Statistics 262: Intermediate Biostatistics

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Statistics 262 Intermediate Biostatistics
May 18, 2004 Cox Regression III residuals and
diagnostics, repeated events

• Jonathan Taylor and Kristin Cobb

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Residuals

• Residuals are used to investigate the lack of fit
  of a model to a given subject.
• For Cox regression, theres 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 outoutdata 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 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 individuals
  likelihood of dying at ti).

• Plot Schoenfeld residuals against time to
  evaluate PH assumption

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Schoenfeld residuals

• In 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. Whats 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 statistics

• In SAS
• 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
• Etc

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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 eventsStrategy 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