b a vctor of paramtrs and th prturbd loglklhood L(β ω) s obtand by rplacng xplanatory varabls X wth Z whch s dfnd as Z X + WV (4.4) whr W (ω ) s a n (p+) matrx of prturbatons and th scalng factor V dag (v,v,...,v p+ ) s usd to convrt th prturbatons ω to th approprat sz and unts so that ω v s consstnt wth th th lmnt of X. Undr ths prturbaton schm, th prturbd loglklhood wll tak th form [ ( )] β ( x + ω v ) ( β ω) y ( x + ω v ) β ln + L (4.5) whr,,...,n and 0,,...,p and to obtan th stmats of th paramtrs, w solv th followng frst dffrntal tratv quatons by NwtonRaphson Mthod: ( β ω) L β ( y ˆ πω ) 0 z ( y ˆ πω )( x + ωv ) 0; whr πˆ ω 0 ( x +ωv ) βˆ ( x +ωv ) + To comput th curvatur dagnostcs, w partton th matrx of ordr (p+) n(p+) nto (p+) submatrcs as (,,..., p+ ) (4.7) whr kth (k,,...,p+) submatrx k of ordr (p+) n s dfnd by k ( β ω) L β ω ( y π )( x + ωv ) ω k ( y πω ) πω ( πω )z β ) k βˆ. v ;,,...,n and 0,,...,p (4.6) ω (4.8) valuatd at ω 0 and β.thus th nflunc matrx H s calculatd by consdrng of (4.7) through th rlaton H I. In ths applcaton, H s a vry larg n(p+) n(p+) matrx. a) Indvdual Explanatory Varabl Th abov rsults can b rstrctd to th stuatons whr only on xplanatory varabl s of ntrst by sttng v 0 for th unprturbd varabls. In partcular, lt only th frst column of X s prturbd. 5

6 Dhaka Unv. J. Sc., 5(): (July) Thus v 0 for and th nflunc matrx s dfnd by H / I (4.9) whr s a (p+) n matrx and C max s calculatd by obtanng u max from H. b) Mor than On Explanatory Varabls Th procdur of calculatng th maxmum curvatur dagnostc C max s dscussd n ths scton for partal prturbaton of th whol dsgn matrx. Ths s th xtnson of stuaton (a) dscussd abov for prturbng two or mor but lss than p+ xplanatory varabls. For xampl, th procdur of calculatng th curvatur dagnostcs s dscussd for th followng stuatons: () Suppos, w want to prturb th frst two columns of Z. Thus v 0 for, and s parttond as (, ); () Smlarly, f w want to prturb columns, and 3, so wll b of ordr 3 3n that s (,, 3 ) 3 3n. 4.3 Indvdual Coffcnt For xamnng th snstvty of th th coffcnt to ach of th prturbaton schm dscussd abov, a curvatur dagnostc s xtndd for th logstc rgrsson modl on th bass of Cook s (986) mthod suggstd for lnar rgrsson. Frst w rarrang th columns of X as X (X (), X () ) so that th frst column X () corrsponds to th coffcnt β of ntrst. Th curvatur s dfnd by Cu ( ) u ( I ) u β (4.0) whr u s th gnvctor of ( I ) corrspondng to th largst gnvalu and th symbols dnot usual manng xcpt whch s gvn by 0 0 (4.) 0 P M N whr P s obtand from th partton of th nformaton matrx I. O P Undr spcfc prturbaton schm, th snstvty of th othr coffcnts can b nvstgatd by followng th abov procdur. 5. Illustraton Th applcaton of th proposd dagnostcs s shown by fttng a logstc rgrsson modl to th Framnghalm Hart Study data subst whch conssts of a random sampl of 00 ndvduals out of 669 ndvduals (takn from Kahn t al. 989). Th rspons s bnary, prsnc or absnc of coronary hart dsas (CHD), and xplanatory varabls ar ag, systolc blood prssur, dastolc blood prssur, cholstrol, Framngham rlatv wght (FRW), and cgartt smokd pr day(cig). To dntfy th nfluntal cass and to assss nflunc on th stmat of β, w frst us th local nflunc dagnostcs for th null prturbatons of th caswghts. Indx plot of U max (Fg.) and 6

7 Dhaka Unv. J. Sc., 5(): (July) valu of / of H (Tabl 4.) dmonstrats that th cas 5 s th most nfluntal along wth th cass 4, 47, 68, 98,7, 93 tc umax Cas numbr Fgur : Indx plot of u max whn ω for all. Snc th cas 5 has largr varanc componnt, th gratst local chang n β ssntally dpnds on th wght gvn to cas 5. Th nxt attmpt has bn mad only to prturbng th most nfluntal cas 5 and w consdr ω Th nfluntal natur of th cass rman unchangd for ths mnor prturbatons. Th ffct for th dlton of th cas 5 and othr nfluntal cass on th stmats s also assssd n Tabl 4. by maxmum curvatur dagnostc and componnts of u max ar shown n Tabl 4.3. Dlton of th cass 4, 47 has changd th nfluntal natur of th cas 5 and C max valu s found to b comparatvly hghr for dlton of th cass 4, 47 and 5. In Tabl 4., th applcaton of th dagnostcs for th prturbaton of an ndvdual xplanatory varabl s also dmonstratd to s th ffct of mnor prturbatons on th stmats of paramtrs and on th natur of th nfluntal cass. Undr th prturbaton schm, w dcras th valu of CIG nto 5 for th followng nfluntal cass: 4, 47, 5, 54, 64, 98, 7, 5, 7, 90, 93 and 97 bcaus th numbr of cgartt smokd pr day s at last 0 by ach of th cass. For such prturbatons, th C max valu s found to b 4.77, whch s much lowr than for othr prturbatons and ndcats th xtnt of nflunc on th stmats. It s also vdnt from th componnts of u max that all th nfluntal cass hav changd thr drcton of dvaton. Thus th prturbaton schm consdrd hr s found to b nfluntal to th modl fttng and dagnostcs suggst that th ont nflunc or 'maskng ffct'of th cass s du to thr hghr numbr of cgartt smokd pr day. Maor prturbaton such as cas dlton s assssd by Curvatur dagnostcs and n ths stuaton u max s also usd to ndcat th drcton of varaton of th cass (Tabl 4. and 4.3). Although th tradtonal dagnostcs for cas dlton such as Cook s D, DFBETA and DFFITS tc. ar not usd hr, w can drv concluson from th rsults that local nflunc dagnostcs can b usd to assss th global nflunc on th stmats. 7

8 Dhaka Unv. J. Sc., 5(): (July) Undr th null prturbatons and th prturbatons of xplanatory varabl CIG, th snstvty of coffcnts s also xamnd by th Curvatur dagnostcs. In both stuatons, coffcnt β 6 shows snstvty as th C max valu s hghr (Tabl 4.4). Hr usually th quston arss about th valu of C max n ordr to dntfy th nflunc contand n data. Th dcson can b takn from mprcal xprnc of usng dagnostcs and undrstandng th natur of th obsrvatons. Howvr, thr s arbtrarnss n makng dcson on th bass of curvatur dagnostcs. Ths ssu s addrssd n ths papr and an altrnatv approach s proposd by usng Chsquar calbraton of C max. Th altrnatv approach of th Chsquar calbraton for C max s dscussd blow. Escobar and Mkr (99) showd that D(ω) can b xprssd approxmatly as ½ u / Hu and suggstd that f D(ω)>χ (α,p), th prturbaton ω rsults n a βˆ ω that ls outsd of th null prturbaton approxmat lklhoodratobasd 00(α)% confdnc rgon for β. Snc C max u / Hu, from th abov rsults w may conclud that C max >4χ (α,p) ndcats nflunc on th stmats whch provds approxmat lklhoodratobasd 00(α)% confdnc rgon for β. On th bass of th chsquar calbraton, w can summarz th rsults of C max n Tabls 4., 4. and 4.4. For th C max valus n Tabl 4. and Tabl 4., consdr α0.05 and p7 for whch th χ valu s.67. Snc th C max valus for th caswght prturbatons ar gratr than 8.67 n Tabl 4., so th prturbaton ω rsults n a βˆ ω that ls outsd of th null prturbaton whch provds 95% confdnc rgon for β. Smlarly, th C max valus for th prturbaton of th xplanatory varabl CIG ndcats no nflunc on th stmats (Tabl 4.), whl C max valus n Tabl 4. ndcats nflunc on th stmats for th dlton of th slctd cass. Th snstvty of th coffcnts for th spcfc prturbatons s also xamnd by th proposd chsquar calbraton for C max. For th C max valus n Tabl 4.4, w consdr α0.05 and p for whch th valu of χ s Snc C max valus of β, β 5 and β 6 ar gratr than 0.036, ths coffcnts ar found snstv to th modl fttng for th null prturbaton. Smlarly all th coffcnts ndvdually shows snstvty to th modl fttng for th prturbaton of CIG. But for ths prturbaton schm, th ovrall ffct was found lss nfluntal on th stmats (Tabl 4.). 5. Summary and Concluson Th applcaton of th proposd dagnostcs for th logstc rgrsson modl undr dffrnt prturbaton schms has bn dscussd n ths papr. Th dagnostcs can b mployd n ordr to dtct nfluntal cass that produc th gratst local changs on th stmats. Th advantags of 8

9 Dhaka Unv. J. Sc., 5(): (July) local nflunc analyss ovr global nflunc analyss ar also xplord. From our mprcal rsults, w s that locally nfluntal cass ar also globally nfluntal. Ths papr also provds nsght nto th ntrprtaton of th valu of curvatur dagnostc on th bass of chsquar calbraton. It s notworthy that rlatvly largr valus of curvatur dagnostc ndcat th nflunc of th prturbatons. Ths s also nvstgatd by th chsquar calbraton. On th bass of th rsults obtand from th curvatur dagnostcs for th prturbaton of xplanatory varabl, CIG rvals that ncrasd numbr of cgartts can nflunc th stmats. Th ovrall ffct of ths prturbatons has lss nflunc on th stmats whl ndvdually th coffcnts show snstvty to th modl fttng. Dlton of th nfluntal cass rsultd n small changs n th paramtr stmats as wll as n th valu of curvatur dagnostcs. Ths changs ar not larg nough to affct th nfrncs drawn from th analyss. Tabl 4. Influnc gn vctor componnts, half of th H valus and stmatd paramtrs for null prturbaton, caswght prturbaton (w 5 0.8), and prturbaton of xplanatory varabl CIG. Cas # C max Egn vctor componnt Half of H valus Paramtr stmats U (null prturbaton) U (w 50.8) U 3 (prturbaton of CIG) V (null prturbaton) V (w 50.8) V 3 (prturbaton of CIG) For null prturbaton β β 0.0 β 0.00 β 3 9 β β β For w 50.8 β β 0.0 β 0.00 β 3 9 β β β For prturbaton of CIG β β 0.00 β 0.00 β 3 6 β β β Tabl 4. : Chang on M.L.Es wth th th cas dltd. 9

10 Dhaka Unv. J. Sc., 5(): (July) Cas M.L.Es wth cas dltd β 0 β β β3 β4 β5 β6 C max Non Tabl 4.3 : Th componnts of u max for th cas dltd. Slctd Cas () u (4dlt d) u (47 dltd) u 3 (5 dltd) u 4 (98 dltd) u 5 (7 dltd) u 6 (5 dltd) Tabl 4.4 : Examnng th snstvty of an ndvdual coffcnt wth null prturbaton and for th prturbaton of varabl CIG. Coffcnts Valu of C max (for null prturbaton) Valu of C max (for prturbaton of CIG) β β β 3 β 4 β 5 β Rfrncs Cook, R.D. (986). Assssmnt of local nflunc, Journal of Royal Statstcal Socty, Srs B; 48,

11 Dhaka Unv. J. Sc., 5(): (July) Escobar, LA and Mkr, W.Q. (99), Assssng nflunc n rgrsson analyss wth cnsord data. Bomtrcs 48, Hosmr, D. W. and Lmshow, S. (989). Appld logstc rgrsson. John Wly and Sons, Nw York. Kahn, H. A. and Smpos, C. D. (989). Statstcal mthods n pdmology. Oxford Unvrsty Prss, Nw York. Prgbon, D (98). Logstc rgrsson dagnostcs. Annals of statstcs, 9, Thomas, W. and Cook, R.D. (989). Assssng nflunc on rgrsson coffcnts n gnralzd lnar modl. Bomtrka 76, Wssfld, L. A. and Schndr, H. (990a). Influnc dagnostcs for normal lnar modl wth cnsord data. Australan Journal of Statstcs, 3, 0. Wssfld, L. A. and Schndr, H. (990b). Influnc dagnostcs for th Wbull modl ft to cnsord data, Statstcs and Probablty Lttrs, 9, Wssfld, L. A. (990). Influnc dagnostcs for th proportonal hazards modls. Statstcs and Probablty Lttrs, 0, 447.

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