Survival analysis models

The study of events involving an element of time has a long and important history in statistical research and practice. Survival analysis is a collection of statistical procedures for the analysis of data, where the response of interest is the time until an event occurs. Though such events may refer to any designated experience of interest, they are generally referred to as ‘failures’, whereas the time to their occurrences is referred to as ‘lifetime’ or ‘failure time’. Examples of failure times include the lifetimes of machine components in industrial reliability, the durations of strikes or periods of unemployment in economics, the times taken by subjects to complete specified tasks in psychological experimentation and the survival or remission times of patients in clinical trials.Generally speaking, the estimation, prediction or otimization of survival probabilities or life expectancies has become an issue of considerable interest in many different fields of human life and activity. Therefore, survival analysis has developed into an important tool for researchers in many areas, particularly, those involving biomedical studies and industrial life testing. This dissertation is occupied with continuous lifetime models. In this context, the first chapter, provides a short overview on the basic concepts o f survival analysis. Distribution representations of the time to failure are given when the life lengths are measured by a continuous nonnegative random variable and special emphasis is placed on the hazard function due to its intuitive appeal. In the sequel, several univariate popular lifetime distributions are presented and two specialized models designed to describe more complicated failure patterns (competing risks and frailty models) are briefly examined. The basic concepts of survival analysis for bivariate populations are considered next and the most popular bivariate lifetime distributions are reported. In the second chapter, various statistical properties and reliability aspects of a two parameter distribution with decreasing and increasing failure rates are explored. The model includes the Exponential-Geometric distribution (Adamidis and Loukas, 1988) as a special case. Characterizations are given and the estimation of parameters is studied by the method of maximum likelihood. An EM algorithm (Dempster et al., 1977) is proposed for computing the estimates and expressions for their asymptotic variances and covariances are derived. Numerical examples based on real data are shown, to illustrate the applicability of the new model. The results of this chapter are included in Adamidis et al. (2005).Though the most popular lifetime models are those with monotone hazard rates, when the entire life span of a biological entity or a manufactured item is under consideration, high initial and eventual failure rates are frequently observed, indicating a bathtub shaped failure rate (Gaver and Acar, 1979). Also, situations involving a high occurrence of early ‘failures’ are best modeled by distributions with upturned bathtub shaped hazard rates (Chhikara and Folks, 1977). In the third chapter, a three parameter lifetime distribution with increasing, decreasing, bathtub and upside down bathtub shaped failure rates is introduced. The new model includes the Weibull distribution as a special case. A motivation for its derivation is given using a competing risks interpretation when restricting its parametric space. Several of its statistical properties and reliability aspects are explored and the estimation of the parameters is studied using the standard maximum likelihood procedures. Applications of the model to real data are also included. The results of this chapter are included in Dimitrakopoulou et al. (2006 b). In the forth chapter, bivariate extensions of the model introduced in the second chapter are presented, along with the physical considerations leading to their derivation. Marginal and conditional distributions are obtained and their corresponding survival and hazard functions are calculated. The dependence in the proposed bivariate distributions is evaluated by means of the Pearson correlation coefficient. The models presented so far, implicitly assume that the population under study is homogeneous, an assumption which is often unrealistic in practice. However, heterogeneity is not only of interest in its own right but actually distorts what is observed. One o f the ways of assessing the impact of heterogeneity in mortality studies is via the concept of frailty introduced by Vaupel et al. (1979). When the multiplicative frailty model is underconsideration (e.g. Hougaard, 1984), the assumption of a gamma distributed frailty leads to the so called gamma frailty model. Chapter five, is devoted to exploiting some aspects of its relevant distribution theory. Failure rate characterizations are obtained and bounds on the survival function are constructed. Moreover, it is shown that the model can serve as a method of constructing lifetime models or extending existing ones (by adding a parameter in the sense of Marsall and Olkin, (1997)). Therefore, the investigation of its reliability aspects, provides a unified approach in studying lifetime distributions in a reliability context and a way of assessing the impact of the ‘average’ individual survival capacity - in the presence of heterogeneity - on what is actually observed. The results of this chapter are included in Dimitrakopoulou et al. (2006 a)