Frailty Models and Other Survival Models Framingham Heart Study

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Frailty Models and Other Survival
Models(Framingham Heart Study)

• June 28, 2003
• Usha Govindarajulu
• Prof. Ralph B DAgostino,Sr.

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Outline of presentation

• Motivation for frailty research
• List of models
• Procedures for selecting models
• Results

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Cox proportional hazards model

• Cox model
• Typically used for survival modeling
• Must satisfy baseline proportional hazards
  assumption (restriction)
• What if have a mixture of hazards in the
  population or heterogeneity of the hazards?

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Hazard Plot for af data (all days)
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Intent of frailty research

• To explore issues of frailty models
• To see if a frailty model would be appropriate to
  model the Framingham Heart Study atrial
  fibrillation data and other relevant data
• To potentially derive a new frailty model

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What is Frailty?

• Frailty models are basically random effect
  survival models
• The random effect, or frailty describes
  unexplained heterogeneity, the influence of
  unobserved risk factors in the model
• The concept of frailty may be used to explain the
  unaccounted for heterogeneity which leads to the
  differential survival patterns of members of a
  population (Keiding et al 1997, Vaupel et al
  1979)

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Basic terminology (contd)

• The proportional hazards model implies that the
  hazard function is fully determined by x, the
  observed covariate vector,
• But there may be unobservable covariates not
  represented in this model, which does not include
  a residual term.
•  
• Therefore, Vaupel (1979) introduced a
  multiplicative term, z, to the hazard rate, to
  account for the unobserved population
  heterogeneity.
• h(t,x) z . h0(t)exp(xb)

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Basic terminology (contd)

• This term, z, called frailty, varies from
  individual to individual and is not observable.
• The distribution of z of the population G(z),
  must be specified.
• Since the hazard function is non-negative, z must
  be restricted to non-negative values.
• Another way to write the above model, showing how
  z fits into the error, e, is
•  
• h(t,x) h0(t)exp(xb e)
• where e log(z).

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Univariate and Multivariate

• Frailty models are either univariate or
  multivariate (Gutierrez, 2002 Hougaard, 1995)
• Univariate frailty model (per subject basis)
• Unexplained heterogeneity varies from individual
  to individual
• The frailty, the random effect, is an individual
  variable
• Multivariate frailty model (grouping factor)
• The unexplained heterogeneity is shared among
  individuals
• The frailty, is a variable common to several
  individuals
• Individuals are in units or groups that are
  chosen at random from the population
• (i.e. families, sibling groups)

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Univariate case

• Focus on univariate case since most of the data
  in our research is on a per subject basis
• Hougaard (1995) points out the impact of
  unmeasured covariates can lead to transformation
  of the hazard function and the coefficients of
  the measured covariates
• Accounting for frailty is important

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Atrial fibrillation (af) dataset

• Time to event defined as time between 1st atrial
  fibrillation event and 1st stroke
• Right censoring if event had not occurred within
  5 years, observation was censored
• Frailty was used on unique ids and assumed a
  particular distribution
• Possible risk factors age, gender, smoking
  status, LVH, sbp, treatment of hypertension,
  diabetes, MI, valvular disease, congestive heart
  failure, previous stroke

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Other survival models

• Interest came about for use as comparisons to
  standard survival models and frailty models
• Also, have tried to develop a set of guidelines
  on how to select the appropriate survival model
  for a particular dataset
• Other survival models that were applicable
• Non-proportional Weibull model
• Bailey-Makeham model

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LIST OF ALL MODELS

• Standard survival models
• Frailty models
• Non-proportional Weibull model
• Bailey-Makeham model
• New model

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How to select best survival model with or without
frailty

• Choose between semi-parametric and parametric
  model
• Test model assumptions
• Decide which variables to incorporate into the
  model
• Choose between frailty vs non-frailty model
• Performance measures

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How to select best survival model with or without
frailty (continued)

• 1)Choose between semi-parametric and parametric
  model
• Must decide if want baseline hazard to roam
  freely (semi-parametric) or baseline hazard to be
  specified (parametric)
• 2)Test model assumptions
• Semi-parametric
• Test if baseline proportional hazards assumption
  met
• Plot of Schoenfeld residuals vs time can be used
  to assumption (Grambsch and Therneau, 1994)

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How to select best survival model with or without
frailty (continued)

• 2)Test model assumptions (continued)
• Parametric models (Klein and Moeschberger, 1997)
• Test of shape parameter
• Test null hypothesis that shape parameter equals
  1 vs it does not by comparing LR of Weibull model
  to LR of exponential model
• If fail to reject H0, then assume model follows
  exponential
• Plot of log(H(tx) vs log(time)
• A straight line indicates Weibull model is a good
  fit
• If slope1, then exponential model is fine

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How to select best survival model with or without
frailty (continued)

• 2)Test model assumptions (continued)
• Parametric models (continued)
• Plot of deviance residuals vs time
• Deviance residuals are basically smoothed out
  Martingale residuals
• A plot of deviance residuals vs time provides a
  check of models adequacy
• Cox-Snell residuals
• Plot estimated cumulative hazard vs Cox-Snell
  residuals

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Example Cox-Snell residuals
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How to select best survival model with or without
frailty (continued)

• 3)Decide which variables to incorporate into the
  model
• Variable selection procedures
• i.e. backwards elimination, stepwise regression
• Statistical significance
• Clinical interest to investigators

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How to select best survival model with or without
frailty (continued)

• 4)Choose between frailty vs non-frailty
• Plot hazard function and see if frailty model is
  appropriate
• See if frailty effect is significant, if not,
  default is non-frailty model
• 5)Performance measures
• Used to judge discrimination and calibration
• c-statistic (for survival model)
• Judges discrimination by testing if predictive
  probability function produces higher predicted
  probabilities for those who develop events than
  for those who survived and did not (Nam, 2000)

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How to select best survival model with or without
frailty (continued)

• 5)Performance measures (continued)
• Used to judge discrimination and calibration
  (ctd)
• Calibration
• A chi-square statistic goodness of fit test
  extended from the generalized linear model to a
  survival model (Nam, 2000)

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A) Usual survival models

• Cox proportional hazards model
• Accelerated failure time (AFT) model
• NOTE Results for these will be presented with
  frailty models for comparison

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B) Frailty models

• Cox frailty model
• Uses Cox proportional hazards model and
  incorporates frailty as a random variable, i.e.
  z, multiplied onto the hazard
• h(t,x) z . h0(t)exp(xb)
• Modeled the frailty as gamma-distributed as
  followed in literature
• Best implemented in Splus

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B) Frailty models (continued) Cox frailty model
(results)
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B) Frailty models (continued)

• Parametric frailty model
• AFT model
• Modeled the frailty again as inverse gaussian
  distributed
• Implemented in STATA

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B) Frailty models (continued)Parametric frailty
models (Results)
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C) Non-proportional Weibull Model

• The natural logarithm of the survival time, T,
  has location, ?, and dispersion, ?, so to compute
  the probability that time to event is less than
  some time t, (Anderson et al, 19911990)
• Anderson (1991) further defined his model
  allowing the dispersion to vary as a function of
  the location

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C) Non-proportional Weibull model (continued)

• Was published by Anderson et al (19901991) and
  revised

• Is an AFT model with a varying scale parameter
• This accounts for non-proportionality

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C) Non-proportional Weibull modelResults
Note LogL(full) -1312.74 plt.05
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D) Bailey-Makeham Model

• Published in Bailey et al (1977)

• Choice of the exponential model constrains these
  parameters to be positive values and consistent
  with the model
• Hazard consists of 3 components

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D) Bailey-Makeham Model (continued)

• Does not assume proportionality of the hazards
• Can show effects of variables in the short-term
  and long-term
• Run in user-created stand alone program to
  estimate model

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D) Bailey-Makeham Model (continued) Results
LRTs of signficance for variables in
Bailey-Makeham model
Note LogL(full) -1300.431
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Performance measures c-statistic (continued)
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E) New Model

• Model incorporating frailty as a function of
  covariates
• Estimated by Monte Carlo Markov Chain
• Results and interpretations are currently being
  finalized and will be summarized in thesis
• Please correspond if want to know more about this
  model later

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References

• Anderson KM (1991). A nonproportional hazards
  Weibull accelerated failure time regression
  model. Biometrics 47, 281-288.
• Anderson KM, Odell PM., Wilson PWF, and Kannel
  WB. (1990). Cardiovascular disease risk profiles.
  AHJ 121, 293-8.
• Grambsch P and Therneau T (1994). Proportional
  hazards tests and diagnostics based on weighted
  residuals. Biometrika. 81 515-26.
• Gutierrez RG (2002). Parametric frailty and
  shared frailty survival models. The Stata
  Journal. 2(1) 22-44.

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References (continued)

• Hougaard P (1995) Frailty models for survival
  data. Lifetime Data Analysis. 1 255-273.
• Klein JP and Moeschberger ML (1997) Survival
  analysis Techniques for censored and truncated
  data. New York Springer.
• Keiding N, Andersen PK, and Klein JP (1997). The
  role of frailty models and accelerated failure
  time models in describing heterogeneity due to
  omitted covariates. Statistics in Medicine. 16
  215-224.
• Nam, B. (2000) Discrimination and calibration in
  survival analysis. Boston Univ.,(Unpublished
  thesis).