Modelling and Analysis of Bivariate Lifetime Data using Reversed Hazard Rates

Department of Statistics, Cochin University of Science and Technology

Regression models for bivariate survival data

Multivariate lifetime data arise in various forms including recurrent event data when individuals are followed to observe the sequence of occurrences of a certain type of event; correlated lifetime when an individual is followed for the occurrence of two or more types of events, or when distinct individuals have dependent event times. In most studies there are covariates such as treatments, group indicators, individual characteristics, or environmental conditions, whose relationship to lifetime is of interest. This leads to a consideration of regression models.The well known Cox proportional hazards model and its variations, using the marginal hazard functions employed for the analysis of multivariate survival data in literature are not sufficient to explain the complete dependence structure of pair of lifetimes on the covariate vector. Motivated by this, in Chapter 2, we introduced a bivariate proportional hazards model using vector hazard function of Johnson and Kotz (1975), in which the covariates under study have different effect on two components of the vector hazard function. The proposed model is useful in real life situations to study the dependence structure of pair of lifetimes on the covariate vector . The well known partial likelihood approach is used for the estimation of parameter vectors. We then introduced a bivariate proportional hazards model for gap times of recurrent events in Chapter 3. The model incorporates both marginal and joint dependence of the distribution of gap times on the covariate vector . In many fields of application, mean residual life function is considered superior concept than the hazard function. Motivated by this, in Chapter 4, we considered a new semi-parametric model, bivariate proportional mean residual life time model, to assess the relationship between mean residual life and covariates for gap time of recurrent events. The counting process approach is used for the inference procedures of the gap time of recurrent events. In many survival studies, the distribution of lifetime may depend on the distribution of censoring time. In Chapter 5, we introduced a proportional hazards model for duration times and developed inference procedures under dependent (informative) censoring. In Chapter 6, we introduced a bivariate proportional hazards model for competing risks data under right censoring. The asymptotic properties of the estimators of the parameters of different models developed in previous chapters, were studied. The proposed models were applied to various real life situations.

‘Reliability Concepts in Quantile-based Analysis of Lifetime Data

Reliability analysis is a well established branch of statistics that deals with the statistical study of different aspects of lifetimes of a system of components. As we pointed out earlier that major part of the theory and applications in connection with reliability analysis were discussed based on the measures in terms of distribution function. In the beginning chapters of the thesis, we have described some attractive features of quantile functions and the relevance of its use in reliability analysis. Motivated by the works of Parzen (1979), Freimer et al. (1988) and Gilchrist (2000), who indicated the scope of quantile functions in reliability analysis and as a follow up of the systematic study in this connection by Nair and Sankaran (2009), in the present work we tried to extend their ideas to develop necessary theoretical framework for lifetime data analysis. In Chapter 1, we have given the relevance and scope of the study and a brief outline of the work we have carried out. Chapter 2 of this thesis is devoted to the presentation of various concepts and their brief reviews, which were useful for the discussions in the subsequent chapters .In the introduction of Chapter 4, we have pointed out the role of ageing concepts in reliability analysis and in identifying life distributions .In Chapter 6, we have studied the first two L-moments of residual life and their relevance in various applications of reliability analysis. We have shown that the first L-moment of residual function is equivalent to the vitality function, which have been widely discussed in the literature .In Chapter 7, we have defined percentile residual life in reversed time (RPRL) and derived its relationship with reversed hazard rate (RHR). We have discussed the characterization problem of RPRL and demonstrated with an example that the RPRL for given does not determine the distribution uniquely

Nonparametric Estimation of Survivor Function in Bivariate Competing Risk Models’

So far, in the bivariate set up, the analysis of lifetime (failure time) data with multiple causes of failure is done by treating each cause of failure separately. with failures from other causes considered as independent censoring. This approach is unrealistic in many situations. For example, in the analysis of mortality data on married couples one would be interested to compare the hazards for the same cause of death as well as to check whether death due to one cause is more important for the partners’ risk of death from other causes. In reliability analysis. one often has systems with more than one component and many systems. subsystems and components have more than one cause of failure. Design of high-reliability systems generally requires that the individual system components have extremely high reliability even after long periods of time. Knowledge of the failure behaviour of a component can lead to savings in its cost of production and maintenance and. in some cases, to the preservation of human life. For the purpose of improving reliability. it is necessary to identify the cause of failure down to the component level. By treating each cause of failure separately with failures from other causes considered as independent censoring, the analysis of lifetime data would be incomplete. Motivated by this. we introduce a new approach for the analysis of bivariate competing risk data using the bivariate vector hazard rate of Johnson and Kotz (1975).

Some concepts and models usefulninthe analysis of discrete life time data

This thesis entitled Reliability Modelling and Analysis in Discrete time Some Concepts and Models Useful in the Analysis of discrete life time data.The present study consists of five chapters. In Chapter II we take up the derivation of some general results useful in reliability modelling that involves two component mixtures. Expression for the failure rate, mean residual life and second moment of residual life of the mixture distributions in terms of the corresponding quantities in the component distributions are investigated. Some applications of these results are also pointed out. The role of the geometric,Waring and negative hypergeometric distributions as models of life lengths in the discrete time domain has been discussed already. While describing various reliability characteristics, it was found that they can be often considered as a class. The applicability of these models in single populations naturally extends to the case of populations composed of sub-populations making mixtures of these distributions worth investigating. Accordingly the general properties, various reliability characteristics and characterizations of these models are discussed in chapter III. Inference of parameters in mixture distribution is usually a difficult problem because the mass function of the mixture is a linear function of the component masses that makes manipulation of the likelihood equations, leastsquare function etc and the resulting computations.very difficult. We show that one of our characterizations help in inferring the parameters of the geometric mixture without involving computational hazards. As mentioned in the review of results in the previous sections, partial moments were not studied extensively in literature especially in the case of discrete distributions. Chapters IV and V deal with descending and ascending partial factorial moments. Apart from studying their properties, we prove characterizations of distributions by functional forms of partial moments and establish recurrence relations between successive moments for some well known families. It is further demonstrated that partial moments are equally efficient and convenient compared to many of the conventional tools to resolve practical problems in reliability modelling and analysis. The study concludes by indicating some new problems that surfaced during the course of the present investigation which could be the subject for a future work in this area.

‘Modeling and Analysis of Competing Risks Data’

there has been much research on analyzing various forms of competing risks data. Nevertheless, there are several occasions in survival studies, where the existing models and methodologies are inadequate for the analysis competing risks data. ldentifiabilty problem and various types of and censoring induce more complications in the analysis of competing risks data than in classical survival analysis. Parametric models are not adequate for the analysis of competing risks data since the assumptions about the underlying lifetime distributions may not hold well. Motivated by this, in the present study. we develop some new inference procedures, which are completely distribution free for the analysis of competing risks data.

An Extended Pearson System Useful In Reliability Analysis

In the present environment, industry should provide the products of high quality. Quality of products is judged by the period of time they can successfully perform their intended functions without failure. The cause of the failures can be ascertained through life testing experiments and the times to failure due to different cause are likely to follow different distributions. Knowledge of this distribution is essential to eliminate causes of failures and thereby to improve the quality and the reliability of products. The main accomplishment expected to the study is to develop statistical tools that could facilitate solution to lifetime data arising in such and similar contexts

Dynamic cumulative residual Renyi’s entropy

Recently, cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon’s entropy (see Rao et al. [Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory. 50(6) (2004), pp. 1220–1228] and Asadi and Zohrevand [On the dynamic cumulative residual entropy, J. Stat. Plann. Inference 137 (2007), pp. 1931–1941]). Motivated by this finding, in this paper, we introduce a generalized measure of it, namely cumulative residual Renyi’s entropy, and study its properties.We also examine it in relation to some applied problems such as weighted and equilibrium models. Finally, we extend this measure into the bivariate set-up and prove certain characterizing relationships to identify different bivariate lifetime models

Quantile based reliability aspects of partial moments

Partial moments are extensively used in literature for modeling and analysis of lifetime data. In this paper, we study properties of partial moments using quantile functions. The quantile based measure determines the underlying distribution uniquely. We then characterize certain lifetime quantile function models. The proposed measure provides alternate definitions for ageing criteria. Finally, we explore the utility of the measure to compare the characteristics of two lifetime distributions

Modelling and Analysis of Recurrent Event Data with Multiple Causes

This thesis Entitled “modelling and analysis of recurrent event data with multiple causes.Survival data is a term used for describing data that measures the time to occurrence of an event.In survival studies, the time to occurrence of an event is generally referred to as lifetime.Recurrent event data are commonly encountered in longitudinal studies when individuals are followed to observe the repeated occurrences of certain events. In many practical situations, individuals under study are exposed to the failure due to more than one causes and the eventual failure can be attributed to exactly one of these causes.The proposed model was useful in real life situations to study the effect of covariates on recurrences of certain events due to different causes.In Chapter 3, an additive hazards model for gap time distributions of recurrent event data with multiple causes was introduced. The parameter estimation and asymptotic properties were discussed .In Chapter 4, a shared frailty model for the analysis of bivariate competing risks data was presented and the estimation procedures for shared gamma frailty model, without covariates and with covariates, using EM algorithm were discussed. In Chapter 6, two nonparametric estimators for bivariate survivor function of paired recurrent event data were developed. The asymptotic properties of the estimators were studied. The proposed estimators were applied to a real life data set. Simulation studies were carried out to find the efficiency of the proposed estimators.

Some bivariate life time models in discrete time"

The term reliability of an equipment or device is often meant to indicate the probability that it carries out the functions expected of it adequately or without failure and within specified performance limits at a given age for a desired mission time when put to use under the designated application and operating environmental stress. A broad classification of the approaches employed in relation to reliability studies can be made as probabilistic and deterministic, where the main interest in the former is to device tools and methods to identify the random mechanism governing the failure process through a proper statistical frame work, while the latter addresses the question of finding the causes of failure and steps to reduce individual failures thereby enhancing reliability. In the probabilistic attitude to which the present study subscribes to, the concept of life distribution, a mathematical idealisation that describes the failure times, is fundamental and a basic question a reliability analyst has to settle is the form of the life distribution. It is for no other reason that a major share of the literature on the mathematical theory of reliability is focussed on methods of arriving at reasonable models of failure times and in showing the failure patterns that induce such models. The application of the methodology of life time distributions is not confined to the assesment of endurance of equipments and systems only, but ranges over a wide variety of scientific investigations where the word life time may not refer to the length of life in the literal sense, but can be concieved in its most general form as a non-negative random variable. Thus the tools developed in connection with modelling life time data have found applications in other areas of research such as actuarial science, engineering, biomedical sciences, economics, extreme value theory etc.