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Statistics 262: Intermediate Biostatistics

Jonathan Taylor and Kristin Cobb

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

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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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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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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 Residuals In SAS, option on the output

statement:Ouput out=outdata resdev=

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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 residualsWhere 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).

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Plot Schoenfeld residuals against time to evaluate PH assumption

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Schoenfeld residualsIn SAS: option on the output statement:ressch=

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Influence diagnostics How would the result change if a

particular observation is removed from the analysis?

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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 statisticsIn SAS: option on the output statement:ld= dfbeta=

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

What about repeated events?

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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.

Repeated events: 1

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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

Repeated events:Strategy 2

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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

Strategy 3