BAYESIAN ANALYSIS OF THE SIMPLE LINEAR REGRESSION WITH MEASUREMENT ERRORS

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1 BAYESIAN ANALYSIS OF THE SIMPLE LINEAR REGRESSION WITH MEASUREMENT ERRORS Marta Yuk BABA Frado Atoo MOALA ABSTRACT: Usually th classcal approach to mak frc lar rgrsso modl assums that th dpdt varabl dos ot cota masurmt rrors. I practc howvr th data ca cota masurmt rrors ad th prsc of ths rrors ca affct th rsults of th aalyss drastcally. Rodrgus ad Baba (994) proposd a Baysa approach to stmat th slop paramtr β lar rgrsso modl wth masurmt rrors cosdrg th rlablty rato K as kow. Thr ar stuatos howvr whr th formato rgardg th rlablty rato K ot always s avalabl. I ths papr our ma trst s to mak a Baysa frc about β udr th assumpto that th rlablty rato K s ukow. To obta th postror dstrbuto w us Gbbs Samplr algorthm. KEYWORDS: Postror dstrbuto; rlablty rato; slop paramtr; Gbbs samplr. Itroducto Th classcal smpl lar rgrsso aalyss assums that th dpdt varabl s dfd by y = α + β + ε =...; () whr (... ) s fd rpatd samplg ad ε ar dpdt N(0 ε) radom varabls. It s assumd that s masurd wthout rror. Howvr practc partcularly socal sccs ad bologcal ssay ths assumpto s oft volatd. Thr s a lot of work o th problm of paramtr stmato wh th cota rrors of masurmt s for ampl Fullr (987). I th prst papr w propos a Bays approach to stmat th modl paramtrs. W shall study modls of typ () wth whr stad of obsrvg o obsrvs th sum α = 0 W mak th assumpto that + u = =... () UNESP Dpartmt of Mathmatcs Prsdt Prudt SP Brazl. E-mal: marta@fct.usp.br / fmoala@fct.usp.br 74 Rv. Bras. Bom. São Paulo v.30. p

2 u ε NI µ dag (3) ε ( ) '~ (( 0 0) ' ( u )); whr ~ NI s a abbrvato for dstrbutd ormally ad dpdtly ad ( u ) dag s a dagoal matr wth th gv lmts o th dagoal. It follows from () () ad (3) that th vctor ( Y )' s dstrbutd as a bvarat ormal Y β µ β + β ~ NI. µ β + u W ca wthout loss of gralty tak µ = 0. A way of frc of th paramtr cossts of aalyzg t subjct to th twodmsoal modl howvr Rodrgus ad Corda (990) aalyz udr othr prspctv workg wth th codtoal dstrbuto of Y gv. Bcaus ( Y ) s dstrbutd as a bvarat ormal th codtoal dstrbuto of Y gv s gv by whr K ~ Y N K K = = β βu+ ; + u s calld rlablty rato. Rodrgus ad Baba (994) proposd a Baysa approach to stmat th slop paramtr β lar rgrsso modl wth masurmt rrors cosdrg th rlablty rato K as kow. Thr ar stuatos howvr whr th formato rgardg th rlablty rato K ot always s avalabl. I ths papr our ma trst s to mak a Baysa frc about β udr th assumpto that th rlablty rato K s ukow.. Baysa aalyss of th rgrsso modl Lt ( Y )... ( Y ) ar dpdt ad dtcally dstrbutd radom varabls agrmt wth th codtoal modl: whrk s ukow. ~ Y N K K β β u + ; W cosdr th ovr dtfabl stuato that s = u = commo varac s kow. Thus th lklhood fucto s gv by 0 (4) (5) whr th L K K Y K ( K β + ) ( β ) ( β + ) p β 0 = ( ). (6) Rv. Bras. Bom. São Paulo v.30. p

3 Dotg th varac of th dstrbuto gv (6) by = K β + = K β + w obta u 0 ( ) L K Y K ( β ) p = β. ( ) (7) Bo ad Tao (973) propos a locally uform jot pror dstrbuto for β ad log gv by: ( ) Cosqutly th jot pror dsty for π β log costat. (8) β ad s: π ( β ). (9) To propos a pror dstrbuto for K w obsrv that a clos valu of zro ca b du to a grat masurmt rror ( = ) a rror of plag of th data that a statstca caot accpt or t ca b du to = 0 stablshd for th fuctoal cas a stuato o studd hr. Thus w cosdr a pror uform (0.3 ) for K that s ( ) K π costat 0.3 K <. (0) < Now supposg th paramtrs ar dpdt w obta a jot pror for K gv by ( β ) π ( β K ). () Thrfor from (7) ad () w ca prss th jot postror dstrbuto as: p ( K data ) p ( ). Y K β β + () = Bcaus our tto s to stmat th slop paramtr w cosdr th applcato of Gbbs Samplr to obta th margal postror dstrbuto of β. For ths w d to obta th complt codtoal dstrbutos. Wrtg ( Y β K ) = Y βk Y + = = = = β K data algbras th postror codtoal dstrbuto ( ) Y p s gv by ad aftr som 76 Rv. Bras. Bom. São Paulo v.30. p

4 Y = p ( β K data ) p K ; β = K = (3) that s Y = β K data ~ N ;. K K = = Th postror codtoal dstrbuto for K gv by Y p K data p K 0.3 K ; ( β ) = β = β = whch s a Normal Trucatd wth K rstrctd to th trval (0.3 ). Bsds codtoal to β ad K th postror codtoal dstrbuto for s gv by ( ) ( ) β K data p Y + β K (4) p ; (5) = that s th postror codtoal dstrbuto for s a Gamma Ivrtd dstrbuto wth paramtrs α = ad = = γ ( Y ) β K 3. Implmtato of Gbbs Samplr Gbbs samplr s a partcular cas of substtuto samplg Glfad ad Smth (990) whch all full codtoal dsts ar supposd kow. A c troducto to th Gbbs samplr s gv by Caslla ad Gorg (99). I ths papr th codtoal dsts ar gv by quatos (3) (4) ad (5). W ow dscrb th Gbbs Samplr mplmtato usd our framwork. Th algorthm procds as follows 0 0. Choos startg valus ( β K 0 ).. At stp +: a) Draw β from th codtoal postror p ( K data). β gv (3); Rv. Bras. Bom. São Paulo v.30. p

5 b) Draw c) Draw K from th codtoal postror p ( K β data) from th codtoal postror p( K data ) + gv (4); β + + gv (5). 3. Ths provds a squc of sampld valus ( β K ) =0... N whch s a ralzato of th Markov cha assocatd wth th full codtoal dsts of p β K data gv (). ( ) 4. Numrcal llustrato I ths scto w llustratd th prformac of th procdur proposd ths papr basd th sampls gratd th softwar R cosdrg α = 0 β = ad varacs = u = 0 = ad =. Two valus for th rlablty rato K ar cosdrd for stac K = 0.5 ad K =0.8 ordr to compar th prformac of stmats th aalyss. I ths cas th paramtr assums valus qual to =.73 ad.05 rspctvly. A practcal ampl wth ral data s also prstd. As w ar ot abl to fd a aalytc prsso for margal postror dstrbutos ad hc to tract charactrstcs of paramtrs such as Bays stmats ad crdbl trvals w d to appal to th Gbbs Samplr algorthm to obta a sampl of valus of paramtrs from th jot postror. Th cha s ru for N=0000 tratos wth a bur- prod of sz 000 whch wr dscardd to lmat th ffct of th tal valus. Fgur prsts th MCMC output plot ad th margal dsts rsultg for th paramtrs of th rgrsso modl cosdrg K = 0.5 ad =0. Th MCMC plots suggst w hav achvd covrgc. Th postror summars of trst ar gv Tabls ad for dffrt sampl szs as =0 ad =50rspctvly. Both Tabls allow a comparso of th stmators of βk ad usg th approachs proposd by Rodrgus ad Corda (990) ad th Baysa procdur proposd ths papr. Th 95% trvals from th mamum lklhood (ML) ad Baysa approachs ar also dsplayd th Tabl. Th rsults from Tabl show that ML ad Baysa approachs do ot provd good stmats for β howvr th cofdc trval by ML approach has gatv valus. Th Baysa stmat for th paramtr K s too clos to th tru valu whl th ML stmat assums a mpossbl valu that s gratr tha. Bsds th ML trval s largr tha th rag (0 ) of K. For th paramtr th ML stmat s bttr tha th Baysa. 78 Rv. Bras. Bom. São Paulo v.30. p

6 Tabl - Estmats valus ad cofdc trvals for th paramtrs β ML Baysa ML ad Baysa approachs for = 0.5 ad =0 β.354 ( ).8087 ( ) K.03 ( ) ( ) K ad by.73 ( ).8658 ( ) β Tm β k p(β ) Tm k p( ) p(k ) Tm Fgur - Estmats Dsts va Gbbs Samplr ad th tracs of th chas gratd for th paramtrs β K ad. Rv. Bras. Bom. São Paulo v.30. p

7 Tabl - Estmats valus ad cofdc trvals of th paramtrs β K ad by ML approach ad Baysa stmator obtad va Gbbs Samplr for K = 0.5 ad =50 β K ML.6666 ( ) ( ).646 ( ) Baysa.837 ( ) ( ).6405 ( ) Wh th sampl sz crass (=50) all th stmats gt bttr ad mor accurat provdg qut smlar rsults as pctd. Not howvr that th ML stmat for K bcoms clos to th 0.5 but wth trval stll larg whl th Baysa stmato for K dos ot mprov. I summary wh sampl sz s small th ML stmats ca assum admssbl valus ad th Baysa stmats wll hav lkly valus but ot vry accurat. Howvr for modrat ( = 50) th most stmats wll hav smlar valus. I Tabls 3 ad 4 w ca aalyz th pot ad trvals stmats of β K ad for dffrt valus of K (K =0.5 ad K =0.8) for a sampl sz =0. W cosdr K = 0.8 rsultd from = = ad = 4. u Tabl 3 -Estmats valus of th paramtrs β K ad by ML approach ad Baysa stmator (va Gbbs Samplr) for =0 β K ML 0.85 (.354) Gbbs (.8087) Not: valus corrspodg to.3598 (.03) (0.584) K =0.5 (ar btw parthss)..646 (.73).768 (.8658) Tabl 4-95% cofdc trvals of th paramtrsβ K ad by ML approach ad Baysa stmator (va Gbbs Samplr) for =0 β K ML ( ) ( ) ( ) Gbbs ( ) ( ) ( ) 80 Rv. Bras. Bom. São Paulo v.30. p

8 As th sampl sz s small th ML ad Baysa stmats hav th smlar bhavor for K =05 howvr by comparg th stmatos for K =0.5 ad K =0.8 w aalyz how th paramtr K ca affct th stmatos. Not that for K =0.8 ( 4) = th stmato of β gts wors ad accurat wth both stmato approachs. For paramtr K th Baysa approach s stll bttr tha ML. Thrfor K s a mportat paramtr of trst th rgrsso aalyss wth masur rrors ad t should b cosdrd ths study maly wh thr ar fw obsrvd data st. 5. A ral llustrato wth ltratur data Cosdr th ampl of rgrsso modl wth masurmt rrors proposd by Fullr (987 pag 8) that volvs yld of cor (Y) for dffrt lvls of sol trog (). Hr th plaatory varabl sol trog lvl has b dtrmd wth masurmt rror. It s assumd thr s a pror stmat of th masurmt rror for sol trog s = 57. W cosdr th ovr dtfabl stuato that s u = u 0. = Th data st gv Tabl 5 rprsts th cor ad dtrmatos avalabl sol trog collctd at sts o Marshall Sol Iowa. Tabl 5 -Ylds of cor o Marshall sol Iowa St Yld (Y) Sol Ntrog () Aftr ru th Gbbs Samplr algorthm for N=30000 tratos w provd th Fgur wth th MCMC output plot ad th margal dsts rsultg for th paramtrs of th rgrsso modl. Comparso of th rgrsso stmats from ML ad Baysa approachs ar provdd at th Tabl 6 blow whch shows th pot stmats ad 95% cofdc trvals for th paramtrsβ K ad. Rv. Bras. Bom. São Paulo v.30. p

9 Fgur - Estmats Dsts va Gbbs Samplr ad th tracs of th chas gratd for th paramtrsβ K ad. Tabl 6 - Estmats valus ad cofdc trvals of th paramtrs β K ad by ML approach ad Baysa stmator obtad va Gbbs Samplr for = ML Baysa β K ( ) ( ) ( ) ( ) ( ) ( ) 8 Rv. Bras. Bom. São Paulo v.30. p

10 Now what would happ f w dd ot cosdr th stmato rror th varabl. I othr words what would b th stmator of β for fd a smpl lar rgrsso modl? W ca spcfy such a lar rgrsso modl asly by R softwar ad stmato rsults would b: Call: lm(formula = Y ~ - + ) Rsduals: M Mda 3Q Ma Estmat Std. Error t valu Pr(> t ) IC 95% ( ;.37844) Cocluso Bcaus th procdurs proposd by Rodrgus ad Corda (990) ar basd asymptotc rsults th classc stmator of β th rgrsso modl wth masurmt rrors dos ot produc satsfactory stmats wh th sz of th sampl s small. I ths cas th Baysa approach for stmato of th studd modl producs bttr rsults tha classcal stmators. Thrfor w vrfd that Baysa mthod usually rqurs lss sampl data to achv th bttr qualty of frcs tha th mthod basd o classc thory. I may cass ths s th practcal motvato for usg Baysa mthods ad rprsts th practcal advatag th us of pror formato. Ths s a spcally mportat cosdrato thos aras of applcato whr sampl data may b thr psv or dffcult to obta t. I addto th statstcal frcs basd o samplg thory ar usually mor rstrctv tha Baysa Ifrc du to th clusv us of sampl data. Th Baysa Ifrc s us of rlvat past prc whch s quatfd by th pror dstrbuto producs mor formatv frcs thos cass whr th pror dstrbuto accuratly rflcts th varato th paramtr. So t was possbl that w put th formato that K assums valus btw 0 ad ths work. W obsrv that K s a mportat paramtr th rgrsso aalyss wth masur rrors ad t should b cosdrd th study maly wh thr ar fw obsrvd data st. Th dgr to whch mor formatv frcs occur othrws dpds upo th qualty of th assssmts mbodd th pror dstrbuto. Thrfor othr prors could b trd to mprov th stmato. Th comparso of prors for th modl wth rrors varabls s a futur study of our trst. BABA M. Y.; MOALA F. A. Aáls Baysaa da rgrssão lar smpls com rros d mdda. Rv. Bras. Bom. São Paulo v.30. p Rv. Bras. Bom. São Paulo v.30. p

11 RESUMO:Gralmt a aáls clássca d frêca do modlo d rgrssão lar assum qu a varávl dpdt ão cotém rros d mdda.na prátca porém os dados podm cotr rros d mdção a prsça dsts rros pod aftar drastcamt os rsultados da aáls. Rodrgus Baba (994) propusram uma abordagm Baysaa para stmar o parâmtro d clação β o modlo d rgrssão lar com rros d mdda cosdrado a razão d cofabldad K como cohcda. Há stuaçõs o tato m qu a formação sobr a razão d cofabldad K m smpr é dspoívl. Nst artgoosso trss prcpal é ralzar uma frêca Baysaa do parâmtro β sob a suposção dqu a razão d cofabldad K é dscohcda. Para obtr a dstrbução a postror usamos o algortmo amostrador d Gbbs. PALAVRAS-CHAVE:Dstrbução a postror; razão d cofabldad; parâmtro d clação; amostrador d Gbbs. Rfrcs BABA M. Y. Ifrêca baysaa para rgrssão lar smpls com rro as varávs.994. Dssrtação (Mstrado) Isttuto d Cêcas Matmátcas d São Carlos Uvrsdad d São Paulo São Carlos 994. CASELLA G.; GEORGE E. I. Eplag th Gbbs samplr. Am. Stat. Baltmor v.46.3 p FULLER. W.A. Masurmt rror modls. Nw York: J. Wly p. GELFAND A.E.; SMITH A.F.M. Samplg basd approachs to calculatg margal dsts. J. Am. Stat. Assoc. Baltmor v.85.4 p RODRIGUES J.; BABA M.Y. Baysa stmato of a smpl rgrsso modl wth masurmt rror. Braz. J. Prob. Stat. São Paulo v.8. p RODRIGUES J.; CORDANI L.K.A ot o lklhood stmato of a smpl rgrsso modl wth masurmt rror va th orthogoal paramtrzato. S. Afr. Stat. J. Prtora.4 p Rcvd Approvd aftr rvsd Rv. Bras. Bom. São Paulo v.30. p