Binary Response Nonparametric Regression Model And Its Application in University Graduation
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1 Proceedgs of IICMA 9 Topc, pp.. Bary Respose Noparametrc Regresso Model Ad Its Applcato Uversty Graduato Jerry Dw Troyo Puromo Statstcs Departmet Isttut Tekolog Sepuluh Nopember Emal: errypuromo@yahoo.com Suhartoo Statstcs Departmet Isttut Tekolog Sepuluh Nopember Emal: suhartoo@statstka.ts.ac.d ABSTRACT I lear logstc regresso model, the epectato of a bary respose varable wth the logt model l(p(/(-p( = `a. The assumpto of learty s ofte volated by smooth fucto of the eplaatory varables, so that, alteratve form are sought. The epectato of k a bary respose varable wth the logt model become l(p(/(-p( = are geeral smooth fuctos of the eplaatory varables. Estmato s achevg usg local mamum lkelhood. The techque s llustrated uversty graduato problem (master degree, wth status of grade pot average (GPA as respose varable, ad toefl pre-test ad Tes Potes Akademk (TPA as eplaatory varables. Key word: Logstc regresso, bary respose, smooth fucto, local mamum lkelhood. INTRODUCTION Oe of the mportat methods statstcs s that of regressg a bary respose varable o a set of eplaatory varables. Ths has specal feld medcal dagoss, educato ad rsk aalyss. A eample wll be used ths paper s uversty graduato (master degree of Isttut Tekolog Sepuluh November Surabaya. The respose varable y s coded f GPA greater tha 3.5 (scale 4., else. We have a sample of master degree graduato s GPA, whch we use to model the probablty of the bary respose as a fucto of the eplaatory varables. Specfcally, we wsh to estmate p( = p(y= = -p(y= for ay vector. A stadar approach to the problem s the lear logstc regresso model (Haste, 987: p( ep(' a ep(' a or p( l logt p( ' a ( p( Mathematcs Subect Classfcato:
2 I words ths says that the log-odds of the model are lear the predctor varables. The ukow parameters a, ca be foud by usg mamum lkelhood (Hosmer ad Lemeshow, 989. Aother approach s to make the assumpto that the predctor varables are otly ormal wth the same covarace matr each group (y= or y= but wth dfferet mea vectors. For ths model, the log-odds are oce aga lear ad the parameters are fucto of the parameters of the ormal dstrbuto. Ths s kow as Fsher s lear dscrmat fucto (Lachebruch, 975. The logt form ( guaratee that the estmated probabltes are postve ad the terval [,]. It s also the form of the atural parameter for Bomal dstrbuto Epoetal Famly (Haste, 987. A ofte uustfed ad msleadg assumpto s that logt p( s lear. The effect of a predctor may be felt oly for a porto of ts rage. Sometmes ths lear effect s requre terms of predctve ablty, however, a lear term mght be approprate ad lead to the wrog terpretato. I order to geeralze (, we propose the model: k logt p( ( ( s a specfed oparametrc smooth fucto of, ad k s the dmeso of. The estmato s performed usg local lkelhood techque troduce by Tbshra (98 the cotet of cesored data ad the proportoal hazards model.. The Lear Logstc Model. METHODS Cosder the lear case whch logt p( = a. The log-lkelhood for depedet observatos (y,,, (y, s L( a [ y l p ( y l ( p ] [ y a ' l ( e ' a ] p = p(. Let X be p matr of the predctor varables, y a vector of resposes ad p a vector of model probabltes wth th elemet p. The mamum lkelhood estmate â mamaze (3 ad the score fucto s gve by X '( y pˆ (4 ' aˆ e pˆ a ' ˆ e The epected formato matr s gve by I(a X' VX V s a dagoal matr wth th etry p (-p. The Newto-Rhapso teratve procedure ca be used to solve the o-lear system (4 wth the estmate at the (t+ st terato a ˆ( t aˆ( t I ( aˆ( t X '( y pˆ( t (5 (3
3 . The Nolear Model Wth Oe Predctor Cosder that the model logt p( = ( s a scalar predctor varable. Let the sample pot,,, be sorted ascedg order. We wsh the estmate at each pot to ehbt the local behavor of the respose. We the cosder oly those pots wth a certa eghborhood of ad base estmato o them. The eghborhood s defed terms of a spa, whch s the proporto of the sample. Usually we take half the spa to the left, ad half to the rght of. At the ed pot we have to cosder asymmetrc eghborhood. The local lkelhood for spa s, (s [,], at pot gve by r(, s a ( a ( L ( a(,, s [ y a ( y a ( l( e ] (6 l(, s l(, s = ma(, - s r(, s = m(, + s Let aˆ ( mamze (6 ad defe ˆ ˆ a ˆ aˆ ( ( (7 The estmate of s oly affected by the s/ earest eghbors to the left ad s/ to the rght, ad thus ehbts local propertes of the data. As we move to estmate, pot l(, s leaves the lkelhood ad pot r(, s+ eters t, ad thus the lkelhood does ot chage much. As a cosequece, aˆ ( s ot much dfferet from a ˆ(, ad hece ˆ s ot much dfferet from ˆ. Ths result a smooth estmated curve ˆ (.. As s crease toward, ˆ wll get smoother ad the lmt s the usual straght le (Haste, 987. Each local lkelhood s mamzed usg the above teratve procedure ad ca be tme cosumg. However, aˆ ( s a ecellet startg value for the (+ st local lkelhood ad covergece s usually acheved or teratos (Haste, The No-Lear Model Wth More Tha Oe Predctor The procedure here s related to the backfttg algortm appled to addtve models Fredma ad Stuetzle (98, ad adapted by Tbshra (98 for local lkelhood estmato the Co model. Suppose we are gve (.,..., p (. ad let ' lkelhood s: p ( p = (,,, p. We eed to estmate p p. The local 3
4 L ( a(,, s r(, s l(, s ( y ( p a ( a ( p ( p y a( y pa( l( e logt ( p ˆ p aˆ aˆ ( ( ( ( p. The local formato s defe smlarly. Thus ˆp (. ca be foud usg the Newto-Rhapso procedure as before..4 Sple Noparametrc Regresso There are varous approaches used to get the estmator f. If the regresso curve s assumed to be smooth, the sese of cotuous ad dfferesable, the estmate for f s obtaed by usg the approach of Pealzed Least Square (PLS, the crtera that cosder goodess-of-ft smoothess. I geeral, sple fucto wth orde (m- wth kots S, S,... S k are ay fucto that ca be preseted the form (Heckma, 986: k h k t δ (t S S(t α (8 (t S k, t S (t k S, t < S ad are real costats ad S, S,...S k are kots..5 Choose of Smoothg Parameter ( λ Smoothg parameter s cotrollg the balace betwee the complace curve to the data ad smoothess curve (Eubak, 988. Parg a very small value or greater value of λ wll provde very coarse resoluto or very smooth fucto. (Wahba, 99; Eubak, 988. Several methods of selecto of smoothg parameter whch have bee developed are Cross Valdato (CV ad Geeralzed Cross Valdato (GCV (Crave ad Wahba (979, Wahba (985, L (986, Koh et al (99, Adrews (99, Shao (993, Veter ad Syma (995, Kauerma ad Opsomer (, Carew, Wahba, Xe, Nordhem, ad Meyerad (3, Ruppert, Wad, ad Carroll (3. I the sple regresso model, GCV crtero s defed as: GCV ( ( MSE( ( tr[ I A( ] 4
5 ( (y tr(i f λ A(λ (I A(λ y tr(i A(λ 3. MAIN RESULTS A observato has bee doe 8 by Fathurahma at Isttut Tekolog Sepuluh November (ITS cocered master degree graduato of ITS. There are 3 graduato studets ad three varables: y = f GPA < 3.5 (scale 4. f GPA 3.5 (scale 4. = toefl pre-test score = TPA score Descrptve statstcs of ITS graduato studets are gve Table 3., ad the summary of parameter estmato of logstc regresso are show Table 3.. Table 3.. Descrptve statstcs of graduato studets Varable Mea St. Dev. Mmum Mamum GPA (y Toefl pre-test ( TPA ( Table 3.. Parameter estmato of logstc regresso Varable a Stadard Error Sg Costat The lear logstc regresso model from ths observato s: ep( p( ep( or p( l p( logt p( The observato made by Fathurahma (8 for master degree graduato of ITS show that betwee GPA, toelf pre-test score ad TPA score have uclear patter. Those are see graduato studets wth hgh GPA, ther toefl score ad TPA score are low, ad vce versa. It s also show from R = 3 %, that relatvely small. Ths show dcato the usage 5
6 GPA GPA PURNOMO AND SUHARTONO of oparametrc models. So that, the lear logstc regresso model s become: logt p( = ( ( 56 Wth R = 98% Toelf Toefl Pre-test Fgure 3.. Correspodece betwee GPA ad toefl-pre-test TPA Fgure 3.. Correspodece betwee GPA ad TPA score 3. CONCLUDING REMARKS The presece of pecular relatoshp the data up above volve that lear logstc regresso method s less sutable practce. Pecular relatoshp betwee respose varable ad eplaatory varables are due to the estece of smooth fuctos eplaatory varables. The use of oparametrc, sple approach, s the soluto that ca be used to overcome ths codto, sce ths approach gves better result tha lear logstc regresso. 6
7 REFERENCES. D.W.K. Adrews, Asymtotc Optmalty of Geeralzed C L, Cross Valdato, ad Geeralzed Cross Valdato Regresso wth Heterokedastc Error, Joural of Ecoometrcs, 47, , 99.. J.D. Carew, G. Wahba, X. Xe, E.V. Nordhem, ad M.E. Meyerad, Optmal Sple Smoothg of fmri Tme Seres by Geeralzed Cross Valdato, NeuroImage, 8, 95-96, Crave, ad G. Wahba, Smoothg Nosy Data Wth Sple Fucto: Estmatg The Correct Degree of Smoothg by The Method of Geeralzed Cross Valdato, Numer. Math,3, , R.L. Eubak, Sple Smoothg ad Noparametrc Regresso, Marcel Dekker, New York, J.H. Fredma, ad W. Stuetzle, Smoothg of Scatterplots, Dept. of Statstcs Tech. Rept. Oro 3, Staford Uversty, T.J. Haste, Noparametrc Logstc ad Proportoal Odds Regresso. Appled Statstcs, 36, 6-76, D.W. Hosmer, ad Lemeshow, Appled Logstc Regresso, New York: Joh Wley, G. Kauerma, ad J.D. Opsomer, A Fast Method for Implemetg Geeralzed Cross Valdato Multdmesoal Noparametrc Regresso, Paper 47,. 9. R. Koh, Et al, The Performace of Cross Valdato ad Mamum Lkelhood Estmators of Sple Smoothg Parameters, Joural of The Amerca Statstcal Assocato, 86, 4-5, 99.. P.A. Lachebruch, Dscrmat Aalyss, New York: Hafer Press, K.C. L, Asymtotc Optmalty of C l ad Geeralzed Cross Valdato Rdge Regresso Wth Applcato to Sple Smoothg, A.Statst., 4, -, D. Ruppert, M.P. Wad, ad R.J. Carroll, Semparametrc Regresso. New York: Cambrdge Uversty Press, R. Tbshra, Noparametrc Estmato of Relatve Rsk. Submtted to the Joural of The Amerca Statstcal Assocato, J.H. Veter, ad J.L.J. Syma, A ote o The Geeralzed Cross Valdato Crtero Lear Model Selecto, Bometrka, 8, 5-9, G. Wahba, A Comparso of GCV ad GML for Choosg the Smoothg Parameter the Geeralzed Sple Smoothg Problem, Joural the Aals of Statstcs, 3, 378-4, G. Wahba, Sple Models for Observasoal Data, SIAM, Pesylvaa, 99. 7
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