Time-dependent%20covariates%20and%20further%20remarks%20on%20likelihood%20construction

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Time-dependent covariates and further remarks on
likelihood construction

• Presenter Li,Yin
• Nov. 24

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Tests of proportionality for cox regrssion

• 1 Kaplan-Meier curves
• Works best for time fixed covariates
  with few levels.
• 2 Including time dependent covariates in the cox
  model
• 3 Tests and graphs based on the Schoenfeld
  residuals

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Data set for example

• The goal of the UIS  data is to model time until
  return to drug use .
• The patients were randomly assigned to two
  different sites (site0 is site A and site1 is
  site B). 
• Herco indicates heroine or cocaine use in the
  past three months (herco1 indicates heroine and
  cocaine use, herco2 indicates either heroine or
  cocaine use and herco3 indicates neither heroine
  nor cocaine use).
• ndrugtx indicates the number of previous drug
  treatments. 

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Cox regression model for uis_small
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Kaplan-Meier curves

• It is very easy to create the graphs in SAS using
  proc lifetest.  The plot option in the model
  statement lets you specify both the survival
  function versus time as well as the
  log(-log(survival) versus log(time).

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Tests and graphs based on the Schoenfeld
residuals

• Testing the time dependent covariates is
  equivalent to testing for a non-zero slope in a
  generalized linear regression of the scaled
  Schoenfeld residuals on functions of time.  A
  non-zero slope is an indication of a violation of
  the proportional hazard assumption.

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Including time dependent covariates in the cox
model

• Generate the time dependent covariates by
  creating interactions of the predictors and a
  function of survival time and include in the
  model.  If any of the time dependent covariates
  are significant then those predictors are not
  proportional.

proc phreg datauis model timecensor(0) age race treat site agesite aget racet treatt sitet aget agelog(time) racet racelog(time) treatt treatlog(time) sitet sitelog(time) proportionality_test test aget, racet, treat, sitet run
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Including time dependent covariates in the cox
model (continued)
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When the proportional hazard assumption is
violated

• If one of the predictors was not proportional
  there are various solutions to consider.
• We can change from using a semi-parametric Cox
  regression model to using a parametric regression
  model.
• Another solution is to include the time-dependent
  variable for the non-proportional predictors.
• Finally, we can use a model where we stratify on
  the non-proportional predictors.

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Two types of time dependent covariates

• External
• Internal
• A covariate that is not external is called
  internal.
• Internal covariates typically arise as
  time-dependent measurements taken on an
  individual study subject, the path of which is
  affected by the survival status.

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Examples

• External covariates
• 1st type fixed or time independent covariates
• 2nd type defined covariates
• eg. Stress factors under control of
  the experimenter that is to be varied
  in a predetermined way
• eg. Age of an individual in a trial of
  long duration
• 3rd type ancillary covariates
• eg. Measures of pollutions as a
  predictor for the frequency of asthma attacks.

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Examples

• Internal covariates
• Eg. Measures of a patients general condition
  take values of 0,1,2,3,4. 0 is assigned to z(t)
  for dead and 4 is for no disease and 1,2,3
  represent levels of decreasing disability.
• Eg. Measures of immune status such as white
  blood count as a predictor for examining the
  effect of immunotherapy on the failure rate in
  cancer.

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Likelihood construction

• Problem setup
• n individuals start on test at t0
• The risk of failure at time t
• where x is a vector of fixed basic
  covariates measured in advance and theta is a
  vector of parameters.
• The data for the ith individual are

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• Under random censoring

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• For any tgt0, let the history be

Then, the likelihood can be constructed as a
product of the conditional terms.
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• Let be the set of
  labels associated with the individuals failing in
  ,and is the set of labels
  associated with the individuals censored in
  .

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• The first factor on the right side arises from
  the failure information

• The remaining factor arises from the censoring
  information

• If this term depends on the , the
  censoring mechanism is said to be informative,
  and otherwise noninformative.

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Likelihood construction for internal covariates

• The probability contribution of the interval
  is written as

Where
has failure, censoring and
covariates information up to time t and
has only the covariates
information.