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Statistics 262: Intermediate Biostatistics. May 18, 2004: Cox Regression III: residuals and diagnostics, repeated events. Jonathan Taylor and Kristin Cobb. Residuals. Residuals are used to investigate the lack of fit of a model to a given subject.

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statistics 262 intermediate biostatistics

Statistics 262: Intermediate Biostatistics

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

Jonathan Taylor and Kristin Cobb

Satistics 262

  • 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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deviance residuals
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:


See Hosmer and Lemeshow for more discussion…

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deviance residuals4
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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deviance residuals5
Deviance Residuals
  • In SAS, option on the output statement:

Ouput out=outdata resdev=

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schoenfeld residuals
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.

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schoenfeld residuals7
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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schoenfeld residuals8
Schoenfeld residuals


option on the output statement:


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influence diagnostics
Influence diagnostics
  • How would the result change if a particular observation is removed from the analysis?

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influence statistics
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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influence statistics11
Influence statistics


option on the output statement:

ld= dfbeta=

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What about repeated events?

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


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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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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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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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