Several Types of Residuals in Cox Regression. Model: An Empirical Study

Transcription

1 Int. Journal of Math. Analyss, Vol. 7, 213, no. 53, HIKARI Ltd, Several Types of Resduals n Cox Regresson Model: An Emprcal Study Anwar Ftranto* Department of Mathematcs, Faculty of Scence/ Unverst Putra Malaysa, Malaysa *anwarstat@gmal.com Laboratory of Appled & Computatonal Statstcs, Insttute for Mathematcal Research, Unverst Putra Malaysa Department of Statstcs, Faculty of Mathematcs and Natural Scences Bogor Agrcultural Unversty, Indonesa Rebecca Loo Tng Jn Department of Mathematcs, Faculty of Scence Unverst Putra Malaysa, Malaysa Copyrght 213 Anwar Ftranto and Rebecca Loo Tng Jn. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract There are several methods for calculatng resdual n survval analyss, especally n Cox regresson model by whch each method has specfc use, such as goodness-of-ft, to dentfy possble outlers and nfluental observatons, or n general to check necessary assumptons. In ths artcle, we study four methods of resduals, namely Schoenfeld, Martngale, devance, and score resduals and we appled those methods on cardovascular data. Keywords: Cox regresson, goodness-of-ft, outlers, nfluental observatons, resduals

2 2646 Anwar Ftranto and Rebecca Loo Tng Jn 1 Introducton Adequacy of a ftted model needs to be assessed after a model has been assessed. Dagnostc procedures for model checkng are known as essental parts of a modelng process. Many for ths procedure are based on resduals. In survval analyss, especally when we buld a Cox s proportonal hazard model of Cox [2], few types of resduals can be consdered for dfferent purposes [6]. Several useful dagnostc tools whch are based on resduals are (1) Schoenfeld resdual for checkng the proportonal hazards assumpton for a covarate, (2) Martngale resdual used to examne overall test of the goodness-of-ft of a Cox model, (3) Devance resdual for detecton of poorly predcted observatons, and (4) Score resdual for determnaton of nfluental observatons. Schoenfeld resdual was purposed by Schoenfeld [5] as partal resdual that s essental to nterpretaton of volaton of the proportonal hazards assumptons. Schoenfeld [5] mentoned that th resdual can be plotted aganst t to test the assumpton n whch resduals do not depend on tme. He defned a partal resdual as the dfferent between the observed value of X and ts condtonal expectaton gven the rsk set R. It can be wrtten as the followng equaton: rˆ k k ( X R ) = X Eˆ (1) k n the vector rˆ = ( rˆ, K, r ) '. He then expanded ( ) obtan E ˆ 1 p 2 ( rˆ ) g( t ) E( X R ) E( X R ) k E about ( ) = X k R g and 2 { k k }. (2) A plot of rˆ k versus t wll be centered about f proportonal hazards holds E( rˆ ). Kumar and Klefsjö [4] revewed the graphcal method purposed by Schoenfeld [5] for testng the assumptons of proportonal hazards model. They found that the plots of the estmated partal resduals aganst the tme should be randomly scattered around zero, snce the condtonal expectaton of these resduals, gven the rsk sets, wll be approxmately zero and asymptotcally uncorrelated. A large value of the estmate of the partal resdual wll ndcate that the correspondng tme to falure s more unlkely to be explaned by the proportonal hazards model. Grambsch and Therneau [3] suggested scale the Schoenfeld resdual by an estmator of ts varance whch yelds a resdual wth greater dagnostc power than the unscaled resduals. The vector of scaled Schoenfeld resdual s the product of the nverse of the covarance matrx tmes the vector of resduals. Let t

3 Several types of resduals n Cox regresson model 2647 rˆ [ Var ˆ ( rˆ )] rˆ = (3) * 1 be the scaled Schoenfeld resdual. Then ( rˆ* ) g( ), E (4) t where the rˆ s the partal resdual at Equaton (1) that was purposed by Schoenfeld [5]. Few years later, Barlow and Prentce [1] proposed another type of resdual, whch then was named as Martngale-based resdual or Martngale resdual. It s defned to as the followng expresson: Mˆ t ˆ ' β X ( s ) () t = N () t Y ( s) e dhˆ (), s (5) Mˆ as a shorthand for ( ) wth Mˆ. The H s the cumulatve baselne hazard, and the N () t s the countng process for the th subject n the set of ndcates the number of observed events experenced over the tme t. The resdual at the Equaton (5) reduces to more smple form as follows: Mˆ ˆ ' β X ( τ ) e. = δ Hˆ (6) whereτ denotes the observaton tme for subject and δ known as the vtal status for the Cox model, can dentfed as uncensored ( δ = 1) or censorng ( δ = ). Martngale resduals compare the observed to expected. The frst part of Equaton (6), δ s consdered as observed number of events for th observaton wth 1 or whle second part of ths equaton. The second part s X Hˆ ˆ β ( τ ) e, whch s the expected number of events for th subject, accountng for censorng. So, the martngale resdual s lkely havng the excess number of events and sum of these resduals whch wll be equal to. Accordng to Therneau et al. [6] one defcency of the martngale resdual M, partcularly n the sngle event settng of Cox model, s that t s heavly skewed. Ths skewness dstorts the appearance of a standard resdual plot, makes them hard to use to dentfy outlers. Another dsadvantage s t has maxmum value of +1 and for ts mnmum value. Due to the drawback of the martngale resdual, Therneau et al. [6] proposed devance resdual of the Cox regresson. It s much more symmetrcally dstrbuted about zero and defned as

4 2648 Anwar Ftranto and Rebecca Loo Tng Jn d 1 ( Mˆ )[ 2{ Mˆ + log( M ˆ )}] 2 = sgn δ δ, (7) where Mˆ s the martngale resdual for the th ndvdual, and the functon of (). sgn s the sgn functon. Observatons whch correspond to relatvely large devance resduals are those that do not well ftted by the model, and then consdered as outlers. The devance resdual transform the symmetrcal of the martngale resduals so that the dstrbuton of the devance resdual s better approxmated by normal dstrbuton than martngale resduals when censorng s mnmal, let say < 25%. The last resdual type to be dscussed n ths paper s score resdual. Accordng to Therneau and Grambsch [7], t can be used n conjuncton wth the varance-covarance matrx of parameter estmates to approxmate the teratve deleton process n parameter estmaton. Moreover, score resdual s qute useful n dagnoss of each subject s leverage on parameter estmates and the score resduals wll sum to zero. Lj ( βˆ ) s the score resdual for the th subject on the jth covarate expressed as L j ( ˆ) = { X () t X ( ˆ, β t) } dmˆ (), t β (8) where the X s defned as j j ( ) ( t) ( ) ' () t exp b X ( t) X j X () t exp b X () t Y X j ( b, t) =, (9) ' and s a functon of tme: the mean over the rsk set at tme t. Ths equaton was evaluated at β = b. Large values of score resdual mply large nfluence of the th subject on the estmate of β j, the coeffcent of X j. The score resdual s not a sngle value of resdual for each observaton, but a set of values, one for each covarate n the ftted Cox regresson model. These resduals also have property of zero for expected value and uncorrelated wth one another. 2 Materal and Methods In ths artcle, we use emprcal data for cardovascular study whch s obtaned from Serdang Hosptal, Malaysa n whch those observatons are patents who experenced cardovascular dsease and stayed n Cardothoracc

5 Several types of resduals n Cox regresson model 2649 Hgh Dependency Ward (CHDW) of Cardology Department for a perod to receve necessary medcal treatment. The Natonal Cardovascular Database (NCVD) s a servce supported by the Mnstry of Health (MOH) of Malaysa to collect nformaton about cardovascular dsease, whch wll enable us to know the ncdence of ths dsease, to evaluate ts rsk factors and treatment n the country. We wll do model buldng usng Cox regresson for the cardovascular hosptalzaton data. Then, from the obtaned model, we do some resdual analyss by graphcal methods based on scaled Schoenfeld resdual, martngale resdual, devance resdual, and score resdual. Also, we checked the volaton of proportonal hazards assumpton by usng numercal method then compared t wth graphcal method. 3 Results and Dscusson We conducted analyss for the Cox regresson model pror to the resdual analyss. Once we obtan a fnal reduced model for the Cox model [2], we can do some exploraton about assumptons checkng for the model. The stepwse model selecton could dentfy that out of 7 covarates ncluded n the analyss, 14 covarates have sgnfcant contrbuton to the hosptalzaton duraton of the patents durng the treatments n the CHDW. Then, we can plot scaled Schoenfeld resduals versus tme. Out of 14 sgnfcant covarates n the Cox model, we choose arbtrarly varable heart rate to create a plot of scaled Schoenfeld resduals versus tme to show proportonal hazards assumpton checkng. From Fgure 1, we can see that scaled Schoenfeld resduals are uncorrelated wth each other and scattered around zero. But, n order to mnmze subjectvty of the graphcal method, we also provde formal test for checkng the ndependence assumpton. It s done by checkng the correlaton of the scaled Schoenfeld resdual of each sgnfcant covarates from Equaton (3) wth the tme(of hosptalzed), log(tme) and tme 2. Schoenfeld [5] clamed that the partal resduals must not depend on tme. Hence, f there s no any correlaton or assocaton between the covarate and tme, whch means faled to rejected the H (The null hypothess s there s no correlaton wthn the covarate and tme), then there are no assumpton volate n ths model. Therefore, we compare the result of proportonal hazards assumpton checkng by usng the graphcal and numercal methods.

6 265 Anwar Ftranto and Rebecca Loo Tng Jn Scaled Schoenfeld Resduals Plot for Heart Rate Standardzed Schoenfeld Resdual B Fgure 1. Scaled Schoenfeldd Resduals Plot for Varable Heart Rate Tme Fgure 2. Martngale Resduals Plot

7 Several types of resduals n Cox regresson model 2651 Table 1. Pearson Correlaton for Scaled Schoenfeld Resduals of Sgnfcant Covarates wth Tme Covarates Tme Varables Ldays Days2 NDay Hypertenson Myocardal Infarcton Hstory Chronc Angna (Onset > 2 weeks) Chronc Lung Dsease Renal Dsease Heart Rate Level of Good Cholesterol Level of Bad Cholesterol Fastng Blood Glucose ACE Inhbtor (Durng Admsson) Other Lpd Lowerng Agent Calcum Antagonst (Pre Admsson) Insuln (After Dscharge) Pulse Pressure Notes: Ldays = log(tme), Days2 = tme 2, and NDay = tme Accordng to the Table 1, all the p-values are > α =. 1 n whch we faled to reject the null hypotheses. It means there s no any correlaton between each covarates of Cox model and tme whch mples that proportonal hazards assumpton s fulflled. The shape of martngale resduals plot n Fgure 2 s skewed to reflect that there s only sngle event settng n the model. The martngale resduals plot shows an solaton pont (wth lnear predctor score 7.3 and martngale resdual -2.11) of patent number 31. Ths observaton looks lke an outler n the

8 2652 Anwar Ftranto and Rebecca Loo Tng Jn martngale resdual plot. But, ths observaton s no longer dstngushable n the t devance plot. It suggests that theree s no ndcaton of a lack of ft for the model of that partcular observaton. Fgure 3 shows devance resduals whch consst of nformaton about nfluental data and outlers. Ths plot s roughly symmetrcally dstrbuted around zero. When proporton of censorng s mnmal (whchh s less than 25%), the t dstrbuton of the resdual can bee approxmated by normal dstrbuton. Snce graphcally they are approxmatelyy normally dstrbuted, we can then move our attenton to possble outlers. Under common defnton, an observaton s consdered as an outlerr f the value of the correspondng resdual s outsde the t range of (-2.5, 2.5). Percentage of censorng of ths cardovascular hosptalzaton data was only 21.5% %. As we can see, there are noo any observatons wth devance resduals less than t -2.5 orr greater than 2.5, so that there s no any outler n ths Cox model. It also means that there s no any observatons have unusual large (large postve devance resdual) or small (large negatve devance resdual) of hosptalzaton duraton regardng ther cardovascula dsease than expected hosptalzaton days. Fgure 3. Devance Resduals Plot Meanwhle, a plot of score resduals aganst covarate s dsplayed n Fgure 4. We need to have randomly scattered ponts, centeredd around zero f the ftted model s adequate. We use ths plot n order to dentfy possble nfluental observatons n the data snce t hass effects on model-based nferences. In general, nfluental observatons are ponts that are solated fromm all other ponts n the t graph. There are no anomales shown n Fgure 4 of score resdualss for patents heart rate. However, the observatons wth the smallest score resdual (-29.61) for f heart rate s from patent 51 wth a low heart rate of 53 was w shown as an solaton pont. Ths pont could be determned as an observaton that wll affect the

9 Several types of resduals n Cox regresson model 2653 estmaton of coeffcent for the parameter of heart rate, known as β 6 of the Cox regresson n ths study. Fgure 4. Score Resduals Plot for Covarate Heart Rate Acknowledgements The authors are grateful to Unversty Grants Scheme by Unverst Putra Malaysa for awardng a research grant for supportng the research and to Department of Cardology of the Serdang Hosptal, Malaysa for provdng necessary nformaton ncludng the data. References [1] W. E. Barlow, and R. L. Prentce, Resduals for relatve rsk regresson, Bometrka, 75(1988), [2] D. R. Cox, Regresson models and lfe-tables, Journal of the Royal Statstcal Socety. Seres B (Methodologcal), 34(1972), [3] P. M. Grambsch and T. M. Therneau, Proportonal hazards tests and dagnostcs based on weghted resduals, Bometrka, 81(1994), [4] D. Kumar and B. Klefsjö, Proportonal hazards model: A revew, Relablty Engneerng and System Safety, 44(1994), [5] D. Schoenfeld, Partal resduals for the proportonal hazards regresson model, Bometrka, 69(1982),

10 2654 Anwar Ftranto and Rebecca Loo Tng Jn [6] T. M. Therneau, P. M. Grambsch, and T. R. Flemng, Martngale-based resduals for survval models, Bometrka. 77(199), [7] T. M. Therneau and P. M.Grambsch, Modellng Survval Data: Extendng the Cox Model, Sprnger, New York, 2. Receved: August 6, 213