Statistics 262: Intermediate Biostatistics

1. Statistics 262: Intermediate Biostatistics May 18, 2004: Cox Regression III: residuals and diagnostics, repeated events Jonathan Taylor and Kristin Cobb Satistics 262

2. 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 Satistics 262

3. Deviance Residuals • Deviance residuals are based on martingale residuals: 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) See Hosmer and Lemeshow for more discussion… Satistics 262

4. 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. Satistics 262

5. Deviance Residuals • In SAS, option on the output statement: Ouput out=outdata resdev= Satistics 262

6. 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 • Based on the individual contributions to the derivative of the log partial likelihood (see chapter 6 in Hosmer and Lemeshow for more math details, p.198-199) • Note: Schoenfeld residuals are not defined for censored individuals. Satistics 262

7. Schoenfeld residuals Where K is the covariate of interest, the Schoenfeld residual is the covariate-value, Xik, for the person (i) who actually died at time ti minus the expected value of the covariate for the risk set at ti (=a weighted-average of the covariate, weighted by each individual’s likelihood of dying at ti). Plot Schoenfeld residuals against time to evaluate PH assumption Satistics 262

8. Schoenfeld residuals In SAS: option on the output statement: ressch= Satistics 262

9. Influence diagnostics • How would the result change if a particular observation is removed from the analysis? Satistics 262

10. Influence statistics • Likelihood displacement (ld): measures influence of removing one individual on the model as a whole. What’s the change in the likelihood when this individual is omitted? • DFBETA-how much each coefficient will change by removal of a single observation • negative DFBETA indicates coefficient increases when the observation is removed Satistics 262

11. Influence statistics In SAS: option on the output statement: ld= dfbeta= Satistics 262

12. What about repeated events? • Death (presumably) can only happen once, but many outcomes could happen twice… • Fractures • Heart attacks • Pregnancy Etc… Satistics 262

13. 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. Satistics 262

14. 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 Satistics 262

15. Strategy 3 • Stratify by individual (“fixed effects partial likelihood”) • In PROC PHREG: strata id; • Problems: • does not work well with RCT data, however • 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 Satistics 262