ABSTRACT
Frailty
models are very useful for analyzing correlated survival data, when observations
are clustered into groups or for recurrent events. The aim of this research is
to present the new version of an R package called frailty pack. This package
allows to fit Cox models and four types of frailty models (shared, nested,
joint, and additive) that could be useful for several issues within biomedical
research. It is well adapted to the analysis of recurrent events such as cancer
relapses and/or terminal events (death or lost to follow-up). The approach uses
maximum penalized likelihood estimation. Right-censored or left-truncated data
are considered. It also allows stratification and time-dependent covariates
during analysis. A frailty model is a random effects model for time variables,
where the random effect (the frailty) has a multiplicative effect on the
hazard. It can be used for univariate (independent) failure times, i.e. to
describe the influence of unobserved covariates in a proportional hazards
model. More interesting, however, is to consider multivariate (dependent)
failure times generated as conditionally independent times given the frailty.
This approach can be used both for survival times for individuals, like twins
or family members, and for repeated events for the same individual. The
standard assumption is to use a gamma distribution for the frailty, but this is
a restriction that implies that the dependence is most important for late
events. More generally, the distribution can be stable, inverse Gaussian, or
follow a power variance function exponential family. Theoretically, large
differences are seen between the choices. In practice, using the largest model
makes it possible to allow for more general dependence structures, without
making the formulas too complicated.
KEYWORDS:
Frailty models, R, penalized likelihood, cross-validation, correlated survival
data, splines, and hazard functions.
TABLE OF CONTENTS
Title
page
Certification i
Dedication ii
Acknowledgment
iii
Table
of contents iv
CHAPTER ONE
1.1 Introduction 1
1.2 Objective 2
CHAPTER TWO
2.1 Literature Review 3
2.2 Mathematical
Definition of Frailty Models 3
2.3 Cox Model 4
2.4 Cox
Proportional Hazards Model with Random Effects 7
2.5 The
shared frailty model 13
2.6 Nested frailty model 17
2.7 Nested
frailty model and inference 17
2.8 Joint
frailty model 20
2.9 Parametric
Survival Models 25
2.10 Methodology for Fitting Frailty Models 26
2.11 Meaning of some of the Common Arguments for
fitting a model
using R Package 30
2.12 Methodology on how to use the above Arguments
in RFor
Fitting Frailty
Models (e.g. Cox and Shared Frailty Models) 33
CHAPTER THREE
3.1 Frailtypack Argumentson survival data 39
CHAPTER FOUR
4.1 Analysis of the Survival Data in Table 3.1
Using Cox and Shared
frailty Models with the
help of their common arguments in
chapter 2.11 on R Package 56
4.2 Analysis of Survival Data in Table 3.1 using
Cox Model on R 59
4.3
Interpretation: 61
4.4 Analysis of Survival Data in Table 3.1
using Shared model on R: 63
4.5 Interpretation: 65
4.6 Estimate of Hazard Ratios: For the Cox
Model 67
4.7 Survival or Hazard Baseline function for
Cox model 67
4.8 Baseline Survival Function of Cox Model 69
4.9 Estimate of Hazard Ratios: for the shared
model 69
4.10 Survival or Hazard Baseline function for
shared model 70
CHAPTER FIVE
5.0 Summary and Conclusion 76
REFERENCES 78
CHAPTER ONE
1.1
INTRODUCTION
Frailty
models (Duchateau and Janssen 2008; Hougaard 2000; Wienke 2010; Hanagal 2011)are
extensions of the Cox proportional hazards model (Cox 1972) which is the most
popular model in survival analysis. In many clinical applications, the study
population needs to be considered as a heterogeneous sample or as a cluster of
homogeneous groups of individuals such as families or geographical areas.
Sometimes, due to lack of knowledge or for economicalreasons, some covariates
related to the event of interest are not measured. The frailty approach is a
statistical modelling method which aims to account for the heterogeneity caused
by unmeasured covariates. It does so by adding random effects which act
multiplicatively on thehazard function. Frailtypack is an R package (R
Development Core Team 2012) which allows to fit four types of frailty models,
for left-truncated and right-censored data, adapted to most survival analysis
issues. The aim of this paper is to present the new version of the R package frailtypack,
which is available from the Comprehensive R Archive Network at http://CRAN.R-project.org/package=frailtypack,
and the various new models proposed. It depends on the R survival package
(Therneau 2012). The initial version of this package (Rondeau andGonzalez 2005)
was proposed for a simple shared frailty model, and was developed for more
general frailty models (Rondeau et al. 2012). The shared frailty model (Rondeau
et al. 2003) can be used, when observations are supposed to be clustered into
groups. The nested frailty model (Rondeau et al. 2006) is most appropriate,
when there are two levels of hierarchical clustering.
The
frailty models discussed in recent literature present several drawbacks.Their
convergence is too slow, they do not provide standard errors for the variance
estimate ofthe random effects and they cannot estimate smooth hazard function. Frailtypackuse
a non-parametric penalized likelihood estimation, and the smooth estimation of
the baseline hazardfunctions is provided by using an approximation by splines. Frailtypackwas
first written inFortran 77 and was implemented for the statistical software R.
1.2
OBJECTIVE
The
main objective is to present the Analysis of correlated Survival Data by making
use of Frailty Models (e.g. Cox, shared etc.) and also to know survival rate of
patients suffering from cancer, using Penalized Likelihood Estimation or
Parametrical Estimation method with the help of R PACKAGE.
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