Influence Diagnostics on Competing Risks Using Cox s Model with Censored Data. Jalan Gombak, 53100, Kuala Lumpur, Malaysia.

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1 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) Influence Dagnostcs on Competng Rsks Usng Cox s Model wth Censored Data F. A. M. Elfak 1, Bn Daud. I, N. A. Ibrahm, M. Y. Abdullah and M. Usman 1 1 Department of Scence n Engneerng, Kullyyah of Engneerng, IIUM Jalan Gombak, 531, Kuala Lumpur, Malaysa. Department of Mathematcs, Faculty of Scence and Envronmental Studes Unversty Putra Malaysa, 434, Serdang, Selangor. Abstract hs paper studes nfluence dagnostcs (Cook s Dstance and Lkelhood Dstance) on competng rsk models when true covarates are observable. he one-step EM and onestep ML methods are used to detect nfluental observatons for Cox s model. It was found that results generated from data analyss usng one-step EM algorthm are better than results obtaned by usng one-step ML method. Moreover, Cook s dstance show better results compare to lkelhood dstance. Key word: Censored data; nfluence measurement; one-step methods; Cox s model; Competng rsks. 1. Introducton Dagnostcs are used to assess the adequacy of assumptons underlyng the modelng process and to dentfy unexpected characterstcs of the data that may serously nfluence conclusons or requre specal attenton. A varety of graphcal and nongraphcal methods are avalable to ad one n lnear regresson analyses (Cook and Wesberg 198) but most of these methods requre the a pror specfcaton of a model. Outlers and nfluental observatons, for example, are always judged relatve to some model, ether mplct or explct. he detecton of nfluental observatons, that s observatons whose deleton, ether sngly or multply, result n substantal changes n parameter estmates, ftted values or tests of hypothess, has receved consderable attenton n recent years. Several methods have been proposed for studyng the mpact of deleton of observatons on parameter estmates obtaned from the lnear model (Belsley, Kuh & Welsch, 198; Cook &Wesberg, 198), the logstc regresson model (Pregbon, 1981; Johnson, 1985), the Webull model for censored data (Pregbon, 1981) and the proportonal hazards model (Red & Crepeau, 1985, Bn Daud 1987, Noor Akma 1994, Elfak ). he focus of ths paper s use of dagnostc technque by performng t on ndependent competng rsks wth case deleton usng proportonal hazards model based on censored data. 1

2 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138). he Proportonal hazards Regresson Model We assume that we have n cases and for each we observe the vector ( t, δ, z) where t s the tme untl falure, f the case s uncensored ( δ = 1), or t s the tme untl removal or censorng ( δ = ). For each case there s a p 1 vector z of the explanatory or regresson varables. he proportonal hazards regresson model s defned as follows; we let the hazard functon of the falure tme to a tme constant vector explanatory varable z by takng h ( t / z) = λ( t)exp( zβ ) (1) where λ (t) s an unspecfed base-lne hazard functon correspondng to the case z = and β s a p 1 regresson parameter. Cox (197; 1975) proposed that we may use the partal lkelhood n exp( β z) L( β ) = () exp( β z = 1 ) R for nference concernng β, where the product s evaluated over all observed uncensored falure tmes, and R s the rsk set for th observed falure, that s, the set of ndvdual survvng and uncensored at t. o fnd βˆ usually U ( β ) = s solved where U (β ) s the p 1 score vector of dervatves of L (β ), the log lkelhood s n LogL( β ) = ( β z) log exp( β z) (3) = 1 R From equaton (3) we obtaned z exp( β z) log L( β ) R U (β ) = = z (4) β R exp( β z) R By takng the second dervatve of equaton (3), an expresson s obtaned whch has the form of a varance. For example, the dervatve of (4) wth respect to β s: z R z R exp( β z) exp( β z) log L 1 = (5) β p R exp( β z) exp( β z) R R Maxmum-lkelhood estmates of β can be obtaned by teratve (EM algorthm or Newton-Raphson methods) use of (4) and (5) n usual way. 3. Influence Measurement A general approach to nfluence s gven n Cook and Wesberg (198). We shall confne our study manly by adoptng the case deleton approach. In lnear regresson, Cook (1977), Cook and Wesberg (198) and others, suggested a sutably weghted p

3 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) combnaton of the changes ˆ β ˆ β as measures of nfluence. hey all gave versons of the Cook s (1977) dstance defne as; ˆ ' ' [( ˆ) ( ˆ D = β β X X β ˆ)]/( β s ), = 1,..., n (6) ( ) ( ) σ where βˆ ndcates an estmate for β wth full data. Full data n ths context refers to the falure tme 1 for all observatons that can be obtaned untl the study s completed, whle β ndcates estmate for β by deletng data pont, X X s a postve (sem-) defnte ˆ ( ) matrx, s s the parameter number, and σ s the varance. Lkewse, equaton (6) becomes the bass for most dstance measurements n detectng the nfluence of an observaton or a case. Influence measurements for the ordnary least square are generally based on the change n parameter estmate when the observaton s deleted, that s, ˆ β ˆ β, where βˆ s the estmate of β when the th observaton s deleted (Cook & Wesberg, 198; Belsley et al., 198). hs dfference n measurement has been appled n other computatonally more complex settngs by usng a one-step estmate of βˆ (Cook & Wang, 1983), and can be mplemented for ether the EM algorthm or Newton-Raphson method that wll be dscussed n a later secton n ths paper. For a sngle deleton wth the th case omtted from data, the change s gven by 1 ˆ ˆ ( X X ) X r β β = (7) m where X s the th row of the desgn matrces X, r s the th resdual and m s the th dagonal element of the projecton matrces m = 1 H wth H = X ( X X ) 1 X. 4. Cook s Dstance o get the nfluence n equaton (7) n a quanttatve form, whch s more meanngful, we used Cook s dstance (Cook, 1977, 1979; Cook and Wesberg, 198), whch was defned earler n ths paper by equaton (6). he usefulness of ths s the avalablty of several ways to measure the scale of change vector nvolved n perturbaton lke equaton (7), as suggested. For normal lnear regresson model wth the least square, Cook (1977) recommended a form of scale measurement known as Cook s dstance, whch s gven as follows, ( ˆ β ˆ) ( ˆ β X X β ˆ) β D = (8) q ˆ σ where βˆ s the estmate of the least square for full data, βˆ s the estmate of β when the observaton s deleted, q s the parameter number and ˆ σ s the estmate of varance. 1 he tme observed on ndvdual or object from one orgnal pont to the tme an antcpated event occurs. 3

4 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) In lne wth the least square technque, Pregbon (1981) has produced a matrx whch has a role lke H. Cook s dstance wll be consdered based on Cook and Wesberg (198), that s, ( ˆ β ˆ) β M ( ˆ β ˆ) β D = (9) g where M s a sem postve exact symmetry matrces, and g a postve scale factor. 5. Lkelhood Dstance A more general method to get measurement of nfluence s by usng contour measurement for the log lkelhood functon. Let L (β ) be the log lkelhood on parameter β based on full data. Lkelhood dstance (Cook and Wesberg, 198) s defned as LD [ L( β ) L( β )] = (1) where L( β ) s the log lkelhood functon of β when the observaton s deleted. LD measurement can also be nterpreted n terms of the asymptotc confdence regon (see Cox and Hnkley, 1974) β : { L( β ) L( β )} χ ( α; s) (11) where χ ( α; s) s the upper α pont from ch-squared dstrbuton wth s degree of freedom. If the log lkelhood contours are approxmately ellptcal, LD s quadratc, and can be drawn closer to aylor s expanson around βˆ, as follows, ˆ ˆ L( β ) 1 ˆ L( β ) L( β ) L( β ) + ( β ˆ) β + ( β ˆ) β, (1) β β β L( β ) snce = at maxmum lkelhood,. hen, β ˆ L( β ) LD ( β ˆ) β ( ˆ β ˆ) β β β (13) L( β ) where, s the observed nformaton matrx. hs approach can be confusng f β β log lkelhood contour s nonellptcal. Hence, the approach of aylor s expanson for log lkelhood, whch s not ellptc, needs a better development, whch wll nvolve a complex expresson. 6. Methods for Estmaton We turn now to a dscusson of methods that can be used to detect nfluental cases under competng rsks model that s ftted to censored data. he β ' s are found by teratve numercal technques. hs s a cumbersome task, snce, apart from the computaton needed to obtan βˆ, we are stll requred to calculate for each β, = 1,,..., n. o reduce 4

5 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) computaton, approxmatons to β are of great value. An obvous choce n such a stuaton s the one-step technque, that s, to compute from the maxmum lkelhood estmate βˆ the frst step of an teratve process to fnd βˆ. Can and Lange (1984) consdered the case-weght perturbaton scheme n proportonal hazard regresson model by n exp( β zδ w ) l( β / w) = (14) w = 1 w exp( β z ) R where z, δ, and β are defned n earler n ths paper, and w s a vector of ones for the complete data set-up. If w = and the other w ' s are one, then l( β / w) corresponds to the partal lkelhood wth the th case omtted, and ˆ β ( w ) corresponds to βˆ. Usng the perturbaton scheme as n equaton (14), we have l( β / w = ) equvalent to the partal lkelhood wth th case omtted from the data set. Our attempt to change the parameter nvolves one-step teratve for maxmzng l (β ) appled to l( β / w = ) startng at βˆ. he one-step teratve method for competng rsks model s complex, due to the expresson to log lkelhood. he one-step methods for both EM algorthm and Newton- Raphson wll be dscussed n the followng secton One-step EM he general arguments that follow are applcable n the scale Cox s fnte, and contnuous models, only by usng the EM and ML methods. he one-step technques, as mentoned earler n ths paper, whch are defned to be the maxmum lkelhood estmate of β wth the th case deleted, are formed by takng one-step of the teratve process for fndng βˆ startng at βˆ. he EM approach wth exponental dstrbuton s the best suted for adaptng to cases that are deleted from the data set. We can defne the resultng teratve weghted least square scheme for the full data by 1 β j+ 1 = β j + ( z z) z b (15) where z s the n p desgn matrces of the covarates, and b has a component where λ = exp( β z). b n δ δ 1 λ 1 + λ t t λ j r r R j n δ j = δ λ (16) λ = t j t r R j he covarate matrces z are column centered, that s, r n z = 1 =. 5

6 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) he change n estmates, after deleton of the th observaton, based on the one-step EM algorthm, s gven by 1 ˆ β ˆ β = z z z (17) [ ] where z s the covarate matrces obtaned from z wth the th row omtted, startng from ˆ β = β. 6.. One-step ML he teratve scheme for the Newton-Raphson s gven by ˆ * * 1 β = β + U ( β ) I ( β * ) (18) * where U ( β ) and I 1 ( β * ) are defned n equaton (4) and (5). Usng the same arguments as n the precedng secton, the one-step change from β, for a Newton- Raphson scheme such as (18), s wrtten as ˆ ˆ β β = U ( β ) I 1 ( β ) (19) where U s the th element of U (β ) and I s the th element of I. Equaton (19) s equvalent to that of the robust method of Cook and Wesberg, (198). he one-step estmate can be computed drectly at the fnal teraton for full data, thus a one-step nfluence measures may be obtaned. SAS software may easly be used to carry out the computaton of (17) and (19). However, for ease of manpulaton of matrces, the computaton n our analyss s programmed n SAS. Results from some smulaton data sets are gven n the next secton. 7. Smulaton Data he two smulatons were performed accordng to two dfferent sample szes wth varyng percentage of censored observaton (smulaton 1 and for sample sze 15 wth 5 percent censorng, sample sze 3 wth 33 percent censorng, respectvely, generated 1 tmes). We took the average for the falure tme and covarate to make the calculaton smple. o generate falure tme, the value of λ =. 45 for the frst type of falure and λ =. 57 for the second type of falure were used. In ths smulaton, the objectve s to fnd Cook s dstance based on equaton (6) and lkelhood dstance based on equaton (1) from the frst rsk and the second rsk for every sample sze. Both dstances were calculated by usng one-step EM algorthm based on equaton (17) and one-step ML based on equaton (19) under competng rsks model. Smulaton 1 (sample sze 15 wth 5 percent censorng), n able 1, shows the Cook s dstance and lkelhood dstance obtaned by the one-step EM algorthm and one-step ML under competng rsk model based on equaton (1). From the frst rsk ( k = 1), based on a large value for Cook's dstance from both methods, and the fve observatons, that s, number 5, 7, 9, 11, and 14, show a large nfluence on parameter estmates compared to other observatons. hs can clearly be seen n Fgures 1 and. Also, based on a large value for lkelhood dstance, the same observatons show to have a large nfluence on parameter estmates, compared wth other observatons, as t can clearly be seen n Fgures 3 and 4. But, Cook's dstance obtaned by both methods was found to have smaller value compared wth lkelhood dstance. 6

7 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) From the second rsk ( k = ), the observatons number 5, 7, 9, 11 and 14 showed to have large value from Cook's dstance and lkelhood dstance obtaned by one-step EM. he same observatons and observatons number 13 and 15 have large value from lkelhood dstance obtaned by one-step ML method. However, Cook's dstance obtaned by both methods has smaller value compared wth lkelhood dstance. Fnally, the Cook's dstance and lkelhood dstance obtaned by one-step EM algorthm showed to be smaller than the others obtaned by one-step ML methods from both rsks. Plots of Cook's dstance ( D ) and Lkelhood dstance ( LD ) one-step EM (frst rsk) aganst were gven n Fgure 1 and 3, respectvely. Inspectng the D ' s, the followng can be nferred: case 7 has D 7 =. 33 and LD 7 =. 63 for D and LD, respectvely, suggestng that case 7 for LD, LD 7 =. 63, may have a large enough nfluence to nduce the anomaly. Smlarly, for D and LD one-step ML (frst rsk), Fgure and 4, respectvely, n cases 5 and 14, that s, D 5 = D14 =. 198 for D, whle LD 7 =.58 for LD for case 7. However, consderng Fgures 5 and 7 for D and LD, respectvely, smlar results wth a possble cause: case 7 wth D =. 7 3 for D and case 11 wth LD = for LD are obtaned by one-step EM from the second rsk, whle D and LD for one-step ML from second rsk (Fgure 6 and 8, respectvely) are obtaned n case 14 wth D = and case 7 wth LD 7 =. 58. Despte the fact that LD ' s have hgher values for both one-step EM and ML from the second rsk, t can clearly be seen that the ranges between the hghest two values s too small compared to the hghest two values of D ' s. Cook's D CD Fgure 1: Cook's Dstance from smulaton 1 obtaned by one-step EM (frst rsk) 7

8 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) able 1: Cook's dstance and Lkelhood dstance obtaned by one-step EM and one-step ML under competng rsks model from smulaton 1 (sample sze 15) Frst Rsk One-step EM One-step-ML case Cook's D Lkelhood D Cook's D Lkelhood D Second Rsk One-step EM One-step-ML case Cook's D Lkelhood D Cook's D Lkelhood D

9 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138).5. CD-ML Cook's D Fgure : Cook's Dstance from smulaton 1 obtaned by one-step ML (frst rsk) Lkelhood D.7.6 LD-EM Fgure 3: Lkelhood dstance from smulaton 1 obtaned by one-step EM (frst rsk) 9

10 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) Lkelhood D LD-ML Fgure 4: Lkelhood dstance from smulaton 1 obtaned by one-step ML (frst rsk) Cook's D CD-EM Fgure 5: Cook's dstance from smulaton 1 obtaned by one-step EM (second rsk) 1

11 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138).5. CD-ML Cook's D Fgure 6: Cook's dstance from smulaton 1 obtaned by one-step ML (second rsk).4.35 LD-EM Fgure 7: Lkelhood dstance from smulaton 1 obtaned by one-step EM (second rsk) Lkelhood D 11

12 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) Lkelhood D LD-ML Fgure 8: Lkelhood dstance from smulaton 1 obtaned by one-step ML (second rsk) Smulaton (sample sze 3, wth 33 percent censorng), ables and 3 show Cook's dstance and lkelhood dstance obtaned by one-step EM algorthm method and one-step ML method under competng rsks model. able shows the frst rsk ( k = 1), and the value of Cook's dstance for 3 obtaned by one-step EM has the largest potental and largest nfluence. Also, the value of Cook's dstance for cases 7, 18 and 4 obtaned by one-step ML have largest potental and largest nfluence and the second largest value of Cook's dstance, D 1 = D13 = D15 = D7 =. 15. Only 1 s ndvdually nfluental, compared wth other cases of the value of lkelhood dstance obtaned by one-step ML method. However, Cook's dstance obtaned by both methods have smaller values than lkelhood dstance. able 3 shows the second rsk ( k = ), and the value of Cook's dstance of 3 only has the largest nfluence, compared wth other cases obtaned by one-step EM. Also, D 1 = D18 = D4 =. 145 obtaned by one-step ML method have the largest potental, and the second largest value of Cook's dstance obtaned by ths method s from s, 5, 8, 1, 14, 17, 1,, 5 and 9. However, based on lkelhood dstance, 1 and 3 obtaned by one-step EM and one-step ML, respectvely, have the largest value, compared wth other cases from both methods. Fnally, the value of Cook's dstance and lkelhood dstance obtaned by one-step EM show to be smaller than the others obtaned by one-step ML. Moreover, Cook's dstances show to have a smaller value than lkelhood dstance. 1

13 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) able : Cook's dstance and Lkelhood dstance obtaned by one-step EM and one-step ML under competng rsks model from smulaton (frst rsk) Frst Rsk One-step EM One-step-ML case Cook's D Lkelhood D Cook's D Lkelhood D

14 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) able 3: Cook's dstance and Lkelhood dstance obtaned by one-step EM and one-step ML under competng rsks model from smulaton (second rsk) Second Rsk One-step EM One-step-ML case Cook's D Lkelhood D Cook's D Lkelhood D Concluson he nfluence measurements are based on case deleton. Wthn ths technque, a vector needs to be ntroduced so that an assessment on the case can be used to obtan Cook s dstance and lkelhood dstance. he dagnostc technques and nfluence used dentfy cases of nfluence constructed for competng rsk model were successful, and these technques were able to dentfy odd cases usng the smulaton data sets(elfak, ). 14

15 Proceedngs of the 8th WSEAS Internatonal Conference on APPLIED MAHEMAICS, enerfe, Span, December 16-18, 5 (pp14-138) One-step technque was used to derve the estmate of parameter, followed by dstance measurement. hese technques are one-step EM and one-step ML. they were ntroduced to reduce the teratve steps n computng the nfluence measurements. It was found that from both smulatons the Cook s dstance was sgnfcant compare to lkelhood dstance because t can show clearly the outler observaton. Also the results obtaned by one-step EM s sgnfcant compare to the one obtaned by one-step ML. Reference Andersen, P. K., Borgan. Q., Gll. R. D. and Kedng, N. (1993). Statstcal Models Based on Countng Processes. Sprnger-Verlag. Belsley, D. A., Kuh, E., and Welsch, R. E. (198). Regresson Dagnostcs: Identfyng Influental Data and Sources of Collnearty. New York. Wley. Bn Daud, I. (1987). Influence Dagnostcs n Regresson wth Censored Data. Ph.D., thess, Loughborough Unversty. Can, K. C. and Lange, N.. (1984). Approxmate Case Influence for the Proportonal hazards regresson Model wth Censored data." Bometrcs. 4: Cook, R. D. (1977). Detecton of Influence Observaton n Lnear Regresson. echnometrcs. 19: Cook, R. D. (1979). Influence Observatons n Lnear Regresson. Journal of Amercan Statstcal Assocaton.74: Cook, R. D. (1986). Assessment of Local Influence (wth Dscusson). Journal of Royal Statstcal Socety. 48: Cook, R. D. and Wange, P. C. (1983). "ransformatons and Influental Cases n Regresson." echnometrcs. Vol. 5. No. 4: Cook, R. D. and Wesberg. S. (198). Characterzatons of an Emprcal Influence Functon for Detectng nfluental Cases n Regresson. echnometrcs. Vol.. No. 4: Cook, R. D. and Wesberg. S. (198). Resduals and Influence n Regresson. New York and London; Chapman and Hall. Cox, D. R. (197). Regresson Models and Lfe ables (wth dscusson). J. R.. Statstc. Soc. 34: Cox, D. R. (1975). Partal lkelhood. Bometrka. 6: Cox, D. R. and. Hnkely, D. V (1974). heoretcal Statstcs. London: Chapman and Hall Cox, D. R. and D. Oakes (1984) Analyss of Survval Data. London: Chapman and Hall. Elfak, F. A. M. () EM Approach on Influence Measures n Competng Rsks Va Proportonal Hazard Regresson Model. M.Sc thess Unversty Putra Malaysa. Noor, Akma. Ibrahm (1994). Influence Dagnostcs n Competng Rsks. Ph. D., thess. Unverst Pertanan Malaysa. Pregbon, D. (1981). Logstc Regresson Dagnostcs. Annals of Statstcs. 9: Red, N. and Crepeau, H. (1985). Influence Functon for Proportonal Hazard Regresson. Bometrka. 7: