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# FRAILTY PACK: AN R-PACKAGE FOR THE ANALYSIS OF CORRELATED SURVIVAL DATA WITH FRAILTY MODELS USING PENALIZED LIKELIHOOD ESTIMATION OR PARAMETRICAL ESTIMATION

## Content

**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.

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**TABLE OF CONTENTS**

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Title page

Certification i

Dedication ii

Acknowledgment iii

Table of contents iv

Abstract vi

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**CHAPTER ONE**

1.1 Introduction 1

1.2 Objective 2

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**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

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**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

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**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.