1. HSRP 734: Advanced Statistical Methods July 31, 2008

2. Objectives Describe the general form of the Cox proportional hazards model extended for time-dependent variables Describe the general form of the Cox proportional hazards model extended for time-dependent variables Describe the analysis for staggered entry Describe the analysis for staggered entry Review for Final exam Review for Final exam

3. Time-Dependent Variables Time-dependent variable: covariate whose value may vary over time. Time-dependent variable: covariate whose value may vary over time. Two options if the proportional hazards assumption is not satisfied for one or more of the predictors in the model. Two options if the proportional hazards assumption is not satisfied for one or more of the predictors in the model. Use a stratified Cox model Use a stratified Cox model Use time-dependent variables Use time-dependent variables

4. Time-Dependent Variables Time-dependent variables may be: Time-dependent variables may be: Inherently time-dependent Inherently time-dependent Internal – only have meaning when subject is alive Internal – only have meaning when subject is alive smoking status at time t smoking status at time t white blood count at time t white blood count at time t External – can be obtained whether or not subject is alive External – can be obtained whether or not subject is alive Air pollution index at time t Air pollution index at time t Part internal and part ancillary Part internal and part ancillary E.g., heart transplant status at time t E.g., heart transplant status at time t Defined to analyze a time-independent predictor not satisfying the PH assumption Defined to analyze a time-independent predictor not satisfying the PH assumption E.g., RACE x time; RACE x log(time+1) E.g., RACE x time; RACE x log(time+1)

5. Internal Time-Dependent Variable Internal time-dependent variables are particularly susceptible to be inappropriately controlled. Internal time-dependent variables are particularly susceptible to be inappropriately controlled. They often lie in the causal pathway about which we want to make inferences. They often lie in the causal pathway about which we want to make inferences.

6. Internal Time-Dependent Variable example Clinical trial for treatment of metastatic colorectal cancer – do we adjust for most recent WBC? Clinical trial for treatment of metastatic colorectal cancer – do we adjust for most recent WBC?

7. Internal Time-Dependent Variable example Clinical trial for treatment of metastatic colorectal cancer – do we adjust for most recent WBC? Clinical trial for treatment of metastatic colorectal cancer – do we adjust for most recent WBC? Treatment comparison among subjects with like prognosis at each time Treatment comparison among subjects with like prognosis at each time But treatment might improve prognosis by improving depressed WBC over time But treatment might improve prognosis by improving depressed WBC over time Adjustment for WBC over time might remove the apparent effect of treatment, since patients with the same WBC in either treatment group might have similar prognosis Adjustment for WBC over time might remove the apparent effect of treatment, since patients with the same WBC in either treatment group might have similar prognosis

8. Extended Cox Model for Time- Dependent Variables Model Model Assumption: Assumption: The effect of a time-dependent variable on the survival probability at time t depends on the value of this variable at that same time t. The effect of a time-dependent variable on the survival probability at time t depends on the value of this variable at that same time t. Statistical inferences: Statistical inferences: Wald, Score, Likelihood ratio tests Wald, Score, Likelihood ratio tests

9. Extended Cox Model for Time- Dependent Variables Even though the values of the time-dependent variable may change over time, the hazard model provides only one coefficient for each time-dependent variable in the model. Even though the values of the time-dependent variable may change over time, the hazard model provides only one coefficient for each time-dependent variable in the model.

10. Hazard Ratio for the Extended Cox Model Let X 1 = smoking yes/no; X 2 (t) = X 1 x t Let X 1 = smoking yes/no; X 2 (t) = X 1 x t The hazard ratio (or RR) is a function of time. The hazard ratio (or RR) is a function of time. PH assumption is not satisfied for the extended Cox model PH assumption is not satisfied for the extended Cox model

11. Hazard Ratio for the Extended Cox Model Coefficient represents the “overall” effect of the corresponding time-dependent variable, considering all times at which this variable has been measured in the study. Coefficient represents the “overall” effect of the corresponding time-dependent variable, considering all times at which this variable has been measured in the study. Another model with a time-dependent variable Another model with a time-dependent variable compares an exposed person to an unexposed person at time t. compares an exposed person to an unexposed person at time t.

12. Time-Dependent Variables in SAS Do not define the time-dependent variable in a data step Do not define the time-dependent variable in a data step The variable will be time-independent The variable will be time-independent Use the programming statements in proc tphreg Use the programming statements in proc tphreg time depend example.sas

13. Left Truncation of Failure Times Also know as staggered entry Also know as staggered entry Left truncation arises when individuals only come under observation some known time after the natural time origin of the phenomenon under study. Left truncation arises when individuals only come under observation some known time after the natural time origin of the phenomenon under study.

14. Left Truncation Examples Ex 1 Atomic bomb survivors study Ex 1 Atomic bomb survivors study Time zero is August 1945 – time is time since radiation exposure Time zero is August 1945 – time is time since radiation exposure Observation of subjects begins with the 1950 census Observation of subjects begins with the 1950 census People who died before 1950 are not in the sample - survival times are left truncated at 5 years People who died before 1950 are not in the sample - survival times are left truncated at 5 years

15. Left Truncation Examples Ex 2 Welsh nickel refiners Time zero is employee’s start date - all were before 1925 Time zero is employee’s start date - all were before 1925 Observation of most subjects begin in 1934, some in 1939, 1944, or 1949 Observation of most subjects begin in 1934, some in 1939, 1944, or 1949 In contrast to example 1, each subject has his own truncation time i.e. staggered entry In contrast to example 1, each subject has his own truncation time i.e. staggered entry

16. Left Truncation Example 2 cont.

17. Left Truncation The risk set just prior to an event time does not include individuals whose left truncation times exceed the given event time. Thus, any contribution to the likelihood must be conditional on the truncation limit having been exceeded. The risk set just prior to an event time does not include individuals whose left truncation times exceed the given event time. Thus, any contribution to the likelihood must be conditional on the truncation limit having been exceeded.

18. Left Truncation Please do not confuse this with left censoring Please do not confuse this with left censoring Recall – left censoring occurs when the true survival time is less than what we observed Recall – left censoring occurs when the true survival time is less than what we observed We may not know a left censored participant's exact survival time, but at least we know he/she existed; i.e. he/she did get observed We may not know a left censored participant's exact survival time, but at least we know he/she existed; i.e. he/she did get observed In a staggered entry situation, we may not know how many participants we missed. In a staggered entry situation, we may not know how many participants we missed.

19. Implications of left truncation Ex. 1 We have no way of making inferences about risk of death before 5 years We have no way of making inferences about risk of death before 5 years In a Cox model, if there are different relationships between the covariates and λ(t) when t 5, we have no way to detect this. In a Cox model, if there are different relationships between the covariates and λ(t) when t 5, we have no way to detect this.

20. Implications of staggered entry Ex. 2 Any subject in the cohort had to survive from initial employment to beginning of observation Any subject in the cohort had to survive from initial employment to beginning of observation If we ignore this in a Cox model, we will compare the covariates of subject 2 (for example) to all other subjects If we ignore this in a Cox model, we will compare the covariates of subject 2 (for example) to all other subjects This is not fair. There would be subjects in the denominator who could not possible be in the numerator This is not fair. There would be subjects in the denominator who could not possible be in the numerator

21. Solution At each event time, include in the risk set only those subjects who have not yet died and who are under observation At each event time, include in the risk set only those subjects who have not yet died and who are under observation Risk sets are not necessarily nested and can get bigger as time progresses Risk sets are not necessarily nested and can get bigger as time progresses Every inferential statement we make must be made conditional on surviving to beginning of observation Every inferential statement we make must be made conditional on surviving to beginning of observation

22. Solution – main assumption The sampling process leading to late entry into the sample does not preferentially select subjects with unusual risks or covariate values The sampling process leading to late entry into the sample does not preferentially select subjects with unusual risks or covariate values

23. ? How are the coefficient estimates from a Cox model for example 1 ( atomic bomb survivors study ) different if we correct for left truncation from those from an uncorrected model? How are the coefficient estimates from a Cox model for example 1 ( atomic bomb survivors study ) different if we correct for left truncation from those from an uncorrected model?

24. ? Answer: Answer: They do not change. No one fails until after everyone has entered. The risk sets do not change. They do not change. No one fails until after everyone has entered. The risk sets do not change.

25. ? 2 What changes in example 2 ( Welsh nickel refiners ) if instead of correcting for left truncation we change the time scale to be time since each subject’s entry into observation? What changes in example 2 ( Welsh nickel refiners ) if instead of correcting for left truncation we change the time scale to be time since each subject’s entry into observation? This is not the same as accounting for left truncation This is not the same as accounting for left truncation

26. New Time Scale

27. ? 2 What changes in example 2 ( Welsh nickel refiners ) if instead of correcting for left truncation we change the time scale to be time since each subject’s entry into observation? What changes in example 2 ( Welsh nickel refiners ) if instead of correcting for left truncation we change the time scale to be time since each subject’s entry into observation? Answer Answer The risk set compositions change. Thus, the coefficient estimates and hazard function changes. The risk set compositions change. Thus, the coefficient estimates and hazard function changes.

28. Left Truncation Coding example from SAS manual Coding example from SAS manual proc tphreg data=one; proc tphreg data=one; model t2*dead(0)=x1-x10/entry=t1; baseline out=out1 survival=s; baseline out=out1 survival=s; run; run;