PubH 7405: REGRESSION ANALYSIS DIAGNOSTICS IN MULTIPLE REGRESSION

Transcription

1 PubH 7405: REGRESSION ANALYSIS DIAGNOSTICS IN MULTIPLE REGRESSION

2 The daa are n he form : {( y ; x, x,, x k )},, n Mulple Regresson Model : Y β 0 β x β x β k x k ε ε N (0, σ ) The error erms are dencally and ndependenly dsrbued as normal wh a consan varance.

3 A smple sraegy for he buldng of a regresson model consss of some, mos, or all of he followng fve seps or phases: () Daa collecon and preparaon, () Prelmnary model nvesgaon, (3) Reducon of he predcor varables, (4) Model refnemen and selecon, and (5) Model valdaon Le say we fnshed sep #3; we enavely have a good model: s me for some fne unng!

4 Dagnoscs o denfy possble volaons of model s assumpons are ofen focused on hese major ssues : () Non-lneary, () Oulyng & nfluenal cases, (3) Mul-collneary, (4) Non-consan varance & (5) Non-ndependen errors.

5 DETECTION OF NON-LINEARITY

6 A lmaon of he usual resdual plos (say, resduals agans values of a predcor varable): hey may no show he naure of he addonal conrbuon of a predcor varable o hose by oher varables already n he model. For example, we consder a mulple regresson model wh ndependen varables X and X ; s he relaonshp beween Y and X lnear? Added-varable plos (also called paral regresson plos or adjused varable plos ) are refned resdual plos ha provde graphc nformaon abou he margnal mporance of a predcor varable gven he oher varables already n he model.

7 ADDED-VARIABLE PLOTS In an added-varable plo, boh he response varable Y and he predcor varable under nvesgaon (say, X ) are boh regressed agans he oher predcor varables already n he regresson model and he resduals are obaned for each. These wo ses of resduals reflec he par of each (Y and X ) ha s no lnearly assocaed wh he oher predcor varables. The plo of one se of resduals agans he oher se would show he margnal conrbuon of he canddae predcor n reducng he resdual varably as well as he nformaon abou he naure of s margnal conrbuon.

8 A SIMPLE & SPECIFIC EXAMPLE Consder he case n whch we already have a regresson model of Y on predcor varable X and s now consderng f we should add X no he model (f we do, we would have a mulple regresson model of Y on (X,X )). In order o decde, we nvesgae smple lnear regresson models: (a) The regresson of Y on X and (b) The regresson of X on X and oban ses of resduals as follows:

9 Regresson# : Y on X ^ Y e b 0 ( Y X (frs or old b ) X Y ^ Y model) These resduals represen he par of Y no explaned by X

10 Regresson# : X ^ X e ( X on X b * 0 X (second ) b * X X or new model) ^ X These resduals represen he par of X no conaned n X

11 We now do a regresson of e (Y X ) as new dependen varable on e (X X ) as ndependen varable: Tha s o see f he par of X no conaned n X can furher explaned he par of Y no explaned by X ; (f can, X should be added o he model for Y whch already has X n ). There are hree possbles:

12 The horzonal band shows ha X conans no addonal nformaon useful for he predcon of Y beyond ha conaned n and provded for by X

13 Ths lnear band wh a non-zero slope ndcaes ha he par of X no conaned n X s lnearly relaed o he par of Y no explaned by X alone. Tha s, X should be added o form a -varable model. Noe ha f we do a regresson hrough he orgn, he slope b should be he same as he coeffcen of X f s added o he regresson model already conanng X.

14 Suppose : [ Y ^ () Y ( X ) X ^ () X ( X ) X and : (3) e( Y Then : X ) ( X AN EXAMPLE 6.9e( X )] 6.9[ X we have he same resul as n mulple regresson fng : Y [50.70 (6.9)(40.78)] 6.9X Y X 4.7X X ) ( X [5.54 (6.9)(.7)] X )],

15 The curvlnear band ndcaes ha, lke he case (b) prevously, X should be added o he model already conanng X. However, furher suggess ha he ncluson of X s jusfed bu some power erms or some ype of daa ransformaon are needed.

16 The fac ha an added-varable plo may sugges he naure of he funconal form n whch a predcor varable should be added o he regresson model s more mporan han ha he varable possble ncluson (whch can be resolved whou graphcal help). The added-varable plos play he role ha we use scaer dagrams for n smple lnear regresson; hey would ell f daa ransformaon or f ceran polynomal model s desrable.

17 ADDED-VARIABLE PLOT PROC REG daa Orgse; Smple Lnear Regresson here Model Y X X/noprn; Oupu Ou SaveMe R YResd XResd; Run; PROC PLOT daa SaveMe; Plo YResd*XResd; Run;

18 Anoher Example: Suppose we have a dependen varable Y and 5 predcor varables X-X5; and assume ha we focus on X5, our prmary predcor varable: () we regress Y on X-X4 and oban resduals (say, YR), and () we regress X5 on X-X4 and oban resduals (say, X5R), hen (3) we regress YR on X5R (SLR, no nercep) lnear assumpon abou X5 can be suded from hs las SLR & s (added-value) plo.

19 MARGINAL SL REGRESSION PROC REG daa Orgse; Model Y X5 X X X3 X4/noprn; Oupu Ou SaveMe R YR X5R; Run; PROC PLOT daa SaveMe; Plo YR*X5R; Run; PROC REG daa SaveMe; Model YR X5R/ non; Run; Mulple Lnear Regresson here New: Margnal Smple Lnear Regresson

20 REMEDIAL MEASURES If a lnear model s found no approprae for he added value of ceran predcor, here are wo choces: () Add n a power erms (quadrac), or () Use some ransformaon on he daa o creae a f for he ransformed daa

21 Each has advanages & dsadvanages: frs approach (quadrac model) may yeld beer nsghs bu may lead o more echncal dffcules; ransformaons (log, recprocal, ec ) are more smple bu may obscure he fundamenal real relaonshp beween Y and ha predcor. One s hard o do and one s hard o explan

22 OUTLYING & INFLUENTIAL CASES

23 SEMI-STUDENTIZED RESIDUALS _ e e e * MSE e MSE If MSE were an esmae of he sandard devaon of he resdual e, we would call e* a sudenzed (or sandardzed) resdual. However, he sandard devaon of he resdual s complcaed and vares for dfferen resduals, and MSE s only an approxmaon. Therefore, e* s call a sem-sudenzed resdual. Bu exac sandard devaons can be found!

24 THE HAT MATRIX ^ Y Xb X[(X' X) X' Y] [X(X' X) X']Y HY H X(X' X) X' s called he "Ha Marx" The Ha marx can be obaned from daa

25 RESIDUALS Model : Y Fed Value : ^ Y Resduals : e Xβ ε Xb ^ Y Y Y Xb Y HY (I H)Y Lke he Ha Marx H, (I-H) s symmerc & dempoen

26 VARIANCE OF RESIDUALS H) (I H) (I H) H)(I (I H) H)(I (I H)' I)(I H)( (I H)' (Y)(I H)σ (I (e) σ H)Y (I e MSE ^ ' σ σ σ σ

27 For he h observao n : σ s σ ( e s( e h h j (e ( e, e, e ) ) ) ( h ) ( h j j h h j j )σ ) MSE σ ; j MSE; j s he h elemen on he man s on h row and jh column of dagonal & he ha marx

28 STUDENTIZED RESIDUALS r e s( e ) ( e h ) MSE Sudenzed resduals s a refne verson of he semsudenzed resduals; n r we use he exac sandard devaon of he resdual e no an approxmaon. Sem-sudenzed resduals and Sudenzed Resduals are ools for deecng Oulers.

29 DELETED RESIDUALS The second refnemen o make resduals more effecve for deecng oulyng or exreme observaons s o measure he h resdual when he fed regresson s based on all cases excep he h case (smlar o he concep of jackknfng). The reason s o avod he nfluence of he h case especally f s an oulyng observaon - on he fed value. If he h case s an oulyng observaon, s excluson wll make he deleed resdual larger and, herefore, more lkely o confrm s oulyng saus: d Y ^ Y ( )

30 STUDENTIZED DELETED RESIDUALS Combnng he wo refnemens, he sudenzed resdual and he deleed resdual, we ge he Sudenzed Deleed Resdual. A sudenzed resdual s also called an nernal sudenzed resdual and a sudenzed deleed resdual s an exernal sudenzed resdual; MSE () s he mean square error when he h case s omed n fng he regresson model. d s( d ) ( h e ) MSE ( )

31 e ( h ) MSE( ) The sudenzed deleed resduals can be calculaed whou havng o f new regresson funcons each me a dfferen case s omed; ha s, we need o f he model only once wh all n cases o oban MSE no he same model n mes, here p s he number of parameers, p k: ( n p) MSE ( n p ) MSE ( ) e h

32 ( n leadng p) MSE o : n p e SSE( h) e whch s dsrbue d as "" wh degreesof ( n p ) MSE freedom under H ( ) 0 e h of (n p ) no oulers. Ths can be used as a sascal es for oulers wh allowance for mulple decsons

33 The dagonal elemens h of he ha marx, called leverages, have he same ype of nfluence on he sudenzed deleed resduals and, herefore, serve as good ndcaors n denfyng oulyng X observaons; he larger hs value he more lkely ha he case s an oulers. A leverage value s usually consdered o be large f s more han wce as large as he mean leverage value (whch s p/n).

34 "no good" s so large ; ) ( ), dempoen s ( ) symmerc, s ( ) ( ^ h h Y σ σ σ σ σ H HH H H H' H H HH H' I H (Y)H' Hσ (Y) σ HY Y ^ ^

35 IDENTIFICATION OF INFLUENTIAL CASES Wha we have done s o denfy cases ha are oulyng wh respec o her Y values (say, usng her sudenzed deleed resduals) and/or X values (usng her leverages). The nex sep s o asceran wheher or no hese oulyng cases are nfluenal; case s nfluenal f s excluson causes major changes n he fed regresson funcon. (If a case s deermned o be oulyng and nfluenal, he nex sep would be nvesgang usng oher sources o see f should be aken ou).

36 INFLUENCE ON A FITTED VALUE Recall he dea of deleed resduals n whch we measure he h resdual from he fed value where regresson fng s based on all cases excep he h case so as o avod he nfluence of he h case self. We can use he very same dea o measure he nfluence of a case on s own fed value; ha s o measure he dfference ( DF ) beween he fed values when all daa are used o he fed value where regresson fng s based on all cases excep he h case; MSE() s he mean square error when he h case s omed ( DFFITS ) ^ ^ Y Y MSE ( ) ( ) h

37 INFLUENCE ON ALL FITTED VALUES Takng he same dea of deleng a case o nvesgae s nfluence bu, n conras o DDFITS whch consder he nfluence of a case on s own fed value, Cook s Dsance shows he effec of h case on all fed values. The denomnaor serves only as a sandardzed measure so as o reference Cook s Dsance o he F(p,n-p) percenles. D n j ^ (Y j ^ Y pmse j() )

38 INFLUENCE ON REGRESSION COEFFICIENTS Anoher measure of nfluence, DFBETAS, s defned smlar o DDFITS bu DFBETAS focuses on he effec of (deleng) h case on he values of all regresson coeffcens; n hs formula, c jj s he jh dagonal elemen of marx (X X) -. As a gudelne for denfyng nfluenal cases, a case s consdered nfluenal f he absolue value of a DFBETAS exceeds for medum daa ses and / n for larger ses: ( DFBETAS) j( ) b j b MSE j( ) ( ) c jj

39 THE ISSUE OF MULTICOLINEARITY

40 When predcor varables are hghly correlaed among hemselves we have mulcollneary : he esmaed regresson coeffcens end o have large sandard errors. Oulyng and nfluenal effecs are case-based whereas mulcollneary effecs are varable-based.

41 CONSEQUENCES OF MULTICOLLINEARITY Addng or deleng a predcor varable changes some regresson coeffcens subsanally. Esmaed sandard devaons of some regresson coeffcens are very large. The esmaed regresson coeffcens may no be sgnfcan even her presence mprove predcon.

42 INFORMAL DIAGNOSTICS Indcaons of he presence of mulcollneary are gven by he followng nformal dagnoscs: Large coeffcens of correlaon beween pars of predcor varables n marx r XX. Non-sgnfcan resuls n ndvdual ess some esmaed regresson coeffcens may even have wrong algebrac sgn. Large changes n esmaed regresson coeffcens when a predcor varable s added or deleed.

43 For he purpose of measurng and formally deecng he mpac of mulcollneary, s easer o work wh he sandardzed regresson model whch s obaned by ransformng all varables (Y and all X s) by means of he correlaon ransformaon. The esmaed coeffcens are now denoed by b* s; and here s no nercep.

44 Correlaon ransformaon: s x x x n x _ * Wh ransformed varables Y* s and all X* s, he resul s called he Sandardzed Regresson Model: * * * * * * * * 0 & ε β β β ε β β β β k k k k x x x Y x x x Y Resuls: YX XX * r r X' Y X) (X' b XX ' * r (X X) ) (b σ * * σ σ

45 σ (b * ) σ σ * ' (X X) * I s obvous ha he varances of esmaed regresson coeffcens depend on he correlaon marx r xx beween predcor varables. The jh dagonal elemen of he marx r xx - be denoed by (VIF) j and s called he varance nflaon facor ): r XX

46 VARIANCE INFLATION FACTORS Le R j be he coeffcen of mulple deermnaon when predcor varable X j s regressed on he oher predcor varables, hen we have more smple formulas for he varance nflaon facor and he varance of he esmaed regresson coeffcen b* j. These formulas ndcae ha: () If R j 0 (ha s X j s no lnearly relaed a all o he oher predcor varables), (VIF) j, () If R j 0, hen (VIF) j > ndcang an nflaed varance for b* j

47 ( VIF ) j ( R j ) σ ( b * j ) σ * R j

48 COMMON REMEDIAL MEASURES () The presence of mulcollneary ofen does no affec he usefulness of he model n makng predcons provded ha he values of he predcor varables for nferences are nended follow he same mulcollneary paern as seen n he daa. One smple remedal measure s o resrc nferences o arge subpopulaons havng he same paern of mulcollneary. () In polynomal regresson models, use cenered values for predcors

49 COMMON REMEDIAL MEASURES (3) One or more predcor varables may be dropped from model n order o lessen he degree of mulcollneary. Ths pracce presens an mporan problem: no nformaon s obaned abou he dropped predcor varables. (4) To add more cases ha would break he paern of mulcollneary; hs opon s no ofen avalable.

50 COMMON REMEDIAL MEASURES (5) In some economc sudes, s possble o esmae he regresson coeffcens for dfferen predcor varables from dfferen ses of daa; however, n oher felds, hs opon s no ofen avalable. (6) To form a compose ndex or ndces o represen he hghly correlaed predcor varables. Mehods such as prncpal componens can help, bu mplemenaon s no easy (7) Fnally, Rdge Regresson has proven o be useful o remedy mulcollneary problems; Rdge regresson s one of he advanced mehods whch modfes he mehod of leas squares o allow based bu more precse esmaors of he regresson coeffcens.

51 NON-CONSTANT VARIANCE

52 Graph of Resduals agans Predcor varable or agans he fed values s helpful o see f he varance of error erms are consan; f model fs, shows a band cenered around zero wh a consan wdh. Lack of f resul n a graph showng he resduals deparng from zeros n a sysemac fashon a megaphone shape. No new fancy mehod/graph are needed here!

53 Example: Resdual Plo for Non-consan Varance

54 I s more me-consumng o deec nonlneary, bu s more smple o fx : A log ransformaon of an X or addon of s quadrac erm would normally solve he problem on non-lneary. I s more smple o deec a non-consan varance; added-value plo s no needed. However, would be more dffcul o fx because, n order o change he varance of Y we o make a ransformaon on Y.

55 Transformaons of Y maybe helpful n reducng or elmnang unequal varances of he error erms. However, a ransformaons of Y also changes he regresson relaon/funcon. In many crcumsances an approprae lnear regresson relaonshp has been found bu he varances of he error erms are unequal; a ransformaon would make ha lnear relaonshp non-lnear whch s a more severe volaon. An alernave o daa ransformaons: whch are more dffcul o fnd - usng mehod weghed leas squares nsead of regular leas squares.

56 Wh he Weghed Leas Squares (WLS), esmaors for regresson coeffcens are obaned by mnmzng he quany Q w where w s a wegh (assocaed wh he error erm); seng he paral dervaves equal o zero o oban he normal equaons : Q w k ( β β X ) 0 w Y

57 The opmal choce for he wegh s he nverse of varance. For example, when sandard devaon s proporonal o X 5 (or varance s kx 5 ), we mnmze: ) ( X Y X Q β β

58 Le consder, n more deph, he generalzed mulple regresson model: And frs look a he case where error varances are known and hen relax hs unrealsc assumpon. ) (0, 0 k k N x x x Y σ ε ε β β β β

59 When error varances are known, esmaors for regresson coeffcens are obaned by mnmzng he quany Q w where w s a wegh (assocaed wh he error erm, opmal choce s nverse of he known varance); seng he paral dervaves equal o zero o oban he normal equaons. The weghed leas squares esmaors of he regresson coeffcens are unbased, conssen, and have mnmum varance among unbased lnear esmaors Q w w β β 0 X β X ) ( Y k k

60 Le he marx W be a dagonal marx conanng (X' WX)b b σ w (X' (b w w ) σ he weghs w WX) X' WY (X' X' WY WX) 's, we have Wh Ordnary Leas squares, W I.

61 If he error varances were known, he use of WLS would be sragh forward: easy and smple. The resulng esmaors exhb less varably han he ordnary leas squares esmaors. Unforunaely, s an realsc assumpon ha we know he error varances. We are forced o use esmaes of he varances and perform Weghed Leas squares esmaon usng hese esmaed varances.

62 The process o esmae error varances s raher edous and can be summarzed as follows: () F he model by un-weghed (or ordnary) leas squares and obaned resduals, () Regress he squared resdual agans he predcor varables o oban a varance funcon (varance as a funcon of all predcors) (3) Use he fed values from he esmaed varance o oban he wegh (nverse of esmaed varance) (4) Perform WLS usng esmaed weghs.

63 A SIMPLE SAS PROGRAM If error varances are known, he WEIGHT saemen (no opon ) allow users o specfy he varable o use as he wegh n he weghed leas squares procedure: PROC REG; WEIGHT W; RUN; MODEL Y X X X3 X4; If error varances are unknown, would ake a few sep o esmae hem before you can use he WEIGHT saemen.

64 A SAMPLE OF SAS FOR WLS proc REG daa SURV; model SurTme LTes ETes PIndex Clong; oupu outemp RSurTLSR; run; daa TEMP; se Temp; sqrsurtlsr*surtlsr; run; proc reg daatemp; model sqrltes ETes PIndex Clong; oupu ou emp3 PEsqr; run; daa Temp4; se emp3; w/esqr; run; proc reg daaemp4; wegh w; model SurTme LTes ETes PIndex Clong; run; To form Varance Funcon

65 The condon of he error varance no beng consan over all cases s called heeroscedascy n conras o he condon of equal error varances, called homoscedascy. Heeroscedascy s nheren when he response n regresson analyss follows a dsrbuon n whch he varance s funconally relaed o he mean (so s relaed o a leas one predcor varable). The remedy for heeroscedascy s complcaed and, somemes, may no worh he effors. Transformaons of Y could ge you no roubles and Weghed Leas Squares s less ofen used because s hard o mplemen

66 THE ISSUE OF CORRELATED ERRORS

67 AUTOCORRELATION The basc mulple regresson models have assume ha he random error erms are ndependen normal random varables or, a leas, uncorrelaed random varables. In some felds for example n economcs, regresson applcaons may nvolve me seres ; he assumpon of uncorrelaed or ndependen error erms may no be approprae. In me seres daa, error erms are ofen (posvely) correlaed over me auo-correlaed or serally correlaed. Y β β x β x β x 0 k k ε

68 PROBLEMS OF AUTOCORRELATION Leas squares esmaes of regresson coeffcens are sll unbased bu no longer have mnmum varance MSE serously under-esmae varance of error erms Sandard errors of esmaed regresson coeffcens may serously under-esmae he rue sandard devaons of he esmaed regresson coeffcens; confden nervals of regresson coeffcens and of response means, herefore, may no have he correc coverage. and F ess may no longer applcable, have wrong sze.

69 FIRST-ORDER AUTOREGRESSIVE ERROR MODEL ),σ N( u where u x x x Y k k 0 0 's are ndependen : < ρ ρε ε ε β β β β

70 ),σ N( u where u x x x Y k k 0 0 's are ndependen : < ρ ρε ε ε β β β β Noe ha each error erm consss of a fracon of he prevous error erm plus a new dsurbance erm u ; he parameer ρ ofen posve - s called he auocorrelaon parameer.

71 Frs, we can easly expand from he defnon of he frs-order auoregressve error model o show ha each error erm s a lnear combnaon of curren and precedng dsurbance erms. Ths s used o prove ha he mean s zero and he varance s consan he varance s larger. 0 s s s 0 s s ρ σ ρ σ ) (ε σ 0 ) E(u ρ ) E(ε u ρε ε s s s u u u u u u u u u u u ) ( ) ( ρ ρ ρ ε ρ ρ ρε ρ ρ ε ρ ρε ρ

72 The error erms of he frs-order auoregressve model sll have mean zero and consan varance bu a posve covarance beween consecuve erms: ρ σ ρ },ε σ{ε ) ( 0 ) ( ρ σ ε σ ε E ),σ N( u where u x x x Y k k 0 0 's are ndependen : < ρ ρε ε ε β β β β

73 The auocorrelaon parameer s also he coeffcen of correlaon beween wo consecuve error erms: ρ ρ σ ρ σ ρ σ ρ ε σ ε σ ε ε σ ) ( ) ( ), (

74 s s s s ρ ρ σ ρ σ ρ σ ρ ε σ ε σ ε ε σ ) ( ) ( ), ( The coeffcen of correlaon below, of wo error erms ha are s perods apar, shows ha he error erms are also posvely correlaed bu he furher apar hey are he less he correlaon beween hem:

75 The ex presens a small smulaed daa se, pages , he regresson lne by leas squares mehod vares from case o case dependng on he value of he frs error erm. The esmaed slope ranges from a negave value o a posve value. I could be a serous problem unless we know how o deal wh auocorrelaon. And one could fnd example of me seres daa no only n economcs and busness bu also n bomedcal longudnal daa as well.

76 DURBIN-WATSON TEST The Durbn-Wason es for auocorrelaon assumes he frs-order auoregressve error model (wh values of predcor varables fxed) The es consss of deermnng wheher or no he auocorrelaon parameer (whch s also he coeffcen of correlaon beween consecuve error erms) s zero (f so, errors erms are equal o dsance erms whch are..d. normal): H 0 H A : : ρ ρ > 0 0

77 The Durbn-Wason es sasc D s obaned by frs usng ordnary leas squares mehod o f he regresson funcon, calculang resduals and hen D. Small values of D suppor he concluson of auocorrelaon because, under he frs-order auoregressve error model, adjacen error erms end o be of he same magnude (because hey are posvely correlaed) leadng o small dfferences and a small oal of hose dfference: D n ( e n e e )

78 Exac crcal values for he Durbn-Wason es sascs D are dffcul o oban, bu Durbn and Wason have provded lower and upper bounds d L and d U such ha an observed value of he es sasc D ousde hese bounds leads o a defnve decson. Values of he bound depend on he alpha level, number of predcor varables, and sample sze n (Table B7, page 675). If d If If L D > d D < d U L < D < d : Daa suppor H : Daa suppor H U : Tes s 0 A nconclus ve

79 SAS mplemenaon s very smple: Use opon DW PROC REG; MODEL Y X X X3/DW; An esmae of he auocorrelaon parameer s provded usng he followng formula: r n n e e e

80 When he presence of auocorrelaon s confrmed, he problem could be remeded by addng n anoher predcor varable or varables: one of he major cause s he omsson from he model of one or more key predcors:

81 BASIC SAS OPTIONS CORR: beween all varables n model saemen P: (or PRED) predced values R: (or RESID) resduals COVB: ( B for esmaed regresson coeffcen) CORRB: Varance-covarance marx (among B) CLB: confdence nerval for B CLM: confdence nerval for Mean Response CLI: confdence nerval for ndvdual predced.

82 SAS OPTIONS FOR SELECTION RSQUARE ADJRSQ MAXR AIC FORWARD BACKWARD STEPWISE

83 SAS OPTIONS FOR DIAGNOSTICS CORR: (ndvdual r >.7, say) STUDENT: sudenzed resduals PRESS: deleed resduals (square & add o oban he PRESS sasc) RSTUDENT: sudenzed deleed resduals ((-α/;n-p-) H: Leverages (p/n) COOKD: Cook s dsance (50% percenle of F(p,n-p) DFFITS: ( or p/n) DFBETAS: ( or / n)

84 Readngs & Exercses Readngs: A horough readng of he ex s secons (pp ) and skm over he example n secon 0.6 s recommended. Exercses: The followng exercses are good for pracce, all from chaper 0: 0.7, 0., 0.7, and 0..

85 Due As Homework #9. Refer o daase Cgarees, le Y log(nnal) and consder a model wh hree ndependen varables, X CPD, X Age, and X 3 Gender: a) F he mulple regresson model wh 3 predcor varables. Does appear ha all predcors should be reaned? b) Prepare an added-varable plo for X and X. Do your plos sugges ha he correspondng regresson relaonshps n (a) are approprae? Explan your answers. c) Oban he varance nflaon facors. Are here ndcaon ha serous mulcollneary problems exs here? Why or why no? d) Oban he resduals and prepare a normal probably plo; Also plo he resduals agans X and X, and draw conclusons. #9. Answer he 4 quesons of Exercse 9. usng daase Infans wh Y Brh Wegh, X Gesaonal Weeks, X Moher s Age, and X 3 Toxema (oxema 0/ s a pregnancy condon resulng from meabolc dsorder).