1. Statistics 262: Intermediate Biostatistics
May 18, 2004: Cox Regression III: residuals and diagnostics, repeated events
Jonathan Taylor and Kristin Cobb
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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
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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…
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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.
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5. Deviance Residuals In SAS, option on the output statement:
Ouput out=outdata resdev=
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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):
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 )
Note: Schoenfeld residuals are not defined for censored individuals.
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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
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8. Schoenfeld residuals In SAS: option on the output statement: ressch=
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9. Influence diagnostics
How would the result change if a particular observation is removed from the analysis?
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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
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11. Influence statistics In SAS: option on the output statement:
ld= dfbeta=
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12. What about repeated events?
Death (presumably) can only happen once, but many outcomes could happen twice…
Fractures
Heart attacks
Pregnancy
Etc…
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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.
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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
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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
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