Cox Regression II

1. Cox Regression II

2. Monday “Gut Check” Problem… • Write out the likelihood for the following data, with weight as a time-dependent variable:

3. SAS code for a time-dependent variable… proc phreg data=example; model time*censor(0) = weight; if time<3 then weight=w0; if time>=3 and time<6 then weight=w3; if time>=6 and time<9 then weight=w6; if time>=9 then weight=w9; run;

4. Model results • Using baseline weight: HR=2.8 • Using weight as time-changing variable: HR=9.3

5. 1. Stratification • Violations of PH assumption can be resolved by: • Adding time*covariate interaction • Adding other time-dependent version of the covariate • Stratification

6. Stratification • Different stratum are allowed to have different baseline hazard functions. • Hazard functions do not need to be parallel between different stratum. • Essentially results in a “weighted” hazard ratio being estimated: weighted over the different strata. • Useful for “nuisance” confounders (where you do not care to estimate the effect). • Assumes no interaction between the stratification variable and the main predictors.

7. ♂ ♀ ♀ ♂ ♂ ♀ Example: stratify on gender • Males: 1, 3, 4, 10+, 12, 18 (subjects 1-6) • Females: 1, 4, 5, 9+ (subjects 7-10)

8. ♂ ♀ The PL

9. 2. Using age as the time-scale in Cox Regression • Age is a common confounder in Cox Regression, since age is strongly related to death and disease. • You may control for age by adding baseline age as a covariate to the Cox model. • A better strategy for large-scale longitudinal surveys, such as NHANES, is to use age as your time-scale (rather than time-in-study). • You may additionally stratify on birth cohort to control for cohort effects.

10. Age as time-scale • The risk set becomes everyone who was at risk at a certain age rather than at a certain event time. • The risk set contains everyone who was still event-free at the age of the person who had the event. • Requires enough people at risk at all ages (such as in a large-scale, longitudinal survey).

11. The likelihood with age as time Event times: 3, 5, 7+, 12, 13+ (years-in-study) Baseline ages: 28, 25, 40, 29, 30 (years) Age at event or censoring: 31, 30, 47+, 41, 43+

12. 3. Residuals • Residuals are used to investigate the lack of fit of a model to a given subject. • For Cox regression, there’s no easy analog to the usual “observed minus predicted” residual of linear regression

13. Martingale residual • ci (1 if event, 0 if censored) minus the estimated cumulative hazard to ti (as a function of fitted model) for individual i: ci-H(ti,Xi,ßi) • E.g., for a subject who was censored at 2 months, and whose predicted cumulative hazard to 2 months was 20% • Martingale=0-.20 = -.20 • E.g., for a subject who had an event at 13 months, and whose predicted cumulative hazard to 13 months was 50%: • Martingale=1-.50 = +.50 • Gives excess failures. • Martingale residuals are not symmetrically distributed, even when the fitted model is correctly, so transform to deviance residuals...

14. Deviance Residuals • The deviance residual is a normalized transform of the martingale residual. These residuals are much more symmetrically distributed about zero. • Observations with large deviance residuals are poorly predicted by the model.

15. Deviance Residuals • Behave like residuals from ordinary linear regression • Should be symmetrically distributed around 0 and have standard deviation of 1.0. • Negative for observations with longer than expected observed survival times. • Plot deviance residuals against covariates to look for unusual patterns.

16. Deviance Residuals • In SAS, option on the output statement: Output out=outdata resdev=Varname **Cannot get diagnostics in SAS if time-dependent covariate in the model

17. Out of 628 observations, a few in the range of 3-SD is not unexpected Example: uis data Pattern looks fairly symmetric around 0.

18. What do you think this cluster represents? Example: uis data

19. Example: censored only

20. Example: had event only

21. Schoenfeld residuals • Schoenfeld (1982) proposed the first set of residuals for use with Cox regression packages • Schoenfeld D. Residuals for the proportional hazards regresssion model. Biometrika, 1982, 69(1):239-241. • Instead of a single residual for each individual, there is a separate residual for each individual for each covariate • Note: Schoenfeld residuals are not defined for censored individuals.

22. The person who died was 56; based on the fitted model, how likely is it that the person who died was 56 rather than older? Schoenfeld residuals • The Schoenfeld residual is defined as the covariate value for the individual that failed minus its expected value. (Yields residuals for each individual who failed, for each covariate). • Expected value of the covariate at time ti = a weighted-average of the covariate, weighted by the likelihood of failure for each individual in the risk set at ti.

23. Example • 5 people left in our risk set at event time=7 months: • Female 55-year old smoker • Male 45-year old non-smoker • Female 67-year old smoker • Male 58-year old smoker • Male 70-year old non-smoker The 55-year old female smoker is the one who has the event…

24. Example Based on our model, we can calculate a predicted probability of death by time 7 for each person (call it “p-hat”): • Female 55-year old smoker: p-hat=.10 • Male 45-year old non-smoker : p-hat=.05 • Female 67-year old smoker : p-hat=.30 • Male 58-year old smoker : p-hat=.20 • Male 70-year old non-smoker : p-hat=.30 Thus, the expected value for the AGE of the person who failed is: 55(.10) + 45 (.05) + 67(.30) + 58 (.20) + 70 (.30)= 60 And, the Schoenfeld residual is: 55-60 = -5

25. Example Based on our model, we can calculate a predicted probability of death by time 7 for each person (call it “p-hat”): • Female 55-year old smoker: p-hat=.10 • Male 45-year old non-smoker : p-hat=.05 • Female 67-year old smoker : p-hat=.30 • Male 58-year old smoker : p-hat=.20 • Male 70-year old non-smoker : p-hat=.30 The expected value for the GENDER of the person who failed is: 0(.10) + 1(.05) + 0(.30) + 1 (.20) + 1 (.30)= .55 And, the Schoenfeld residual is: 0-.55 = -.55

26. Schoenfeld residuals • Since the Schoenfeld residuals are, in principle, independent of time, a plot that shows a non-random pattern against time is evidence of violation of the PH assumption. • Plot Schoenfeld residuals against time to evaluate PH assumption • Regress Schoenfeld residuals against time to test for independence between residuals and time.

27. Example: no pattern with time

28. Example: violation of PH

29. Schoenfeld residuals In SAS: option on the output statement: Output out=outdata ressch= Covariate1 Covariate2 Covariate3

30. Summary of the many ways to evaluate PH assumption… 1. Examine log(-log(S(t)) plots PH assumption is supported by parallel lines and refuted by lines that cross or nearly cross Must use categorical predictors or categories of a continuous predictor 2. Include interaction with time in the model PH assumption is supported by non-significant interaction coefficient and refuted by significant interaction coefficient Retaining the interaction term in the model corrects for the violation of PH Don’t complicate your model in this way unless it’s absolutely necessary! 3. Plot Schoenfeld residuals PH assumption is supported by a random pattern with time and refuted by a non-random pattern 4. Regress Schoenfeld residuals against time to test for independence between residuals and time. PH assumption is supported by a non-significant relationship between residuals and time, and refuted by a significant relationship

31. 4. Repeated events • Death (presumably) can only happen once, but many outcomes could happen twice… • Fractures • Heart attacks • Pregnancy Etc…

32. Repeated events: 1 • Strategy 1: run a second Cox regression (among those who had a first event) starting with first event time as the origin • Repeat for third, fourth, fifth, events, etc. • Problems: increasingly smaller and smaller sample sizes.

33. Repeated events: Strategy 2 • Treat each interval as a distinct observation, such that someone who had 3 events, for example, gives 3 observations to the dataset • Major problem: dependence between the same individual

34. Strategy 3 • Stratify by individual (“fixed effects partial likelihood”) • In PROC PHREG: strata id; • Problems: • does not work well with RCT data • requires that most individuals have at least 2 events • Can only estimate coefficients for those covariates that vary across successive spells for each individual; this excludes constant personal characteristics such as age, education, gender, ethnicity, genotype

35. 5. Competing Risks

36. BMT: Related vs. Unrelated Donor

37. SAS Output Patients with related donors survive longer. 37

38. Related/Unrelated Donor is significant. • Can you say definitively to a patient: • If you find a related donor, you will have longer survival time. • What variables could be confounders? 38

39. Survival Analysis categorizes subjects Event of interest was observed Censored Competing risk was observed 39

40. Competing Risk • an event that either precludes the event of interest or alters its probability 40

41. BMT Example Interested in Time to Relapse Competing Risks (preclude or alter probability of relapse) Non-relapse mortality Graft-vs-host disease (GVHD) 41

42. Who failed from the event of interest? Yes Maybe No • Common Pitfall: treating competing risks as censoring • Treats nos as maybes • Puts them partially in the numerator of occurrence when they shouldn’t be there • Thus overestimates risk (underestimates S) Event of interest was observed Censored Competing risk was observed 42

43. What to do instead KM estimate of event free survival (EFS) Cumulative Incidence Analysis 43

44. Event-Free Survival In cancer, often Progression-Free Survival (PFS) Treats competing risks as events Can use KM For each subject, the first event to occur “Survival” implies death is considered an event BMT: first of relapse, GVHD or death Is this of interest? May not be, e.g., Local progression and metastasis 44

45. Cumulative Incidence Analysis Separates competing risks from event of interest If no competing risks, equivalent to KM Estimates occurrence probability: F(t) = 1 – S(t) Each event goes into one bin (event type) 45

46. BMT CumulativeIncidence Curves GVHD Relapse Death

47. 6. Considerations when analyzing data from an RCT…

48. Intention-to-Treat Analysis Intention-to-treat analysis: compare outcomes according to the groups to which subjects were initially assigned, regardless of which intervention they actually received. Evaluates treatment effectiveness rather than treatment efficacy

49. Why intention to treat? • Non-intention-to-treat analyses lose the benefits of randomization, as the groups may no longer be balanced with regards to factors that influence the outcome. • Intention-to-treat analysis simulates “real life,” where patients often don’t adhere perfectly to treatment or may discontinue treatment altogether.

50. Both women on placebo and women on active treatment discontinued study medications. Women on placebo “dropped in” to treatment because their regular doctors put them on hormones (dogma= “hormones are good”). Women on treatment “dropped in” to treatment because their doctors took them off study drugs and put them on hormones to insure they were on hormones and not placebo. Drop-ins and Drop-outs: example, WHI