Chapter 4 Model Adequacy Checking

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1 Capter 4 Model Adequacy Ceckng Te fttng of lnear regresson model, estmaton of parameters testng of ypotess propertes of te estmator are based on followng maor assumptons: Te relatonsp between te study varable and explanatory varables s lnear, atleast approxmately Te error term as zero mean 3 Te error term as constant varance 4 Te errors are uncorrelated 5 Te errors are normally dstrbuted Te valdty of tese assumpton s needed for te results to be meanngful If tese assumptons are volated, te result can be ncorrect and may ave serous consequences If tese departures are small, te fnal result may not be canged sgnfcantly But f te departures are large, te model obtaned may become unstable n te sense tat a dfferent sample could lead to a entrely dfferent model wt opposte conclusons So suc underlyng assumptons ave to be verfed before attemptng to regresson modelng Suc nformaton s not avalable from te summary statstc suc as t-statstc, F-statstc or coeffcent of determnaton One mportant pont to keep n mnd s tat tese assumptons are for te populaton and we work only wt a sample So te man ssue s to take a decson about te populaton on te bass of a sample of data Several dagnostc metods to ceck te volaton of regresson assumpton are based on te study of model resduals wt te elp of varous types of grapcs Ceckng of lnear relatonsp between study and explanatory varables Case of one explanatory varable If tere s only one explanatory varable n te model, ten t s easy to ceck te exstence of lnear relatonsp between y and X by scatter dagram of te avalable data If te scatter dagram sows a lnear trend, t ndcates tat te relatonsp between y and X s lnear If te trend s not lnear, ten t ndcates tat te relatonsp between y and X s nonlnear For example, te followng fgure ndcates a lnear trend Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur

2 wereas te followng fgure ndcates a nonlnear trend: Case of more tan one explanatory varables To ceck te assumpton of lnearty between study varable and explanatory varables, te scatter plot matrx of te data can be used A scatterplot matrx s a two dmensonal array of two dmenson plots were eac form contans a scatter dagram except for te dagonal Tus, eac plot seds some lgt on te relatonsp between a par of varables It gves more nformaton tan te correlaton coeffcent between eac par of varables because t gves a sense of lnearty or nonlnearty of te relatonsp and some Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur

3 awareness of ow te ndvdual data ponts are arranged over te regon It s a scatter dagram of ( y versus X ), ( y versus X ),, ( y versus X k ) Anoter opton to present te scatterplot s - present te scatterplots n te upper trangular part of plot matrx - Menton te correspondng correlaton coeffcents n te lower trangular part of te matrx Suppose tere are only two explanatory varables and te model s y Xβ+ Xβ + ε, ten te scatterplot matrx looks lke as follows Suc arrangement elps n examnng of plot and correspondng correlaton coeffcent togeter Te parwse correlaton coeffcent sould always be nterpreted n conuncton wt te correspondng scatter plots because - te correlaton coeffcent measures only te lnear relatonsp and - te correlaton coeffcent s non-robust, e, ts value can be substantally nfluenced by one or two observatons n te data Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 3

4 Te presence of lnear patterns s reassurng but absence of suc patterns does not mply tat lnear model s ncorrect Most of te statstcal software provde te opton for creatng te scatterplot matrx Te vew of all te plots provdes an ndcaton tat a multple lnear regresson model may provde a reasonable ft to te data It s to be kept s mnd tat we get only te nformaton on pars of varables troug te scatterplot of ( y versus X ), ( y versus X ),, ( y versus X k ) wereas te assumpton of lnearty s between y and ontly wt ( X, X,, X k ) If some of te explanatory varables are temselves nterrelated, ten tese scatter dagrams can be msleadng Some oter metods of sortng out te relatonsps between several explanatory varables and a study varable are used Resdual analyss Te resdual s defned as te dfference between te observed and ftted value of study varable Te t resdual s defned as e y ~ yˆ y yˆ,,,, n were y s an observaton and y ˆ s te correspondng ftted value Resdual can be vewed as te devaton between te data and te ft So t s also a measure of te varablty n te response varable tat s not explaned by te regresson model Resduals can be tougt as te observed values of te model errors So t can be expected tat f tere s any departure from te assumptons on random errors, ten t sould be sown up by te resdual Analyss of resdual elps s fndng te model nadequaces Assumng tat te regresson coeffcents n te model y Xβ + ε are estmated by te OLSE, we fnd tat: Resduals ave zero mean as Ee ( ) ( ˆ Ey y) E( Xβ + ε Xb ) X β + 0 Xβ 0 Approxmate average varance of resduals s estmated by Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 4

5 n n ( e e) e SSres MS n k n k n k res Resduals are not ndependent as te n resduals ave only n k degrees of freedom Te nonndependence of te resduals as lttle effect on ter use for model adequacy ceckng as long as n s not small relatve to k Metods for scalng resduals Sometmes t s easer to work wt scaled resduals We dscuss four metods for scalng te resduals Standardzed resduals: Te resduals are standardzed based on te concept of resdual mnus ts mean and dvded by ts standard devaton Snce Ee ( ) 0 and MS res estmates te approxmate average varance, so logcally te scalng of resdual s e d,,,, n MS res s called as standardzed resdual for wc Ed ( ) 0 Var( d ) So a large value of d ( > 3, say) potentally ndcates an outler Studentzed resduals Te standardzed resduals use te approxmate varance of exact varance of e e as MS res Te studentzed resduals use te We frst fnd te varance of e In te model y Xβ + ε, te OLSE of β s b ( X ' X) X ' y and te resdual vector s Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 5

6 e y yˆ y Xb y Hy I H y H X X X X ( I H)( Xβ + ε) X β HX β + ( I H ) ε Xβ Xβ + ( I H) ε ( I H) ε Hε ( ) were ( ' ) ' Tus e Hy Hε, so resduals are te same lnear transformaton of y and ε Te covarance matrx of resduals s Ve () VH ( ε ) HV ( ε ) H σ H σ ( I H) and V ( ε) σ I Te matrx ( I H) s symmetrc and dempotent but generally not dagonal So resduals ave dfferent varances and tey are correlated If s te t dagonal element of at matrx H and s te (, ) t element of H, ten Var e σ Cov e e ( ) ( ) (, ) σ Snce 0, so f MS res s used to estmate te Var( e ) ten Var e ˆ σ ( ) ( ) MSr es ( ) MS overestmates te Var( e ) res Now we dscuss tat s a measure of locaton of te t pont n x-space Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 6

7 Regresson varable ull (RVH): It s te smallest convex set contanng all te orgnal data x ( x, x,, xk ),,,, n Te depend on te Eucldan dstance of x from te centrod and on te densty of te ponts n RVH In general, f a pont as largest value of, say, max ten t wll le on te boundary of te RVH n a regon of te x -space In suc regon, were te densty of te observatons s relatvely low Te set of ponts x (not necessarly te data ponts used to ft te model) tat satsfy x'( X ' X) x max s an ellpsod enclosng all ponts nsde te RVH So te locaton of a pont, say, x 0 ( x 0, x 0,, x 0 k ), relatve to RVH s reected by x ( X ' X) x ' Ponts for wc 00 > max are outsde te ellpsod contanng RVH If 00 < max ten te pont s nsde te RVH Generally, a smaller te value of 00 ndcates tat te pont x 0 les closer to te centrod of te x - space Snce s a measure of locaton of te t pont n x -space, te varance of e depends on were te pont x les If s small, ten Var( e ) s larger wc ndcates a poorer ft So te ponts near te center of te x -space ave poorer least squares ft tan te resduals at more remote locatons Volaton of model assumptons are more lkely at remote ponts and tese volatons may be ard to detect from te nspecton of ordnary resduals e (or te standardzed resduals d ) because ter resduals wll usually be smaller So a logcal procedure s to examne te studentzed resduals of te form r n place of e MS ( ) r e s e (or d ) For r, Er ( ) 0 Var( r ) regardless of te locaton of x wen te form of te model s correct Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 7

8 In many stuatons, te varance of resduals stablzes (partcularly n large data sets) and tere may be lttle dfference between d and r In suc cases d and r often convey equvalent nformaton However, snce any pont wt a - large resdual and - large s potentally gly nfluental on te least-squares ft, so examnaton of r s generally recommended If tere s only one explanatory varable ten e r,,,, n ( x x) MSre s + n sxx Wen x s close to te mdpont of x -data, e, x x s small ten estmated standard devaton of e s large Conversely, wen x s near te extreme ends of te range of x -data, ten x x s large and estmated standard devaton of e s small Wen n s really large, te effect of ( x ) x s relatvely small So n bg data sets, r may not dffer dramatcally from d PRESS resduals: ˆ Te PRESS resduals are defned as ( y ˆ y() ) were y () s te ftted value of te t response based of all te observaton except te t one Reason: If y s really unusual, ten te regresson model based on all te observatons may be overly nfluenced by ts observatons Ts could produce a y ˆ tat s very smlar to y and consequently e wll be small So t wll be dffcult to detect any outler If ˆ y s deleted, ten y () cannot be nfluenced by tat observaton, so te resultng resdual sould be lkely to ndcate te presence of te outler Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 8

9 Procedure Delete te t observaton, Ft te regresson model to remanng ( n ) observatons, Calculate te predcted value of y correspondng to te deleted observaton Te correspondng predcton error e y y () () Calculate e () for eac,,, n Tese predcton errors are called PRESS resduals because tey are used n computng te predcton error sum of squares Tey are also called as deleted resduals Now we establs a relatonsp between e and e () Relaton between e and e () Let b () be te vector of regresson coeffcents estmated by wt oldng te t observatons Ten ' ( ) b X X X y ' () () () () () were X () s te X -matrx wtout te vector of t observaton and () y s te y -vector wtout te t observaton Ten e y yˆ () () y xbˆ () y x ( X X ) X y ' ' () () () () We use te followng result n furter analyss Result: If X ' X s a k k matrx and x be ts t row vector ten ( X' X xx ' ) denotes te X ' X matrx wt te t row wteld Ten ( X' X) xxx ' ( ' X) X ' X x' x ( X ' X) + xx ( ' X) x' [ ] Usng ts result, we can wrte ' ( X ' X) xx ( X ' X) X() X() ( X ' X) + ' Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 9

10 x X X x ' were ( ' ) Ten ' ( ) e y x X X X y ' () () () () () y x X X ( X ' X) + xx ( X ' X) X y ' ' ( ' ) () () x ( X ' X) xx ( X ' X) X y ' y x( X ' X) X() y() ' ' () () x( X' X) X y ' y x( X ' X) X() y() ' () () ' ' ( ) y ( ) x ( X ' X) X y x ( X ' X) X y ( ) y x ( X ' X) X y () () () () ' () () Usng X' y X y + xy (as x s kvector), we can wrte ' ' () () e () y x X X X y xy ' ( ) ( ' ) ( ' ) ( ) y x ( X ' X) X ' y+ x ( X ' X) x y ' ( ) y xb + y y xb e Lookng at te relatonsp between e and e (), t s clear tat calculatng te PRESS resduals does not requre fttng n dfferent regressons Te e () ' s are ust te ordnary resduals wegted accordng to te dagonal elements of H It s possble to calculate te PRESS resduals from te resduals of a sngle least squares ft to all n observatons Resduals assocated wt ponts for wc s large wll ave large PRESS resduals Suc ponts wll generally be g nfluence ponts Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 0

11 Large dfference between ordnary resdual and PRESS resdual ndcate a pont were te model fts to te data well and a model wtout tat pont predcts poorly Now e Var( e() ) Var Var( e ) ( ) ( ) σ ( ) σ Te standardzed PRESS resdual s e e e Var e () ( () ) σ σ ( ) ( ) wc s same as te Studentzed resduals 4 R-student Te studentzed resdual r s often consdered as an outler dagnostc and n computng r Ts s referred to as nternal scalng of te resduals because MS res s used as an estmate of σ MS res s an nternally generated estmate of σ obtaned from te fttng te model to all n observaton Anoter approac s to use an estmate of σ based on a data set wt t observaton removed, say s () Frst we derve an expresson for s () Usng te dentty ' ( X ' X) xx ( X ' X) X() X() ( X ' X) + ' Post multply bot sdes by ' ( X' y xy ), we get Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur

12 b b ( X' X) xy + ' () ' ' ( X ' X) xx ( X ' X) ( X ' y xy) ' ' ' ' [ ] ' ' ( X ' X) xx ( ' ) ' b X X xy b b() ( X' X) xy b () Now consder ( )( X ' X) x y ( X ' X) xxb+ ( X ' X) x y ' ( X' X) x y xb ( X ' X) ( X' X) b xe xe ' Tus n () () ( n k ) s ( y xb ) ' n x( X' X) xe e y xb y xb + + n e e e + ( ) e e e + + ( ) ( ) n n n e e e n e ( ) ( ) + e e (usng Hy Hy, e 0, as H s dempotent) n n n ˆ ( n k) MSre s e e e s() ( n k) MSres n k Ts estmate of R-student gven by () σ s used nstead of e t,,,, n s ( ) MS res to produce an externally studentzed resdual, usually called Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur

13 In many stuatons, t wll dffer lttle wt r However, f t observaton s nfluental, ten s can dffer () sgnfcantly from MS res and te R student statstc wll be more senstve to ts pont Under usual regresson assumpton, t follows a t -dstrbuton wt ( n k ) degrees of freedom Resdual plots Te grapcal analyss of resduals s a very effectve way to nvestgate te adequacy of te ft of a regresson model and to ceck te underlyng assumptons Varous types of grapcs can be examned for dfferent assumptons and tese grapcs are generated by regresson software It s better to plot te orgnal resduals as well as scaled resduals Typcally, te studentzed resduals are plotted as tey ave constant varance Normal probablty plot Te assumpton of normalty of dsturbances s very muc needed for te valdty of te results for testng of ypotess, confdence ntervals and predcton ntervals Small departures from normalty may not affect te model greatly but gross nonnormalty s more serous Te normal probablty plots elp n verfyng te assumpton of normal dstrbuton If errors comng from a dstrbuton wt tcker and eaver tals tan normal, ten te least squares ft may be senstve to a small set of data Heavy taled error dstrbuton often generates outlers tat pull te least squares too muc n ter drecton In suc cases, oter estmaton tecnques lke robust regresson metods sould be consdered Te normal probablty plots s a plot of te ordered standardzed resduals versus te so called normal scores Te normal scores are te cumulatve probablty P,,,, n n If te resduals e, e,, e n are ordered and ranked n an ncreasng order as e[] < e[] < < e[ n], ten te e [] ' s are plotted aganst P and te plot s called normal probablty plot If te resduals are normally dstrbuted, ten te ordered resduals sould be approxmately te same as te ordered normal scores So te resultng ponts sould le approxmately on te stragt lne wt an ntercept zero and a slope of one (tese are te mean and standard dstrbutons of standardzed resduals) Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 3

14 Te ratonales bend plottng e [] aganst P s as follows: n Dvde te wole unt area under normal curve nto n equal areas We ave a sample of sze n data sets We mgt except tat one observatons les s eac secton, so marked out Frst secton as one pont, so cumulatve probablty s P /n Second secton as one pont, so Ten cumulatve probablty upto second secton s P (/n ) + (/n) /n and so on t ordered resdual observaton s plotted aganst te cumulatve area to te mddle of t secton wc s n Te factor ½ s used for end correcton as all te observatons scattered nsde te strpe are assumed to be concentrated at te md pont of te strpe Dfferent software use dfferent crteron For example, BMDP uses P 3 n + 3 wc produces detrended normal probablty plots from wc slope s removed Mntab uses 3 P 8 and converts to a normal score n + 4 Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 4

15 Suc dfferences are not mportant n real use Te stragt lne s usually determned vsually wt empass on te central values rater tan te extremes Substantal departure from a stragt lne ndcates tat te dstrbuton s not normal Sometmes te normal probablty plots are constructed by plottng te ranked resduals e [] aganst te expected normal value follows from te fact tat Φ n were Φ denotes te standard normal cumulatve dstrbuton Ts n [] Φ E e Varous nterpretatons to te grapc patterns s as follows (a) Ts fgure as an deal normal probablty plot Ponts le approxmately on te stragt lne and ndcate tat te underlyng dstrbuton s normal Cumulatve probablty 05 0 e [] Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 5

16 (b) Ts fgure as sarp upward and downward curves at bot extremes Ts ndcates tat te underlyng dstrbuton s eavy taled, e, te tals of underlyng dstrbuton are tcker tan te tals of normal dstrbuton Cumulatve probablty 05 0 e [] (c) Ts fgure as flattenng at te extremes for te curves Ts ndcates tat te underlyng dstrbuton s lgt taled, e, te tals of te underlyng dstrbuton are tnner tan te tals of normal dstrbuton Cumulatve probablty 05 0 e [] Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 6

17 (d) Ts fgure as sarp cange n te drecton of trend n upward drecton from te md Ts ndcates tat te underlyng dstrbuton s postvely skewed Cumulatve probablty 05 0 e [] (e) Ts fgure as sarp cange n te drecton of trend n downward drecton from te md Ts ndcates tat te underlyng dstrbuton s negatvely skewed Cumulatve probablty 05 0 e [] Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 7

18 Some experence and expertse s requred to nterpret te normal probablty plots because te samples taken from a normal dstrbuton wll not plot exactly as a stragt lne Small sample szes ( n 6) often produce normal probablty plots tat devate substantally from lnearty Larger sample szes ( n 3) produces plots wc are muc better beaved Usually about n 0 s requred to produce stable and easly nterpretable normal probablty plots If resduals are not from a random sample, normal probablty plots often exbt no unusual beavour even f te dsturbances ( ε ) are not normally dstrbuted Suc resduals are oftenly remnants of a parametrc estmaton process and are lnear combnatons of te model errors ( ε ) Tus fttng te parameters tends to destroy te evdence of nonnormalty n te resduals and consequently, we can not rely on te normal probablty plots to detect te departures from normalty Commonly seen defect found s normal probablty plots s te occurrence of one or two large resduals Sometmes, ts s an ndcaton tat te correspondng observatons are outlers Plots of resduals aganst te ftted value A plot of resduals ( e ) or any of te scaled resduals ( d, r or t ) versus te correspondng ftted values y ˆ s elpful n detectng several common type of model nadequaces Followng types of plots of y ˆ versus ave partcular nterpretatons: (a) If plot s suc tat te resduals can be contaned s a orzontal band fason (and resdual fluctuates s more or less n a random fason nsde te band) ten tere are no obvous model defects e Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 8

19 (b) It plot s suc tat te resduals can be contaned s an outward openng funnel ten suc pattern ndcates tat te varance of errors s not constant but t s an ncreasng functon of y (c) If plots s suc tat te resduals can be accommodated n an nward openng funnel, ten suc pattern ndcates tat te varance of errors s not constant but t s a decreasng functon of y Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 9

20 (d) If plot s suc tat te resduals can be accommodated nsde a double bow, ten suc pattern ndcates tat te varance of errors s not constant but y s a proporton between 0 and Te y ten may ave a Bnomal dstrbuton Te varance of a Bnomal proporton near 05 s greater as compared to near zero or So te assumed relatonsp between y and X ' s s nonlnear Usual approac to deal wt suc nequalty of varances s to apply a sutable transformaton to eter te explanatory varables or te study varable or use te metod of wegted least squares In practce, transformatons on study varable are generally employed to stablze te varance (e) If plot s suc tat te resduals are contaned nsde a curved plot, ten t ndcates nonlnearty Te assumed relatonsp between y and X ' s s non lnear Ts could also mean tat some oter explanatory varables are needed necessary Transformatons on explanatory varables and/or tese cases n te model For example, a squared error term may be study varable may also be elpful s Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 0

21 Note: A plot of resduals aganst y ˆ may also reveal one or more unusually large resduals Tese ponts are potental outlers Large resduals tat occur at te extreme y ˆ values could also ndcate tat eter te varance s not constant or te true relatonsp between y and X s nonlnear Tese possbltes sould be nvestgated before te ponts are consdered outlers Plots of resduals aganst explanatory varable Plottng of resduals aganst te correspondng values of eac explanatory varable can also be elpful We proceed as follows Consder te resdual on Y axs and values of t explanatory varable x ' s, (,,, n) on X axs Ts s te same way as we ave plotted te resduals aganst y ˆ In place of yˆ ' s, now we consder x ' s Interpretaton of te plots s same as n te case of plots of resduals versus y ˆ Ts s as follows If all te resduals are contaned n - a orzontal band and te resduals fluctuates more or less n a random fason wtn ts band, ten t s desrable and tere are no obvous model defects - an outward openng funnel sape or nward openng funnel sape, ten t ndcates tat te varance s nonconstant - a double bow pattern or nonlnear pattern ten t ndcates te assumed relatonsp between y and x s not correct Te possbltes lke y may be a proporton, ger ordered term s X (eg X ) are needed or a transformaton s needed are to be consdered n suc a case Note : In te case of smple lnear regresson, t s not necessary to plot resduals versus y ˆ and explanatory varable Te reason s tat te ftted values y ˆ are lnear combnatons of te values of explanatory varable X, so te plots would only dffer s te scale for te abscssa ( X axs) Note : It s also elpful to plot te resduals aganst explanatory varables tat are not currently s te model, but wc could potentally be ncluded Any structure n te plot of resduals versus an omtted varable ndcates tat ncorporaton of tat varable could mprove te model Note 3: Plottng resduals versus explanatory varable s not always te most effectve way to reveal weter a curvature effect (or a transformaton) s requred for tat varable n te model Partal regresson plots are more effectve n nvestgatng te relatonsp between te study varable and explanatory varables Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur

22 Plots of resduals n tme sequence If te tme sequence s wc te data were collected s known, ten te resduals can be plotted aganst te tme order We proceed as follows: Consder te resduals on Y -axs and tme order on X axs Ts s te same way as we ave plotted te resduals aganst y ˆ In place of y ˆ, ust use te tme order Interpretaton of te plots s same as n te case of plots of resduals versus y ˆ Ts s as follows If all te resduals are contaned n a - orzontal band and te resduals fluctuate more or less n a random fason wtn ts band, ten t s desrable and ndcates tat tere are no obvous model deflects - outward openng funnel sape or nward openng funnel sape, ten t ndcates tat te varance s not constant but cangng wt tme - Double bow pattern or nonlnear pattern, ten t ndcates tat te assumed relatonsp s nonlnear In suc a case, te lnear or quadratc terms n tme sould be added to te model Te tme sequence plot of resduals may ndcate tat te errors at one tme perod are correlated wt tose at oter tme perods Te correlaton between model errors at dfferent tme perods s called autocorrelaton If we ave a plot lke followng, ten t ndcates te presence of autocorrelaton Followng type of fgure ndcates te presence of postve autocorrelaton Resdual (e ) Tme Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur

23 Followng type of fgure ndcates te presence of negatve autocorrelaton Resdual (e ) Tme Te metods to detect te autocorrelaton and to deal wt te tme dependent data are avalable under tme seres analyss Some measures are dscussed furter n te module on autocorrelaton Partal regresson and partal resdual plots Partal regresson plot (also called as added varable plot or adusted varable plot) s a varaton of te plot of resduals versus te predctor It elps better to study te margnal relatonsp of an explanatory varable gven te oter varables tat are n te model A lmtaton of te plot of resduals versus an explanatory varable s tat t may not completely sow te correct or complete margnal effect of an explanatory varable gven te oter explanatory varables n te model Te partal regresson plot n elpful n evaluatng weter te relatonsp between study and explanatory varables s correctly specfed Tey provde te nformaton about te margnal usefulness of a varable tat s not currently n te model In partal regresson plot - Regress y on all te explanatory varable except te t explanatory varables resduals e y/ X ( ), say were X ( ) denotes te X -matrx wt X removed - Regress X on all oter explanatory varables and obtan te resduals e X / X ( ), say X and obtan te - Plot bot tese resduals aganst e X / X ( ) Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 3

24 Tese plots provde te nformaton about te nature of te margnal relatonsp for t explanatory varable X under consderaton If X enters nto te model lnearly, tem te partal regresson plot sould sow a lnear relatonsp, e, te partal resduals wll fall along a stragt lne wt a nonzero scope See ow: Consder te model y Xβ + ε X β + X β + ε ( ) ( ) ten resdual s e ( I H) were ' ' ' ( ' ) ' and ( ) ( ) ( ( ), ( ) ) ( ) s te H -matrx based on X ( ) H X X X X H X X X X Premultply y Xβ + ε by ( I H( ) ) and notng tat ( I H( )) X( ) 0, we ave ( I H ) y ( I H ) X β + β ( I H ) X + ( I H ) ε ( ) ( ) ( ) ( ) ( ) 0 + β ( I H ) X + ( I H ) ε ( ) ( ) ( ) ( ) * e y/ X β e X / X + ε were * ε ( I H ) ε ( ) Ts suggests tat a partal regresson plot wc s a plot between e y/ X ( ) and e X / X ( ) (lke between y and X ) sould ave slope β Tus f X enters te regresson n a lnear fason, te partal regresson plot sould sows lnear relatonsp passng troug orgn For example, lke Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 4

25 If te partal regresson plot sows a curvlnear band, ten ger order terms n be elpful X or a transformaton may e (y / X ) e (X / X ) If X s a canddate varable wc s consdered for ncluson n te model, ten a orzontal band on te regresson plot ndcates tat tere s no addtonal useful nformaton n X for predctng y Ts ndcates tat β s nearly zero e (y / X ) e (X / X ) Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 5

26 Example: Consder a model y β + β X + β X + ε 0 We want to know about te nature of margnal relatonsp for X and also want to know weter te relatonsp between y and X s correctly specfed or not? To obtan te partal regresson plot Regress y on X and obtan te ftted values and resduals yˆ ˆ ˆ ( X) θ0 + θx e( y/ X ) y yˆ ( X ),,,, n Regress X on X and fnd te resduals Xˆ ˆ ˆ ( X) α0 + αx e( X / X ) x Xˆ ( X ),,,, n Plot e ( y/ X ) aganst te X resduals e ( X/ X ) If X enters nto te model lnearly, ten te plot wll look lke as follows: Te slope of ts lne s te regresson coeffcent of X n te multple lnear regresson model e (y / X ) Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur e (X / X ) 6

27 If te partal regresson plot sows a curvlnear band, ten ger order terms n X or a transformaton X may be elpful If X s a canddate varable wc s consdered for ncluson n te model, ten a orzontal band on te regresson plot ndcates tat tere s no addtonal useful nformaton for predctng y e (y / X ) Some comments on partal regresson plots: e (X / X ) Partal regresson plots need to be used wt cauton as tey only suggest possble relatonsp between study and explanatory varables Te plots may not gve nformaton about te proper form of te relatonsp of several varables tat are already n te model are ncorrectly specfed Some alternatve forms of relatonsp between study and explanatory varables sould also be examned wt several transformatons Resdual plots for tese models sould also be examned to dentfy te best relatonsp or transformaton Partal regresson plots wll not, n general, detect nteracton effect among te regressors 3 Partal regresson plots are affected by te exstence of exact relatonsp among explanatory varables (Problem of multcollnearty) and te nformaton about te relatonsp between study and explanatory varables may be ncorrect In suc cases, t s better to construct a scatter plot of explanatory varables lke Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur X verus X If tey are gly correlated, multcollnearty s ntroduced and propertes of estmators lke ordnary least squares of regresson coeffcents are dsturbed 7

28 Partal resdual A resdual plot closely related to te partal regresson plot n te partal resdual plot It s desgned to sow te relatonsp between te study and explanatory varables Suppose te model as k explanatory varable and we are nterested n t explanatory varable X Ten X X X ( ( ), ) were ( ) y Xβ + ε X β + X β + ε ( ) ( ) X s te X matrx wt X removed Te model s were β ( ) s te vector of all β, β,, β k except β Te ftted model s y X ˆ + X ˆ + e ˆ ( ) β( ) β y X ˆ X ˆ + e or ˆ ( ) β( ) β were e s te resdual based on all k explanatory varables Ten partal resdual for X (,,, k) s gven by y X ˆ X ˆ + e ˆ ( ) β( ) β or or ey ( / X) e+ ˆ β X e ( y/ X ) e + x,,,, n * ˆ β Partal resduals plots A resdual plot closely related to te partal regresson plot n te partal resdual plot It s desgned to sow te relatonsp between te study and explanatory varables Suppose te model as k explanatory varables X, X,, X k Te partal resduals for X are defned as e ( y/ X ) e + x,,,, n * ˆ β were e are te resduals from te model contanng all te k explanatory varables and ˆ β s te estmate of t regresson coeffcent Wen e * ( y/ X ) are plotted aganst x, te resultng dsplay as slope ˆ β Te nterpretaton of partal resdual plot s very smlar to tat of te partal regresson plot Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 8

29 Statstcal tests on resduals We may apply certan statstcal tests to te resduals to obtan quanttatve measure of some of te model nadequaces Tey are not wdely used In many applcatons, resdual plots are more nformatve tan te correspondng tests However, some resdual plots do requre some skll and experence to nterpret In suc cases, te statstcal tests may prove useful Te PRESS statstc Te PRESS resduals are defned as e y yˆ,,,, n () () were y ˆ() s te predcted value of te t observed study varable based on a model ft to te remanng ( n ) ponts Te large resduals are useful n dentfyng tose observatons were te model does not ft well or te observatons for wc te model s lkely to provde poor predctons for future values Te predcton sum of squares s defned as te sum of squared PRESS resduals and s called as PRESS statstc as PRESS y ˆ y n () n e Te PRESS statstc s a measure of ow well a regresson model wll perform n predctng new data So ts s also a measure of model qualty A model wt small value of PRESS s desrable Ts can also be used for comparng regresson models R for predcton based on PRESS Te PRESS statstc can be used to compute an were R predcton PRESS SS T R -lke statstc for predcton, say SS T s te total sum of squares Ts statstc gves some ndcaton of te predctve capablty of te regresson model For example, f R n predctng new observatons 089, ten t ndcates tat te model s expected to explan about 89% of te varablty Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 9

30 Detecton and treatment of outlers An outler s an extreme observaton Resduals tat are consderably larger n absolute value tan te oters, say, 3 or 4 tmes of standard devaton from te mean ndcate potental outlers n y -space Ts dea s derved from te 3-sgma or 4-sgma lmts Dependng on ter locaton, outlers can ave moderate to severe effects on te regresson model Outlers may ndcate a model falure for tese ponts Resdual plots aganst y ˆ and normal probablty plots elp n dentfyng outlers Examnaton of scaled resduals, eg, studentzed and R-student resduals are more elpful as tey ave mean zero and varance one Outlers can also occurs n explanatory varables n X -space Tey can also affect te regresson results Sometmes outlers are bad values occurrng as a a result of unusual but explanable events For example, faulty measurements, ncorrect recordng of data, falure of measurng nstrument etc Bad values need to be dscarded but sould ave strong nonstatstcal evdence tat te outler s a bad value before t s dscarded Dscardng bad values s desrable because least squares pull te ftted equaton toward te outler Sometmes outler s an unusual but perfectly plausble observaton If suc observatons are deleted, ten t may gve a false mpresson of mprovement n ft of equaton Sometmes te outler s more mportant tan te rest of te data because t may control many key model propertes Te effect of outlers on te regresson model may be cecked by droppng tese ponts and refttng te regresson equaton Te value of t -statstc, F -statstc, R and resdual mean square may be senstve to outlers Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 30

31 An outler test based on R-student A common way to model an outler s te mean sft outler model Suppose we ft a model y Xβ + ε wen te true model s y Xβ + δ + ε were δ s a n vector of zeros except for te δ (0,0,,0, δ u,0,,0) t u observaton wc as a value u δ Tus Assume ε ~ N(0, σ I) for bot te models we ft Our obectve s to fnd an approprate statstc for testng H : δ 0 verus H : δ 0 Ts procedure assumes tat we are specfcally nterested s 0 u 0 e, tat we ave a pror nformaton tat te u t u observaton may be an outler t u observaton, Frst we fnd an approprate estmate of δ u Consder t u resdual as ts estmate Te n resdual vector s Ten [ ] e I H y I X( X ' X) X ' y E() e Hy HE( y) H( Xβ + δ) HX β + Hδ [ I H] 0 + δ I X X X X δ ( ' ) Tus Ee ( ) ( ) δ u uu u ˆ eu δu uu s an unbased estmator of δ u were uu s te It may be observed tat ˆu δ s smply te t u dagonal element of H t u PRESS resdual Furter, te covarance matrx of e s Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 3

32 So σ I [ ] Ve () V( I Hy ) Var e ( I H) V( y)( I H) ( H) ( u ) ( uu ) σ ( ˆ e u Var δu ) Var uu ( uu ) σ ( ) σ uu uu Also e s a lnear combnaton of normally dstrbuted y So e s also normally dstrbuted Tus ˆu δ s also normally dstrbuted Consequently, under H : 0 0 δ u, e u uu e u σ σ uu uu ~ N(0,) Te quantty e u σ uu s smply an example of studentzed resdual Snce σ s unknown and MS r e s σ s a C-square random varable, so a canddate test statstc s e u MS ( ) e r s wc follows a t-dstrbuton f e [ I H] y [ ] σ [ ] I H I I H σ ( I H) 0, uu and SS e y '( I H ) y are ndependent Snce r s so e and SS res are not actually ndependent We already ave developed S () wc s related to resdual mean square n a regresson model wt t observaton wteld gven by e ( n k) MSre s S() n k Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 3

33 Ts estmate of σ s ndependent of e u by te basc ndependence assumpton on random errors So σ can be replaced by s and an approprate test statstc for te mean sft outler model s ( u) s e u ( u) uu wc s te externally studentzed resdual or R -student eu Under H0 : δu 0, ~ tn ( k ) s ( u) eu and under H0 : δu 0, ~ noncentral t ( n k, s wt noncentralty parameter ( u) uu uu [ γ] δ δ u γ σ /( ) uu σ uu Note tat te power of ts test depends on uu If we ft an ntercept to our model, ten uu n So maxmum power occurs wen uu, e, at te center of te data cloud s terms of te X ' s As uu, n te power goes to 0 In oter words, ts test as less ablty to detect outlers at te g leverage data ponts (Note tat te concept of leverage pont s dscussed n later sectons) Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 33

34 Test for lack of ft of a regresson model Ts test for lack of ft of a regresson model s based on te assumptons of normalty, ndependence and constant varance wc are satsfed Only te frst order or stragt lne caracter of te relatonsp s n doubt For example, te data n te followng scatter plot were te ndcaton s tere tat stragt lne ft s not very satsfactory Te test procedure determnes f tere s systematc curvature s present Te test requres replcate observatons on y for at least one level of x and tey sould be true replcatons and not ust te duplcate readngs or measurement of y Te true replcatons conssts of runnng n separate experments at x x and observe y It s not ust runnng a sngle experment at x x and measurng y n tmes n wc te nformaton only on te varablty of te metod of measurng y s obtaned Tese replcated observatons are used to obtan a model-ndependent estmate of σ Suppose we ave n observatons on y at te t level of x,,,, m Let y be te t observaton on y at x, m,,, m,,,, n; n n s te total number of observatons Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 34

35 Consder te model y β + β x + ε 0 Let y be te mean of ( y yˆ ) ( y y ) + ( y yˆ ) n observatons on x Ten te (, ) t resdual s m n m n m ( y ˆ ) ( ) ( ˆ y y y + n y y ) (obtaned by squarng and summng over and ) SSr es SSPE + SS LOF Resdual Sum of Sum of sum of squares due squares due to squares to pure error lack of ft Measures Measures pure error lack of ft If assumpton of constant varance s satsfed, ten SS PE s a model ndependent measure of pure error because only te varablty of ys ' at eac x level s used to compute SS PE Snce tere are ( n ) degrees of freedom for pure error at eac level of x, te number of degrees of freedom assocated wt m SSPE s ( n ) n m LOF level of x and correspondng ftted value SS s a wegted sum of squared devatons between y at eac If y ˆ are close to y, ten tere s a strong ndcaton tat te regresson functon s lnear If y ˆ devate greatly from freedom assocated wt y ten t s lkely tat te regresson functon s not lnear Te degrees of SS LOF s m because tere are m levels of x and two degrees of freedom are lost because two parameters must be estmated to obtan y Computatonally, Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 35

36 SS SS SS LOF r es PE Te test statstc for lack of ft s SSLOF /( m ) F0 SS /( n m) MS MS PE LOF PE n [ ( ) β β x ] n E y 0 ( LOF ) σ + E MS ( m ) If true regresson s lnear, ten E( y ) β0 + βx and E MS ( LOF ) σ If true regresson s nonlnear, ten E( y ) β0 + βx and If true regresson functon s lnear, ten F ~ Fm (, n m ) 0 E( MSLOF ) > σ So to test for lack of ft, compute F 0 and conclude tat regresson functon s not lnear f F0 > Fα ( m, n m) at α level of sgnfcance If we conclude tat regresson functon s not lnear ten te tentatve model must be abandoned and we attempt to fnd a more approprate model If F0 < F ( m, n m) ten tere s no strong evdence of lack of ft Tey MS PE and MS LOF are often α combned to estmate σ If F rato for lack of ft s not sgnfcant and H0 : β 0 s reected, ten ts does not guarantee tat model wll be satsfactory for predcton It s suggested tat te F -rato must be at least four or fve tmes te Fα ( m, n m) f te regresson model s to be useful for predcton A smple measure of potental predcton performance s found by comparng te range of ftted values, e, ( yˆ yˆ ) max to ter average standard error Regardless of te term of te model, te average varance of te mn ftted values s n kσ Var( yˆ) Var( yˆ ) n n Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 36

37 were k s te number of parameters s te model In general, te model s not lkely to be satsfactory predctor unless te range of y ˆ s large relatve to estmated standard error k ˆ n σ were ˆ σ s a model-ndependent estmate of error varance Estmaton of pure error from near-negbours: In test of lack of ft SS SS + SS r es PE LOF SS PE s computed usng responses at repeat observatons at some level of x Ts s model ndependent estmate of σ Ts general prncple can be appled to any regresson model Calculaton of SS PE requres repeat observatons on te response y at te same set of levels on te explanatory varables x, x, x k,, e, some of te rows of X -matrx must be same In practce, repeat observatons do not often occur n multple regresson and te procedure of lack of ft s not often useful A metod to obtan a model ndependent estmate of error wen tere are no exact repeat ponts are te procedures wc searc for tose ponts s x -space tat are near-negbours Ts s te sets of observatons tat ave been taken wt near dentcal levels of x, x,, x k Te response y from suc near-negbours can be consdered as repeat ponts and used to obtan an estmate of pure error As a measure of te dstance between any two ponts, x, x,, x k and x', x',, x k ', use wegted sum of squared dstance (WSSD) D k ˆ β ( x x ' ) ' MSre s Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 37

38 Te pars of ponts wt small values of D ' are near negbours, e, tey are relatvely close togeter n x -space Pars of ponts for wc D ' s large (eg, D ' >> ) are wdely separated s x -space Te resduals at two ponts wt a small value of D ' can be used to obtan an estmate of pure error Te estmate s obtaned from te range of resduals at te ponts and ', say E e e' Tere s a relatonsp between te range of a sample from a normal populaton and te populaton standard devaton For example, for sample sze, ts relatonsp s E σ 0886E 8 Te quantty σ so obtaned s an estmate of standard devaton of pure error An effcent algortm may be used to compute ts estmate lke as follows: - Frst arrange te data ponts x,, x k n order of ncreasng y ˆ Ten - Note tat ponts wt dfferent values of y ˆ cannot be near negbour but tose wt smlar values of y ˆ could be negbours (or tey could be near te same contour of constant ŷ but for apart n some x -coordnates) Compute te values of D ' for all ( n ) pars of ponts wt adacent values of ŷ Repeat ts calculaton for te pars of ponts separated by one, two and tree ntermedate ŷ values Ts wll produce (4n 0) values of D ' Arrange te (4n 0) values of D found s step Let E, u,,,(4n 0) be te range of te ' resduals at tese ponts 3 For te frst m values of E u, calculate an estmate of te standard devaton of pure error as m 0886 ˆ σ Eu m u u Note tat ˆ σ s based on te average range of te resduals assocated wt te m smallest values of D, ' m must be cosen after nspectng te values of D ' One sould not nclude values of for wc te wegted sum of squared dstance s too large E u s te calculaton Regresson Analyss Capter 4 Model Adequacy Ceckng Salab, IIT Kanpur 38

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