Lecture 4: Cox PH model continued

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Lecture 4 Cox PH model continued Nested case
control studies

• 513-668 Statistical Models in Epidemiology

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Outline

• Time dependent explanatory variables
• Impact on computation
• Switching of exposed/unexposed status
• Multiple time-scales
• Continuous/discrete
• Nested case control studies
• Sampling risk sets
• Matching and counter matching

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Time-dependent explanatory variables

• If the follow-up period is sufficiently long,
  most explanatory variables are likely to change.
  ex. changing of exposure to hazardous chemicals,
  changing of treatments for a chronic disease

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Time-dependent explanatory variables Computation

• ?i in log(?ti / Sj in riskset ?tj) is a
  function of time
• Calculation of denominator at t1 no longer
  simple update of value at t
• One solution is to use a nested case control
  design, i.e. sample from the risk set

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Time-dependent explanatory variables changing
exposure

• Divide an individuals person-time according to
  exposed/nonexposed and fit CoxPH model,
• i.e. ?it ?Ct ?i does not change
• This simplification is based on strong
  assumptions 1) reason for change in exposure is
  random, 2) or at least random within strata
  defined by a covariate, and model for failure
  adjusts for this covariate
• - heart transplant ex

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Time-dependent explanatory variables Multiple
time scales

• Not an issue if the study is of a short duration,
  ex. use age at start of study as a covariate.
• Longer study - Fig 31.2
• ?it ?Ct ?i becomes ?it ?Ct ?(age i)
• discussion of which time scale is to be included
  in the baseline rate - follow-up time or age? ex.
  31.3?

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Time-dependent explanatory variables Dependence
between multiple time scales

• Different time scales (ex. Time since exposure
  and age) are the same variable with different
  origins
• Cannot increase one time scale by 1 unit while
  keeping other constant

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Time-dependent explanatory variables multiple
time scales - continuous

• log(?it) log(?Ct) (age at diagnosis) log ?i
• Therefore, log(?itage at d61) - log(?itage at
  d47)
• (61-47) log ?i 14 log ?i
• log(?it) log(?Ct) (age) log ?i
• Therefore, log(?itagex14) - log(?itagex)
  14 log ?i
• Thus linear effects of age and age at d not
  distinguishable. However, non-linear effects are

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Time-dependent explanatory variables multiple
time scales - discrete

• Might falsely appear that you can estimate the
  linear effect see Fig 31.3
• Essentially dont use discrete measures of time
  when you have multiple time scales
• Age-period-cohort problem

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Nested case control study

• Sample controls from the cohort rather than use
  all controls
• save labour. Ex. Coding diary records
• collect additional covariates on a subset.
  Two-stage case-control design
• avoid computational burden due to time-depdt
  covariates
• Sample drawn from risk-set of each case
• Predates CoxPH model

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Sampling risk sets

• A subject can be a control in multiple risk sets
  and can eventually be a case
• If a subject is selected as a control in every
  risk set an adjustment must be made for the
  dependence between risk sets. Case-cohort
  design?
• SD of sample/SD of entire riskset sqrt(11/m)

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Matching/Counter-matching

• Matching Riskset reduced to set of matched
  individuals at risk
• Increases precision of estimate for exposure
• Cant estimate effect of matching variable
• Counter-matching Exposure status available for
  all, but not confounders
• Measure confounders in sample selected for NC
  study.
• How do you use the info on exposure? 5050 split
  most efficient. When n1, Case E matched with
  Control UE etc
• This would improve efficiency by increasing of
  discordant pairs. Application Pharmacoepidemiolog
  y
• Likelihood adjusted to reflect change in E
  distribution
• Ex 33.2 and 33.3