TESTINGGOODNESSOFFITINTHECOX AALEN MODEL
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1 ROBUST 24 c JČMF 24 TESTINGGOODNESSOFFITINTHECOX AALEN MODEL David Kraus Keywords: Counting process, Cox Aalen model, goodness-of-fit, martingale, residual, survival analysis. Abstract: The Cox Aalen regression model for the intensity of counting processes suggested by Scheike and Zhang[13] extends the Cox proportional hazards model as well as the Aalen additive model. We study goodness-of-fit tests based on the stratified martingale residual process. Asymptotic distribution of this process is complicated(a Gaussian process with a complex covariance structure). Therefore, a direct use of, e.g., the Kolmogorov Smirnov typetestisimpossible.weshowtwowaysoutofthisproblem.onepossibility is to simulate realisations from the limiting distribution of the residual process,assuggestedbylin,weiandying[7]forthecoxmodel.another approach consists of tranforming(compensating) the limiting process to a martingale(following ideas of Khmaladze[4]). Both methods are compared in a simulation study. 1 Introduction In survival analysis, regression models are used to explain occurence of events (failures) in time by the influence of explanatory variables(covariates). Let Z i = {(Z i1 (t),..., Z ip (t)) T, t [, τ]}beavectorofcovariates(possibly time-dependent, i.e. predictable stochastic processes) for the i-th observed individual, Y i beanindicatorprocess(indicatingbyitsvalueattime twhetherthe i-thindividualisatriskoftheevent)and λ i betheintensityprocess ofthecorrespondingcountingprocess N i.thenthemostpopularmodelfor the intensity process is the Cox proportional hazards model of the form λ i (t)=y i (t)exp{β T Z i(t)}λ (t), where λ isanunknownbaselinehazardfunctionand β isap-vectorof unknown regression parameters. Another frequently used model is the Aalen additive model λ i (t)=y i (t)x i (t) T α(t) withavectorofunknownfunctions α={(α 1 (t),..., α q (t)) T, t [, τ]}and avectorofcovariates X i. There are several ways of combining these two models(e.g.[8],[12]). Here westudythecox Aalenmodel,whichisduetoScheikeandZhang[13].Their model follows the form λ i (t)=y i (t)exp{β T Z i (t)}x i (t) T α(t), t τ, (1)
2 226 David Kraus where(x T i, ZT i )T isa(q+p)-vectorofpredictablecovariates(usually X i1 1). Somecomponentsof α(t)or X i (t)canevenbenegative,providedthewhole term X i (t) T α(t)isnonnegative(anintensityalwayshastobeso). In[13] an estimation procedure was suggested and asymptotic properties oftheestimatorswerederived.hereweproposeatestofgoodnessoffit.for the Cox model many goodness-of-fit tests were developed. One approach, whichishereadaptedforthesituationofthecox Aalenmodel,isbasedon the stratified martingale residual process. The idea was originated by Arjas [3] who suggested a graphical procedureforassessinggoodnessoffitofthecoxmodel.themethodisbased on comparison of observed and expected number of failures within a given stratum.foreachobservedindividual i {1,...,n}thisdifferenceisexpressedbytheprocess ˆM i = N i ˆΛ i.hence,ifastratum I {1,...,n} is chosen, the process Ξ I = ˆM i I i should fluctuate around zero. Let T (I,1) T (I,2) T (I,K) τ beorderedtimesoftheactual failuresin I.ThenArjas splotsareplotsofthevalues i IˆΛ i (T (I,k) )against k= i I N i(t (I,k) ).Thegraphshouldbeclosetothelinewithslope1 whenthefitisgood,andshoulddifferotherwise. Asymptotic behaviour of the residual process for the Cox model was studied by Marzec and Marzec[9]. Under certain conditions they showed that the limitingdistributionof n 1/2 Ξ I isthatofacontinuouszero-meangaussian process. Later, in[1] they presented some generalisations and used a transformation of the limiting process to a martingale which enables construction of the Kolmogorov Smirnov type test. The idea of Arjas s plots and stratified residual processes was successfully used by Volf[14] also for the Aalen additive model. In the situation of the Cox Aalen model of(1), the stratified martingale residual process has the form Ξ I (t)= i I(N i (t) ˆΛ i (t)) [ dni (s) Y i (s)exp{ˆβ T Z i (s)}x i (s) T dâ(s)] (2) = i I (whereˆβand Â(t)areestimatesof β and A(t)= α(s)ds,respectively). The paper is organised as follows. Section 2 establishes asymptotic propertiesoftheresidualprocess.section3isdevotedtothestudyofseveraltesting procedures.insection4asimulationstudyispresentedandinsection5we mention some generalisations. Due to the lack of space, some results are only sketched in a symbolic way and many details are omitted; the author refers an interested reader to his paper[5].
3 Testing goodness of fit in the Cox Aalen model Asymptotic distribution of the residual process Theestimationprocedurefortheparameters β and A(t) = α(s)dsis describedindetailin[13].hereweonlymentionthatitconsistsoftwosteps: firsttheparameter β isestimatedbysolvingascoreequationandthen the increments of A(t) are estimated by a weighted least squares principle. Denotetheestimatesbyˆβand Â.Thesetwostepscanberepeatedtoobtain better estimates. ByTaylor sexpansionaround β andaftersomerearrangement,ξ I can be expressed as where Γ 1 (t)= Ξ I (t)=γ 1 (t) Q( β, β, t) T Γ 2 (τ), (3) K 1 (β, s) T dm(s) and Γ 2 (t)= K 2 (β, s) T dm(s) aremartingales.here M(t)=(M 1 (t),..., M n (t)) T, βand βlieontheline segmentbetween β and ˆβ,and K 1, K 2 and Qaresomeprocessesofdimensions n 1, n pand p 1,respectively.Theprocess Q( β, β, )converges uniformlyinprobabilitytoafunction,say q(β, ).Theweakconvergence in(d[, τ]) 1+p ofthemartingale n 1/2 (Γ 1,Γ T 2 )T toazero-meancontinuous Gaussianmartingale γ=(γ 1, γ2 T)T followsbyrebolledo smartingalecentral limit theorem[2, Theorem I.2]. Therefore we have the weak convergence n 1/2 Ξ I ( ) D n ξ I( )=γ 1 ( ) q(β, ) T γ 2 (τ) in(d[, τ]). (4) Thelimitingprocess ξ I isagaussianprocess,which,however,isneither a martingale nor a process with a well-known distribution. The(co)variance function of γ is the limit of the predictable(co)variation of n 1/2 Γandcanbeestimateduniformlyconsistentlybythequadratic(optional)(co)variationof n 1/2 Γ.Detailsconcerningthecovariancestructureas well as explicit forms of all of the processes, a further notation, assumptions, exactformulationoftheresultsandproofscanbefoundin[5]. 3 Testing procedures A graphical technique, an analogue of the residual plots of Arjas[3], has already been explained in Section 1. In this section we describe visual and mainly numerical methods of investigation of the martingale residual process. Exceptforaspecialcaseofthemodelwithonedichotomouscovariatein thecoxpart[5,section4.1],adirectuseoftheasymptoticdistributionisnot possible because of its complexity. For instance, the limiting distribution of the Kolmogorov Smirnov type statistic is untractable. We describe how some approximations and transformations enables us to cope with this difficulcy.
4 228 David Kraus 3.1 The simulation approximation We shall describe how to approximate the asymptotic distribution of the martingale residual process through simulations. The simulations can be performed to obtain a sample from the limiting distribution, and hence to assess both graphically and numerically how unusual the observed residual process is.ourapproachisbasedontheideaof[7]. In(3) we have found the martingale representation of the residual process oftheform Ξ I (t)=γ 1 (t)= K 1 (β, s) T dm(s) Q( β, β, t) T K 2 (s) T dm(s), The limiting distribution can be approximated by plugging in the consistent estimate ˆβinplaceof β, β, β,andbyreplacingthemartingaleincrements dm i (t)bytheirsimulatedvalues.forthemartingales M i, i=1,...,nitholds thate M i (t)=andvarm i (t)=e[m i (t) 2 ]=EΛ i (t)=e N i (t).therefore Lin,WeiandYing[7]suggestedtoapproximate M i by M i = G i N i (i.e. M=diag[G]N),where G=(G 1,..., G n )isarandomsampleofstandard normal variables independent of the data. We obtain the approximation Ξ I (t)= K 1 (ˆβ, s) T diag[g]dn(s) τ Q(ˆβ,ˆβ, t) T K 2 (s) T diag[g]dn(s). Itcanbeshown(bymeanssimilartothatof[7])thattheasymptotic distributionof n 1/2 Ξ I isequaltothatof n 1/2 Ξ I,i.e.thedistributionof ξ I.Thus,generatingrepeatedlyastandardnormalsample Gandcomputing Ξ I,weobtainasamplefromthedesiredlimitingdistribution.Wecanvisually assessgoodnessoffitbyplottingξ I togetherwithanappropriatenumberof simulated Ξ I.Totestthehypothesisnumericallywegenerateanadequately largenumberofrealisationsof Ξ I andestimatecriticalvaluesorthe p-value. Note that the simulation of the asymptotic distribution is conditional onthedataandthereforewedonotobtainuniversalcriticalvaluesofthis distribution. In other words, the test is not distribution-free and we have to carry out the simulations for each particular data set separately. This can be computationally demanding. 3.2 The transformation method Another way of overcoming the problem with complexity of the asymptotic distributionoftheresidualprocessisbasedontheideaofkhmaladze[4].in the framework of testing whether the distribution of a random variable follows a parametric form, he suggested a transformation of empirical processes with plugged-in estimated parameters in order to obtain a well-known asymptotic distribution which does not depend on the distribution of the data. The idea
5 Testing goodness of fit in the Cox Aalen model 229 wasthenusedin[1,vi.3.3.4]fortestinggoodnessoffitofparametricmodels forintensitiesandin[1]forassessmentofthecoxmodel. Herewewillfindthecompensatorof ξ I,say ξ I,andusetheempirical counterpartψ I ofthegaussianmartingale ψ I = ξ I ξ I asabasisfortesting. Theprocess ψ I shouldbeamartingalewithrespecttothefiltrationgenerated by ξ I,i.e.withrespectto G t = σ{γ(s), s t; γ 2 (τ)}, t [, τ].astheterm q(β, ) T γ 2 (τ)in(4)ismeasurablew.r.t. G,weonlyneedtocompensate γ 1. Thecompensatorof γ 1,say γ 1,canbederivedratherheuristicallyasfollows (cf.[1, VI.3.3.4, pp ]). Since the process γ is a Gaussian martingale, ithasindependentincrementsand(dγ 1 (t), γ 2 (τ) γ 2 (t)) T isjointlynormally distributed. Therefore E[dγ 1 (t) γ(s), s t; γ 2 (τ)]=e[dγ 1 (t) γ 2 (τ) γ 2 (t)] =cov {dγ 1 (t), γ 2 (τ) γ 2 (t)}[var {γ 2 (τ) γ 2 (t)}] 1 [γ 2 (τ) γ 2 (t)] = c(β, dt) T θ(β, t) 1 [γ 2 (τ) γ 2 (t)], wherethedefinitionof cand θisobvious.henceanaturalcandidateforthe compensatorof γ 1 is γ 1 (t)= E[dγ 1 (s) γ(u), u s; γ 2 (τ)] = The compensated residual process is then ψ I (t)=γ 1 (t) [γ 2 (τ) γ 2 (s)] T θ(β, s) 1 c(β, ds), t [, τ]. [γ 2 (τ) γ 2 (s)] T θ(β, s) 1 c(β, ds). Formalverificationofthefactthat ψ I isactuallyagaussianmartingale(with thesamevariancefunctionas γ 1 )isanalogoustotheproofoflemma3.2 in[1].finally,theempiricalcounterpartof ψ I canbe Ψ I (t)=ξ I (t) [ τ T K 2 (s) dˆm(s)] T Θ(ˆβ, s) 1 C(ˆβ, ds) (5) s with CandΘbeingsomeestimatesof cand θ.theissuewhether n 1/2 Ψ I trulyconvergesto ψ I isdiscussedin[5].ourcomputationalexperienceshows that the convergence can be considered valid generally. HencetheKolmogorov Smirnovtypestatisticsup Ψ I (t) /{ varγ 1 (τ)} 1/2 is asymptotically distributed as the variable sup W(t) (where W denotes the Brownian motion).
6 23 David Kraus 4 Simulations We performed a small simulation study in order to investigate performance of the tests. We generated survival data following various models with various censoring patterns and estimated the Cox Aalen model of the form λ i (t)={α 1 (t)+α 2 (t)x i }exp{β 1 Z i }. Underthenullhypothesis H thesamplescamefromthedistributionwith the intensity λ i (t)={.5+.2tx i }exp{.5z i }, where X i wasuniformlydistributedon[,1]and Z i hadthestandardnormal distribution. Then two alternatives with additional covariates were considered: H 1 with λ i (t)={.5+.2tx i +.7tX i }exp{.5z i }, where Xi hadthealternativedistributionon {,1}withprobability.5,and H 2 having λ i (t)={.5+.2tx i }exp{.5z i +.7Zi } with Z i havingthesamedistributionas X i intheprevioussituation.the covariates were generated independently. Three censoring schemes were considered: no censoring, moderate censoring(with censoring times having the uniform distribution on[, 5]) and heavy censoring(with uniform censoring times on[, 2.5]). The censoring times were mutually independent and independent of the survival times and covariates. The corresponding censoring ratesareindicatedinthetables.thesamplesizeswere n=1and2.for the null hypothesis, stratification with respect to both covariates was studied,i.e.thestratumwasfirst I= {i:x i >.5}andthen I= {i:z i >}. Under the alternatives, the data were stratified with respect to the missing covariate: I= {i:x i =1}under H 1,and I= {i:z i =1}under H 2. Under the null hypothesis as well as under the alternatives, we generated the sample, estimated the model and tested goodness of fit. Two tests were performed: the test of Subsection 3.1 based on the simulation approximation (with 2 realisations of the residual process) and the test of Subsection 3.2 based on the transformation. This was repeated 1 times and empirical levelsandpowersofthetestsonthenominallevelof.5werecomputed. Since the estimation of the model is highly time-consuming, we were able to carry out only 1 repetitions in each situation, hence our results give only abroadimageofthebehaviourofthetests. Table1reportsthesizesofthetestsintheabovementionedsituations under H.Itisseenthatthetestsmaintainapproximatelytheirnominalsignificance levels which is not seriously affected by censoring. Table 2 confirms thatthetestshavegoodpoweragainstthealternatives H 1 and H 2 ofmissing
7 Testing goodness of fit in the Cox Aalen model 231 covariates. When censoring is present, the power decreases. There is no important difference between the two versions(simulation and transformation) ofthetest. I= {i:x i >.5} I= {i:z i >} n=1 n=2 n=1 n=2 Without Simulation censoring Transform Censoring Simulation U[, 5](31%) Transform Censoring Simulation U[, 2.5](51%) Transform Table1:Empiricalsizesofthetwotestsonthenominallevelof.5. H 1 H 2 n=1 n=2 n=1 n=2 Without Simul censoring Transf Censoring U[, 5] Simul (H 1 25%, H 2 24%) Transf Censoring U[, 2.5] Simul (H 1 45%, H 2 42%) Transf Table2:Empiricalpowersofthetwotestsonthenominallevelof.5. 5 Generalisations Inthispaperweusedtheideaofstratification.Thetestsbasedonthemartingaleresidualprocesses ˆM i canbegeneralisedinthefollowingdirection. Insteadofthesumofthemartingaleresiduals ˆM i (t)=n i (t) ˆΛ i (t)over astratumwecanusesumsoftheirtransformsoftheform Ξ(t)= n i=1 H i (s)dˆm i (s), where H i arevectorsofsomepredictableprocesses.thistypeoftestswas studiedin[6].aparticularlyimportantchoiceis H i (t)=z i (t)leadingtothe score process which is used for detection of departures from the proportional hazardsassumption.thechoice H i =1 {i I} correspondstothestratified martingale residual process presented in this paper.
8 232 David Kraus References [1] Andersen P.K., Borgan Ø., Gill R.D., Keiding N.(1993). Statistical models based on counting processes. Springer, New York. [2] Andersen P.K., Gill R.D.(1982). Cox s regression model for counting processes: a large sample study. Ann. Statist. 1, [3]ArjasE.(1988).AgraphicalmethodforassessinggoodnessoffitinCox s proportional hazards model. J. Amer. Statist. Assoc. 83, [4] Khmaladze E.V.(1981). Martingale approach in the theory of goodnessof-fit tests. Teor. Veroyatnost. i Primenen. 26, In Russian. English translation in Theory Probab. Appl. 26, [5] Kraus D.(24). Goodness-of-fit inference for the Cox Aalen additivemultiplicative regression model. Statist. Probab. Lett. To appear. [6] Kraus D.(24). Testing goodness of fit of hazard regression models. In WDS 24 Proceedings of Contributed Papers, Part I: Mathematics and Computer Sciences, Matfyzpress, Prague, [7]LinD.Y.,WeiL.J.,YingZ.(1993).CheckingtheCoxmodelwithcumulative sums of martingale-based residuals. Biometrika 8, [8] Lin D.Y., Ying Z.(1995). Semiparametric analysis of general additivemultiplicative hazard models for counting processes. Ann. Statist. 23, [9] Marzec L., Marzec P.(1993). Goodness of fit inference based on stratification in Cox s regression model. Scand. J. Statist. 2, [1] Marzec L., Marzec P.(1997). Generalized martingale-residual processes for goodness-of-fit inference in Cox s type regression models. Ann. Statist.25, [11] McKeague I.W., Utikal K.J.(1991). Goodness-of-fit tests for additive hazards and proportional hazards models. Scand. J. Statist. 18, [12] Scheike T.H., Martinussen T.(22). A flexible additive multiplicative hazard model. Biometrika 89, [13] Scheike T.H., Zhang M.-J.(22). An additive-multiplicative Cox Aalen regression model. Scand. J. Statist. 29, [14] Volf P.(1996). Analysis of generalized residuals in hazard regression models. Kybernetika 32, Acknowledgement: The author acknowledges that this paper is an abbreviated version of his paper[5] published elsewhere. The research was partly supported by the GAČR Grants 21/2/49 and 42/4/1294. Address: D. Kraus, Department of Statistics, Charles University, Sokolovská 83, CZ Prague, Czech Republic david.kraus@karlin.mff.cuni.cz
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