Chapter 9

1 Chapter 9 Time varying (or time-dependent) covariatesReferences:Allison (*) & LemeshowChapter 7, Section 3 Kalbfleisch & PrenticeSection 7 KleinbaumChapter 6 Cox & OakesChapter 8 Andersen & GillPage 168 (Advanced!)12 Chapter 9. TIME VARYING (OR TIME-DEPENDENT) COVARIATESOur Goal hereSo far, we ve been considering the following Cox PH model: (t,Z)= 0(t) exp( Z)= 0(t)exp p j=1 jZj where the covariatesZjare measured at study entry (t=0).Important feature of this model:The hazard ratio (t,Z=z) (t,Z=0)=exp( z)depends on the covariatesz1, .., zp, but not on relax this assumption, and allow the hazard ratio to dependon timet. allow to incorporate time-varying TO MOTIVATE TIME-DEPENDENT to motivate time-dependent covariatesStanford Heart transplant example:Variables: survival- days since program enrollment until death or censoring dead- indicator of death (1) or censoring (0) transpl- whether patient ever had transplant(1 if yes, 2 if no) surgery- previous heart surgery prior to program (1=yes, 0=no) age- age at time of acceptance into program wait- days from acceptance into program until transplant surgery (=.)

2 For those withouttransplant)4 Chapter 9. TIME VARYING (OR TIME-DEPENDENT) COVARIATESI nitially, a Cox PH model was fit for predicting survival time: (t,Z)= 0(t) exp( 1 transpl+ 2 surgery+ 3 age)Does this fit in the framework we have seen so far?Why or why not? TO MOTIVATE TIME-DEPENDENT COVARIATES5 (t,Z)= 0(t) exp( 1 transpl+ 2 surgery+ 3 age)( ) As the covariate transpl really changes over time and gets a value de-pending on how long the patient has been followed ..this is not a regular Cox PH model as we know it. This model could give misleading results, since patients who died morequickly had less time available to get transplants. A model with a time-dependent indicator of whether a patient had a transplant at each pointin time might be more appropriate: (t,Z)= 0(t)exp( 1 trnstime(t)+ 2 surgery+ 3 age)( )wheretrnstime(t)=1iftranspl=1 andwait<t6 Chapter 9.

3 TIME VARYING (OR TIME-DEPENDENT) COVARIATESSAS code for these two modelsTime-independent covariate fortranspl:proc phreg data=stanford;model survival*dead(0)=transpl surgery age;run;Time-dependent covariate fortranspl:proc phreg data=stanford;model survival*dead(0)=trnstime surgery age;if wait>survival or wait=. then trnstime=0;else trnstime=1;run; TO MOTIVATE TIME-DEPENDENT COVARIATES7If we add time-dependent covariates or interactions with time to the Coxproportional hazards model, then it is not a proportional hazards modelany Cox with a single binary predictor (like heart transplant): The Cox PH model would compare the survival distributions betweenthose without a transplant (ever) to those with a transplant.

4 A subject stransplant status at the end of the study would determine which categorythey were put into for the entire study follow-up. This does not makemuch sense. An extended Cox model would compare the risk of an event between al-ready transplanted and non-yet-transplanted at each event time, and wouldre-evaluate which risk group each person belonged in based on whetherthey d had a transplant by that 9. TIME VARYING (OR TIME-DEPENDENT) COVARIATESR ecidivism Example:(see Allison, ) 432 male inmates werefollowed for one year after release from prison, to evaluate riskof re-arrest as function of financial aid (fin), age at release (age),race (race), full-time work experience prior to first arrest (wexp),marital status (mar), parole status (paro=1 if released with pa-role, 0 otherwise), and number of prior convictions (prio).

5 Datawere also collected on employment status over time during model:includes employment status of the individual at the beginning ofthe study (1 if employed, 0 if unemployed), or perhaps at anypoint during the model:However, employment status changes over time, and it may bethe more recent employment status that would affect the hazardfor re-arrest. , we might want to define a time-dependentcovariate for each month of the study that indicates whether theindividual was employed during the past COX Cox ModelFramework:For individuali, suppose we have their observation time, failureindicator, and a summary of their covariate values over time:(Xi, i,{Zi(t),t [0,Xi]}),{Zi(t),t [0,Xi]}represents the covariate pathfor thei-th indi-vidual while they are in the study, and the covariates can takedifferent values at different : conditional on an individual s covariate history, the Cox modelfor the hazard holds: (t;{Zi(u),u [0,t]})= (t;Zi(t)) = 0(t)e Zi(t)This means we record inZ(t)the part of the history that influ-ences the hazard at 9.

6 TIME VARYING (OR TIME-DEPENDENT) COVARIATESS urvivor function:S(t;Z) = exp{ t0exp( Z(u)) 0(u)du}and depends on the values of the time dependent variables overthe interval from 0 is the classic formulation of the time varying Cox regressionsurvival (u)is step function with one change point att1<t:S(t;Z) = exp t10exp( Z(u)) 0(u)du+ tt1exp( Z(u)) 0(u)du =exp exp( Z(0)) t10 0(u)du+ exp( Z(t1)) tt1 0(u)du And prediction?? COX MODEL11 Kinds of time-varying covariates: internal covariates:variables that relate to the individuals, and can only be mea-sured when an individual is alive, white blood cell count,CD4 count external covariates: variable which changes in a known way, age, dose ofdrug (if non-dynamic drug regime) variable that exists totally independently of all individuals, air temperature time itself12 Chapter 9.

7 TIME VARYING (OR TIME-DEPENDENT) of applications and ExamplesThe extended Cox model is used:I. When important covariates changeduring a study Framingham Heart study5209 subjects followed since 1948 to examine relationship be-tween risk factors and cardiovascular disease. A particularexample:Outcome: time to congestive heart failurePredictors: age, systolic blood pressure, # cigarettes per OF APPLICATIONS AND EXAMPLES13 Liver Cirrhosis (Andersen and Gill, )Clinical trial comparing treatment to placebo for outcome of interest is time to death. Patients were seenat the clinic after 3, 6 and 12 months, then covariates: treatment, gender, age (at diagnosis)Time-varying covariates: alcohol consumption, nutritional sta-tus, bleeding, albumin, bilirubin, alkaline phosphatase andprothrombin.

8 The paper on obesity and heart Recidivism Study (Allison, )14 Chapter 9. TIME VARYING (OR TIME-DEPENDENT) COVARIATESII. For cross-over studies, to indicate change in treatment Stanford heart study (Cox and Oakes )Between 1967 and 1980, 249 patients entered a program atStanford University where they were registered to receive aheart transplant. Of these, 184 received transplants, 57 diedwhile waiting, and 8 dropped out of the program for other rea-sons. Does getting a heart transplant improve survival? Hereis a sample of the data:Waiting transplant? survival post total finaltime transplant survival status---------------------------------- --------------------------49 2.

9 152 ..10 1 15 15 135 1 3 38 117 2 .. 111 1 46 57 1etc(survival is not indicated above for those without transplants, but was available in the dataset)Naive approach: Compare the total survival of transplanted and : Length Bias! OF APPLICATIONS AND EXAMPLES15 III. For Surrogate Outcome AnalysisFor example, in cancer clinical trials, tumor response (or shrink-ing of the tumor) is often used as an outcome. However, clinicianswant to know whether tumor response correlates with this purpose, we can fit an extended Cox model for time todeath, with tumor response as a time dependent : association =causation !

10 16 Chapter 9. TIME VARYING (OR TIME-DEPENDENT) COVARIATESIV. For testing the PH assumptionFor example, we can fit these two models:(1) Time independent covariateZ1 (t,Z)= 0(t) exp( 1 Z1)The hazard ratio forZ1isexp( 1).(2) Time dependent covariateZ1 (t,Z)= 0(t)exp( 1 Z1+ 2 Z1 t)The hazard ratio forZ1isexp( 1+ 2t).(note: we may want to replacetby(t t0), so thatexp( 1)represents HR at some convenienttime, like the median survival time.)A test of the parameter 2is a test of the PH assumption.(how do we get the test? ..using the Wald test from the out-put of second model, or LR test formed by comparing the log-likelihoods of the two models) LIKELIHOOD WITH TIME-VARYING likelihood with time-varying covariatesStarting out just as before, relying on non-informative censoringSuppose there areKdistinct failure (or death) times, and let( 1.)