Robust Regression Analysis for Non-Normal Situations under Symmetric Distributions Arising In Medical Research

Transcription

1 Joual of Mode Appled Statstcal Methods Volume 3 Issue Atcle Robust Regesso Aalyss fo No-Nomal Stuatos ude Symmetc Dstbutos Asg I Medcal Reseach S S. Gaguly Sulta Qaboos Uvesty, Muscat, Oma, gaguly@squ.edu.om Follow ths ad addtoal woks at: Pat of the Appled Statstcs Commos, Socal ad Behavoal Sceces Commos, ad the Statstcal Theoy Commos Recommeded Ctato Gaguly, S S. (04) "Robust Regesso Aalyss fo No-Nomal Stuatos ude Symmetc Dstbutos Asg I Medcal Reseach," Joual of Mode Appled Statstcal Methods: Vol. 3 : Iss., Atcle 9. DOI: 0.37/masm/ Avalable at: Ths Regula Atcle s bought to you fo fee ad ope access by the Ope Access Jouals at DgtalCommos@WayeState. It has bee accepted fo cluso Joual of Mode Appled Statstcal Methods by a authozed edto of DgtalCommos@WayeState.

2 Joual of Mode Appled Statstcal Methods May 04, Vol. 3 No., Copyght 04 JMASM, Ic. ISSN Robust Regesso Aalyss fo No-Nomal Stuatos ude Symmetc Dstbutos Asg I Medcal Reseach S. S. Gaguly Sulta Qaboos Uvesty Muscat, Oma I medcal eseach, whle cayg out egesso aalyss, t s usually assumed that the depedet (covaates) ad depedet (espose) vaables follow a multvaate omal dstbuto. I some stuatos, the covaates may ot have omal dstbuto ad stead may have some symmetc dstbuto. I such a stuato, the estmato of the egesso paametes usg Tku s Modfed Maxmum Lkelhood (MML) method may be moe appopate. The method of estmatg the paametes s dscussed ad the applcatos of the method ae llustated usg eal sets of data fom the feld of publc health. Keywods: Maxmum lkelhood, modfed maxmum lkelhood, studet s t- dstbuto, ode statstcs, delta method Itoducto Ofte medce, a elatoshp s establshed betwee a espose vaable y, whch depeds o the covaates x, x,, x, whch ae depedet of each othe, so that, total, thee may be ( + ) vaables. I classcal egesso model, the espose vaable y s teated as a adom vaable whose mea depeds upo fxed vaables of the x s. The mea s assumed to be lea fucto of the egesso coeffcets α, β, β,, β. The lea egesso model also ases a dffeet settg. Suppose all the vaables y, x, x,, x ae adom ad have a ot dstbuto f ( y, x, x,..., x ), whch s ot ecessaly omal so that S. S. Gaguly s a Pofesso the Depatmet of Famly Medce ad Publc Health. Emal at: gaguly@squ.edu.om. 446

3 S. S. GANGULY f ( y, x, x,..., x ) g( y x, x,..., x ) h( x ). () It s assumed hee that the codtoal dstbuto of y gve, x, x,, x s omal ad s gve by g( y x, x,..., x ) ( ) o o o exp y o o x ( ) o o () wth mea o E( y x) o ( ) o x (3) ad vaace o o V y x. (4) The magal desty coespodg to the covaate x s assumed to be symmetc about mea of the fom: x f (5) Hee o E( x ), V( x ) ad (,,..., ) s the coelato coeffcet betwee y ad x. Relato () povdes fo the measuemet of depedecy of the espose adom vaable o the adom covaates x (=,,,). The lea elatoshp may also be wtte the fom of classcal egesso model as E y x x (6) 447

4 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS whee ad o o o (7) o o,,,..., (8) ae the egesso coeffcets. It may be oted that E y x s the best lea pedcto of the espose vaable y whee the populato s N (, ). I medcal epdemology, oe ofte ecoutes stuatos whee some (f ot all) covaates x have o-omal symmetc dstbutos. Ths atcle s estcted to a stuato whee the covaates have o-omal symmetc T dstbutos. The obectve, theefoe, s to estmate the paametes, fom sample values y x,. Fo ths, cosde the famly of studet s t-, dstbutos. The method, whch has bee developed hee, s, of couse, geeal ad ca be used fo othe famles of locato-scale dstbutos of the type (5). Lkelhood equatos Suppose that the covaate x (=,,,) has the symmetc dstbuto wth the desty gve by p ( x ) h x k, x k (9) whee k p 3, p ; E( x ) ad vx ( ). Assume that p s kow. Fo p = 5, (9) s almost dstgushable fom logstc dstbuto, because the two dstbutos ae both symmetc ad have fst fou momets commo (Peaso, 963). If the two dstbutos ae plotted, t wll be see that oe sts almost o top of the othe. It may be oted that 448

5 S. S. GANGULY t ( x ) ( k) has Studet s t dstbuto wth (p) degees of feedom. Fo p, k s equal to whch case (9) s smply a scale paamete. Gve the data matx ( > +) of the fom ( y ; x,..., x,... x ),,,..., (0) k whee y s the espose vaable ad the x tems as explaatoy vaables o covaates. The the lkelhood fucto L based o elato () ca be wtte as usual ad s gve by L o ( x ) * k o *exp o y o o x ( ) o o p () whee x( ), (,,..., ;,,..., ) ae the ode statstcs of x obsevatos, ad y (,.., ) ae the coespodg cocomtat y obsevatos. The maxmum lkelhood estmatos ae the solutos of the lkelhood equatos,.e, of the devatves of L. These equatos ae, howeve, tactable. Solvg them by teatve pocedues may be poblematc, fo example, oe may ecoute multple oots, slow covegece, o covegece to wog values (see specfcally Baett, 966; Lee et al., 980; Tku ad Suesh, 99; Vaugha, 99). Istead the Tkus method of modfed lkelhood (MML) estmato was employed, whch gves explct estmatos ad volves eplacg tactable tems by lea appoxmatos. Because ths method s aleady well establshed 449

6 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS ad s kow to poduce estmatos whch ae fully effcet fo lage (Tku, 970; Bhattachayya, 985) ad almost fully effcet fo small (Tku et al, 986; Tku ad Suesh, 99; Vaugha, 99, 994). Modfed Maxmum Lkelhood Cosde the th covaate of a adom sample of sze deoted by x, x,,x fom ay locato-scale dstbuto wth desty gve by x f,,,...,. Fo smplcty of otato, suppess the suffx ad cosde f to be a studet t desty. The the lkelhood equatos fo estmatg ad coespodg to each covaate ae L p gz ( ) 0 k ad whee ad L p z g( z ) 0 k z ( x ) z gz ( ). ( kz ) Equatos () do ot povde explct solutos. Followg Tku-Suesh (99); Vaugha ad Tku (000), the fst step s to expess these equatos tems of ode statstcs x () x ()... x ( ). Because complete sums ae vaat to odeg () 450

7 S. S. GANGULY ad whee L p k gz ( ) 0 ( ) L p k ( x ) z ( ) ( ),,,...,. z( ) g( z( ) ) 0 (3) Ude appopate egulaty cosdeatos whch ae vey geeal atue, gz ( ) ca be eplaced by lea appoxmatos gve by the fst two tems of ( ) Taylo sees expasos (Tku, 967, 968; Tku ad Suesh, 99; Tku ad Kambo, 99, Vaugha, 99; Vaugha ad Tku, 000), so that whee d gz( ) g t( ) z( ) t ( ) g( z) dz t z,,,..., ( ) E z ( ) ( ). zt( ) (4) Thus, the modfed equatos ae obtaed,.e. ad * L L p z( ) 0 k * L L p z( ) z( ) 0 k Equatos (5) have explct solutos, whch ae called modfed maxmum lkelhood (MML) estmatos. Note that the ML ad MML estmatos ae asymptotcally equvalet. Fo dstbuto ( p, k p 3) (5) 45

8 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS p x ( ) ( ) ( ), h x k x k (6) Ths method gves the followg MML estmatos (see Tku ad Suesh, 99; Tku ad Kambo, 99; Vaugha, 99; Vaugha ad Tku, 000; Tku et al, 008) x ( m ) m (7) ( ) ad B ( B 4 c) ( ) (8) whee p p B x ad C m k ( ) y ( ) k (9) The coeffcets ad ae obtaed fom the equatos k t 3 ( ) k t ( ) ad,,,..., k t ( ) t ( ) k (0) Fo p (.e. fo omal dstbuto), 0 ad, because k=p 3. Note that, ( ) ad 0. Tables of the value of t ( ) ae avalable fo p=(.5) 0 ad 0 (Tku ad Kuma, 985). Fo > 0, t ( ) ae obtaed fom the equato 45

9 S. S. GANGULY t( ) f ( z) dz ( ). () I evaluatg (), t should be oted that ( k/ ) z has studet s t- dstbuto wth p degees of feedom. It may be of teest to ote that devg the estmatos ad gve by the equatos (7)-(0), the method of MML estmato fo p < automatcally gves small weghts to exteme ode statstcs close to the cete. It s pecsely due to ths fact these estmatos ae obust to easoable depatues fom the tue value of p (6). I most applcatos, theefoe, t s ot vey mpotat to ppot the tue value of p ad use t all devatves. Ay easoable value of p gves almost optmal esults. A Q-Q plot ca be employed to gve a easoable value closue (f ot exactly) the tue value of p coespodg to covaate x (Tku et al, 986, p.77). The ode statstc x ( ) s plotted agast the values t E( z ), z ( x ) /,,,...,, ude the assumed model,.e. fo a ( ) ( ) ( ) patcula value of p (6). If the plot gves a staght le (o ealy so), the model s take to be vald fo the MML estmato. Followg the above pocedue, the paametes ad (,,..., ) ae estmated. I ode to estmate the emag paametes vz., o, o, o (,,..., ), the lkelhood fucto () s cosdeed. Because L L ad, (=,,,) ae expessed tems of gz ( ), the lkelhood L L L equatos 0, 0 (,,..., ) ad 0 (,,..., ) o have o explct solutos. The modfed lkelhood equatos ae * * * L L L 0, 0, (= 0,,,) ad 0 (,,..., ), ad ae obtaed by eplacg ( ) o g z wth the lea appoxmatos gve by (4). The solutos of these equatos ae the followg MML estmatos: o o y o ( x ) () 453

10 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS s o (3) o s o s s Hee, o s s o,,,..., o (4), ( ) (5) y y y x x x s (6) ( ) y y y y s o ( ) (7) ( ) x x x x ad o s y yx x ( ) y y x x,,,...,. (8) Relato () povdes fo the measuemet of depedecy of the espose adom vaable o the adom covaates x (,,..., ). The lea elatoshp s also epeseted the fom of classcal model (6). The asymptotc vaaces ad covaaces of the estmatos o,, o, ad o (,,..., ) ae obtaed wth the use of the secod patal devatves of the lkelhood fucto (). The matx fomed by the egatve of the expected values of the secod patal devatves gves the fomato matx, whch may be expessed as the pattoed matx 454

11 S. S. GANGULY V V O O V (9) whee the matx s of the ode (3+) (3+) ad V * L E of ode (+) (+) ad * L V E,,,,..., of ode (+) (+) wth ( o,,..., ok ). The vese of V ad V matces povdes the elemets of the pecso ad covaace stuctue of the estmated coeffcets. The estmated values of the paametes obtaed above ae used elato (7) ad (8) whch gve the estmated values of the egesso coeffcets α ad (,,..., ) of the model (6). The asymptotc covaace stuctue of the estmated egesso coeffcets ad (,,.., ) ae obtaed usg delta method (Seflg, 980) as:,,,, the g ad Let N G G T ( ),, (30) whee g G o,..., ok of ode (3+) (+) ad 455

12 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS V O O V of ode (3+) (3+). Note that whe p the dstbuto (6) educes to the deal omal dstbuto whch case x (sample mea) ad (sample vaace), x ad s beg optmal ude the assumpto of omalty. s Examples Example Cosde the pat of the data set petag to 0 male sul-depedet dabetc patets as povded Dobso (990, p. 69), whch s epoduced Table. Table. Cabohydate, age ad weght fo twety sul-depedet dabetcs y = Cab. (gm) x = Age (ys) x = Wgt (kg) y = Cab. (gm) x = Age (ys) x = Wgt (kg) I ths sample, the goal s to establsh the elatoshp betwee the espose vaable y (amout of cabohydate) ad the two covaates x (age) ad x (body weght, elatve to deal weght fo heght) usg the lea egesso model (6) whch takes the fom E y x, x x x (3) Hee, t s assumed that, elato (), the codtoal dstbuto of the espose adom vaable y s omal; howeve, the covaates follow 456

13 S. S. GANGULY depedetly o-omal symmetc dstbuto. The model (3) s ftted usg above descbed modfed maxmum lkelhood method. Fst obta the values of p ad pcoespodg to the two covaates x ad x usg Q-Q plots, whee the ode statstcs x ( ) ad x ( ) wee plotted sepaately agast t ( ) ad t ( ) espectvely, =,, fo dffeet values of p as gve Tku ad Kuma (985). The values of p 5 ad p 7 povded a appoxmate staght le pattes whch detemed the appopate types of destes (6). Oce p ad p ae kow, the usg the equatos (7)-(0), the MML estmates of the paametes, ad, ae obtaed. Usg these values equatos ()-(8) the est of the paametes o, o, o ad o ae estmated. Solutos of the fomato matx (9) povded the elemets of the pecso ad covaace stuctue of the estmated paametes. The estmated values ad the stadad eos ae peseted Table. Table. MML estmates of the paametes ad the stadad eos fo the data set Table Table 3. MML ad ML estmates of the paametes ad the stadad eos fo the data set Table Paam. Est. Std. E. Paam. Est. Std. E. W μo Costat (α) μ MMLCoeffcet (β) μ Coeffcet (β) σo σ σ Costat (α) ρo ML Coeffcet (β) ρo Coeffcet (β) Usg the estmated values Table elato (7) ad (8), obta MML estmates of the egesso paametes, ad. Use of delta method as descbed (30) povded the asymptotc stadad eos; also these paametes based o usual maxmum lkelhood method wee estmated. The esults, obtaed ude the two methods ae summazed Table 3. The aalyss Table 3 eveals that the MML estmates of the egesso paametes fo the data set Table ae vey close to the values obtaed usg maxmum lkelhood method, as expected. Moeove, the two methods gave appoxmately the same esults fo the Wald statstcs W, whch pemts to test the 457

14 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS ull hypothess Ho : 0 ad 0. Fo lage, the ull dstbuto of W s efeed to a stadad omal dstbuto. Example Cosde aothe data set fom Muay (937), epoduced El-Sad (995, p. 4) as show Table 4. The data povdes obsevatos o the umbe of male fles ded afte twety mutes exposue to pyethum at vaous cocetatos. The ma obectve s to descbe the pobablty of success of dose p as a fucto x. I lteatue, such type of aalyss ae caed out usually cosdeg ethe pobt o logt models (Cox, 970). Howeve, the logt model s pefeed to a pobt model due to two pmay easos (Hosme ad Lameshow, 989): fom mathematcal pot of vew, t s a easly used fucto, ad t leads to tself to a bologcal meagful tepetato. Table 4. Motalty of male fles afte twety mutes exposue to pyethum Cocetato (log0) Exposed Numbe of fles Ded Popotos Ded The logt model s a famly of Geealzed Lea Models (GLMs) wth lk p fucto g( p ) as (Nelde ad Weddebu, 97; McCullagh ad p Nelde, 989). The lk fucto g( p ) s cotuous ad maps the 0, age of pobabltes oto, ad s epeseted by 458

15 S. S. GANGULY p g( p ) x,,,..., p (3) so that exp( x ) p,,,..., exp( x ) (33) The elato (33) s kow as bay logstc model wth pobablty of success p, ths belogs to the stadadzed logstc dstbuto whch s symmetc atue (Rao ad Toutebug, 995, p. 63). I ode to estmate the ukow paametes α ad β (3), usually ML method s used. The techque volves the soluto of the lkelhood equatos, whch have o explct solutos ad have to be solved by teactve pocedues. Solvg these equatos s, theefoe, tedous ad tme cosumg. Theefoe, these paametes ae estmated usg MML method. Fo ths, cosde the lk fucto.e. log odds as a espose vaable ad x as a covaate. Fst estmate ad fo p=5 dstbuto (6). Usg these values equatos ()-(8), the est of the paametes o, o ad o volved the lkelhood fucto () wee obtaed. The estmated values of the vaaces ad co-vaaces wee obtaed usg these values secod patal devatves of the lkelhood fucto () ad solvg fo the vese of the fomato matx (9). The estmated values of the paametes ad the stadad eos volved the lkelhood fucto () wth p =5 fo the data set Table 4 ae show Table 5. Usg these estmated values of the paametes elato (7) ad (8), obta the MML estmates of the paametes ˆ ad ˆ of the logstc model (33). The use of delta method (30) gave the asymptotc vaaces of ˆ ad ˆ. The ML estmates of these paametes ad the vaaces ude the logt model (3) wee also obtaed usg teatve pocedues vz; Newto-Raphso method (Cox, 970, Chapte ). The esults obtaed ude the two pocedues ae summazed Table 6. These aalyses also eveal that the MML estmates of the egesso paametes α ad β fo the data set Table 4 ae vey close to the values obtaed usg maxmum lkelhood method, as expected. 459

16 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS Table 5. MML estmates of the paametes ad the stadad eos fo the data set Table 4 Table 6. MML ad ML estmates of the paametes ad the stadad eos fo logt model (3) Paam. Est. Std. E. Paam. Est. Std. E. μo Costat (α) MML μ Coeffcet (β) σo σ Costat (α) ML ρo Coeffcet (β) Ths study used Tku s modfed maxmum lkelhood method fo cayg out egesso aalyss whe the udelyg dstbutos of the data set have oomal symmetc dstbutos. The method yelds estmatos whch ae explct fuctos of sample obsevatos ad ae umecally vey close to the maxmum lkelhood estmatos ad equally effcet. Refeeces Baet, V. D. (966). Ode statstcs estmatos of the locato of the Cauchy dstbuto. Joual of Ameca Statstcal Assocato, 6(36): Bhattachayya, G. K. (985). The asymptotcs of maxmum lkelhood ad elated estmatos based o type II cesoed data. Joual of Ameca Statstcal Assocato, 80(390): Cox, D. R. (970). The aalyss of bay data. Methue: Lodo. Dobso, A. J. (990). A toducto to geealzed lea models. Chapma ad Hall: New Yok. El-Sad, M. A. (995). A symmetc exteded logstc model wth applcatos to expemetal toxcty data. Bometcal Joual, 37(), Hosme, D. W. ad Lemeshow, S. (989). Appled logstc egesso. Joh Wley: New Yok. Lee, K. R., Kapada, C. H. ad Dwght, B. B. (980). O estmatg the scale paametes of the Raylegh dstbuto fom doubly cesoed samples. Statstsche Hefte, (): 4-9. McCullagh, P ad Nelde, J. A. (989). Geealzed lea models. Chapma ad Hall: Lodo. 460

17 S. S. GANGULY Muay, C.A. (937). A statstcal aalyss of fly motalty data. Soap, 3(8): Nelde, J. A. ad Weddebu, R. W. N. (97). Geealzed lea models. Joual of Royal Statstcal Socety, Sees A, 35(3): Peaso, E. S. (963). Some poblems asg appoxmatg to pobablty dstbutos usg momets. Bometka, 50, 95-. Rao, C. R. ad Toutebug, H. (995). Lea models: least squaes ad alteatves. Spge-Velag: New Yok. Seflg, R. J. (980). Appoxmato theoems of mathematcal studes. Wley: New Yok. Tku, M. L. (967). Estmatg the mea ad stadad devato fom a cesoed omal sample. Bometka, 54(/): Tku, M. L. (968). Estmatg the paametes of omal ad logstc dstbutos fom cesoed samples. Austala Joual of Statstcs, 0(): Tku, M. L., Islam, M. Q. ad Sazak, H.S. (008). Estmato bvaate o-omal dstbutos wth stochastc vaace fuctos. Computatoal Statstcs & Data Aalyss, 5(3): Tku, M. L. (970) Mote Calo study of some smple estmatos cesoed omal samples. Bometka, 57(): 07-. Tku, M. L. ad Kambo, N. S. (99) Estmato ad hypothess testg fo a ew famly of bvaate o omal dstbutos. Commucatos Statstcs Theoy ad Methods, (6): Tku. M. L. ad Kuma, S. (985). Expected values ad vaaces ad covaaces of ode statstcs fo a famly of symmetc dstbutos (Studet s t). I B. J. Tawsk, R. E. Bechhofe, S. Kuma, M. L. Tku, & A. C. Tahmae (Eds.) Selected tables mathematcal statstcs, Vol. 8. Povdece, R.I.: Ameca Mathematcal Socety: pp Tku, M. L. ad Suesh, R. P. (99). A ew method of estmato fo locato ad scale paametes. Joual of Statstcal Plag ad Ifeece, 30(): 8-9. Tku, M. L., Ta, W. Y. ad Balaksha, N. (986). Robust Ifeece. Macel Dekke : New Yok. Vaugha, D. C. (99). O the Tku-Suesh method of estmato. Commucatos Statstcs Theoy ad Methods, ():

18 ROBUST REGRESSION ANALYSIS FOR NON-NORMAL SITUATIONS Vaugha, D. C. (994). The exact values of the expected values, vaaces ad covaaces of the ode statstcs fom the Cauchy dstbuto. Joual of Statstcal Computato ad Smulato, 49(-): -3. Vaugha, D. C. & Tku, M. L. (000). Estmato ad hypothess testg fo a o-omal bvaate dstbuto wth applcatos. Mathematcal ad Compute Modellg, 3: