2017 | Book

# Statistical Modelling of Survival Data with Random Effects

## H-Likelihood Approach

Authors: Il Do Ha, Jong-Hyeon Jeong, Youngjo Lee

Publisher: Springer Singapore

Book Series : Statistics for Biology and Health

2017 | Book

Authors: Il Do Ha, Jong-Hyeon Jeong, Youngjo Lee

Publisher: Springer Singapore

Book Series : Statistics for Biology and Health

This book provides a groundbreaking introduction to the likelihood inference for correlated survival data via the hierarchical (or h-) likelihood in order to obtain the (marginal) likelihood and to address the computational difficulties in inferences and extensions. The approach presented in the book overcomes shortcomings in the traditional likelihood-based methods for clustered survival data such as intractable integration. The text includes technical materials such as derivations and proofs in each chapter, as well as recently developed software programs in R (“frailtyHL”), while the real-world data examples together with an R package, “frailtyHL” in CRAN, provide readers with useful hands-on tools. Reviewing new developments since the introduction of the h-likelihood to survival analysis (methods for interval estimation of the individual frailty and for variable selection of the fixed effects in the general class of frailty models) and guiding future directions, the book is of interest to researchers in medical and genetics fields, graduate students, and PhD (bio) statisticians.

Advertisement

Abstract

The likelihood, introduced by R.A Fisher (PTRS 222: 309–368, 1922), plays an important role in statistical inference about fixed unknowns, namely parameters. The beauty of the likelihood is that once the statistical model is specified parametrically or nonparametrically, the associated inference procedures for the parameters of interest are straightforward.

Abstract

Let T be time-to-event (failure time), which is a nonnegative random variable. In medicine, a typical example is time from the onset of a condition or an initiation of treatment to death. In studies of reliability of products (or components), time to failure of light bulbs, for example, is often of interest. Rather than using such specific terms, economists refer to durations between events (e.g., duration of unemployment). The distribution of failure time is usually non-normal and skewed.

Abstract

In this chapter, we introduce an h-likelihood approach to the general class of statistical models with random effects. Consider a linear mixed model (LMM), for \(i=1, \ldots , q\) and \(j=1, \ldots , n_{i}\), \(y_{ij}=x_{ij}^{T}{\varvec{\beta }}+v_{i}+e_{ij}\), where \(y_{ij}\) is an observed random variable (response), \( x_{ij}=(x_{ij1}, \ldots , x_{ijp})^{T}\) is a vector of covariates, \({\varvec{ \beta }}\) is a vector of fixed effects, \(v_{i}\sim N(0,\alpha )\) is an i.i.d. random variable for the random effects, \(e_{ij}\sim N(0,\phi )\) is an i.i.d random error or measurement error, and \(v_{i}\) and \(e_{ij}\) are independent. Parameters \(\phi \) and \({\varvec{\alpha }}\) are the variance components. In this model, there are two types of unknowns; the fixed unknowns \(\theta =(\beta ,\phi ,\alpha )^{T} \) and the random unknowns \(v=(v_{1}, \ldots , v_{q})^{T}\).

Abstract

The concept of frailty was first introduced by Vaupel et al. (1979) (Demography 16:439–454, 1979) to account for the impact of individual heterogeneity in univariate (independent) survival data. In this chapter, we introduce the frailty model, an extension of the Cox PH model, for analyzing correlated survival data. The frailty is modeled by an unobserved random effect acting multiplicatively on the individual hazard rate to describe the individual heterogeneity and the correlation (dependency) among survival data from the same subject or cluster (Clayton 1978 (Clayton DG in Biometrika 65:141–151, 1978); Hougaard 2000 (Hougaard P in Analysis of multivariate survival data. Springer, New York, 2000); Duchateau and Janssen 2008 (Duchateau L, Janssen P in The frailty model. Springer, New York, 2008)). Even if heterogeneity and correlation are different concepts, they can both be modeled by frailties.

Abstract

Time-to-event data (recurrent or multiple event times) are often observed on the same subject (or cluster), and the frailty models are useful for analysis of such data. In practice, the multicomponent frailty models are of interest with complicated frailty structures, nested or crossed.

Abstract

Competing risks data arise when an occurrence of an event precludes other types of events from being observed. In this chapter, we extend the h-likelihood inference procedures for frailty models to competing risks models. We first review existing methods for competing risk models without the frailty.

Abstract

Including only the relevant variables in the model is crucial in statistical inference, improving the quality of estimation, prediction, and interpretation. If there exist many potential variables with equal status, i.e., no prior preference among them, then having as few variables as possible in the model would often facilitate clearer interpretation. When there are many potential variables, over-fitting can become a serious problem. However, missing relevant variables would be also undesirable.

Abstract

The frailty model accounts for dependence between survival times, by including a random effect acting multiplicatively on the individual hazard rate.

Abstract

In this chapter, through the h-likelihood approach, we study the joint model, for which the response variables of interest would involve repeated measurements over time on the same subject as well as time to an event of interest with or without competing risks.

Abstract

We have previously presented the h-likelihood procedures for the analysis of survival data under competing risks. There are still many unresolved problems in this area. In this chapter, we present some further topics to highlight that the h-likelihood approach can be extended to more complex multistate survival data. We deal with competing-risks data with missing causes of failure and the semi-competing-risks data.