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Frailty Models and Other Survival Models Framingham Heart Study

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Title: Frailty Models and Other Survival Models Framingham Heart Study

1
Frailty Models and Other Survival
Models (Framingham Heart Study)
• June 28, 2003
• Usha Govindarajulu
• Prof. Ralph B DAgostino,Sr.

2
Outline of presentation
• Motivation for frailty research
• List of models
• Procedures for selecting models
• Results

3
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?

4
Hazard Plot for af data (all days)
5
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

6
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)

7
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)

8
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).

9
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)

10
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

11
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

12
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

13
LIST OF ALL MODELS
• Standard survival models
• Frailty models
• Non-proportional Weibull model
• Bailey-Makeham model
• New model

14
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

15
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)

16
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

17
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
• Cox-Snell residuals
• Plot estimated cumulative hazard vs Cox-Snell
residuals

18
Example Cox-Snell residuals
19
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

20
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)

21
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)

22
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

23
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

24
B) Frailty models (continued) Cox frailty model
(results)
25
B) Frailty models (continued)
• Parametric frailty model
• AFT model
• Modeled the frailty again as inverse gaussian
distributed
• Implemented in STATA

26
B) Frailty models (continued) Parametric frailty
models (Results)
27
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

28
C) Non-proportional Weibull model (continued)
revised
• Is an AFT model with a varying scale parameter
• This accounts for non-proportionality

29
C) Non-proportional Weibull model Results
Note LogL(full) -1312.74 plt.05
30
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

31
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

32
D) Bailey-Makeham Model (continued) Results
LRTs of signficance for variables in
Bailey-Makeham model
Note LogL(full) -1300.431
33
Performance measures c-statistic (continued)
34
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
model later

35
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.

36
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).