The Lindley distribution applied to competing risks lifetime data

Abstract

Competing risks data usually arises in studies in which the death or failure of an individual or an item may be classified into one of k ≥ 2 mutually exclusive causes. In this paper a simple competing risks distribution is proposed as a possible alternative to the Exponential or Weibull distributions usually considered in lifetime data analysis. We consider the case when the competing risks have a Lindley distribution. Also, we assume that the competing events are uncorrelated and that each subject can experience only one type of event at any particular time.

Introduction

Let us consider the situation where an individual or unit is exposed to k ≥ 2 possible causes of death or failure, such that the exact cause is fully or partially known (see, for example, Refs. [6], [11], [5], [19], [3]). The model for lifetime in the presence of such competing risks structure is known as the competing risks model. If Y_j, j = 1, …, k denotes the time to failure due to the jth cause then the observed random variable is T = min  (Y₁, …, Y_k). Under the assumption that the causes are independent, the hazard and survival functions are given, respectively, by:

$$h(t)=\sum _{j=1}^k{h}_j(t)\text{and}S(t)=\prod _{j=1}^k{S}_j(t)\text{.}$$

where h_j(t) and S_j(t) are the hazard and survival functions related to the jth cause of death or failure.

From the hazard and survival functions, (1), the probability density function for T = min  (Y₁, …, Y_k) can be written as:

$$f(t)=\sum _{j=1}^k{h}_j(t)\prod _{j=1}^k{S}_j(t)=\sum _{j=1}^k{f}_j(t)\prod _{l=1\text{,}l\ne j}^k{S}_l(t)\text{,}$$

where f_j(t) is any density function used in time-to-event modeling.

The literature presents many papers in competing risks modeling assuming that the competing risks data follow different standard lifetime distributions such as Exponential or Weibull (see, Refs. [16], [17] and references within). Some non-standard distributions have been recently used in the competing risks scenario [22], [18], [2], [21], [9], [4], however, as we observe in the literature, the Lindley distribution has not been considered.

The Lindley distribution was introduced by Lindley [13], [14] as a new distribution useful to analyse lifetime data especially in applications modeling stress-strength reliability. In a recent paper Ghitany et al. [8] studied the properties of the Lindley distribution under a carefully mathematical treatment. They also showed in a numerical example that the Lindley distribution gives better modeling than the one based on the Exponential distribution. A generalized Lindley distribution was proposed in Zakerzadeh and Dolati [28]. In Ref. [20] it was proposed the Poisson–Lindley distribution to model count data. The zero-truncated Poisson–Lindley distribution and the generalized Poisson–Lindley distribution were considered in Refs. [7], [15], respectively. The negative binomial-Lindley distribution was considered in Ref. [29].

As pointed out in Ref. [8] due to the popularity of the Exponential distribution in statistics and many applied areas, the Lindley distribution has not been very well explored in the literature. The aim of this paper is to study a competing risks model when the causes of failure follow the Lindley distribution. The paper is organized as follows. In Section 2 the competing risks model for the Lindley distribution is formulated. A simulation study is introduced in Section 3 where we consider two and three causes of competing failures. An example considering a real data set where the Lindley distribution presents a better fit than the Exponential distribution is presented in Section 4. Some conclusions are presented in Section 5.