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### Statistics 262: Intermediate Biostatistics

May 18, 2004: Cox Regression III: residuals and diagnostics, repeated events

Jonathan Taylor and Kristin Cobb

Satistics 262

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

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

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

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

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

Influence diagnostics

- How would the result change if a particular observation is removed from the analysis?

Satistics 262

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

- Death (presumably) can only happen once, but many outcomes could happen twice…
- Fractures
- Heart attacks
- Pregnancy

Etc…

Satistics 262

- 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

- 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

- 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

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