Parametric Regression Models Using Reversed Hazard Rates

Proportional hazard regression models are widely used in survival analysis to understand and exploit the relationship between survival time and covariates. For left censored survival times, reversed hazard rate functions are more appropriate. In this paper, we develop a parametric proportional hazard rates model using an inverted Weibull distribution. The estimation and construction of confidence intervals for the parameters are discussed. We assess the performance of the proposed procedure based on a large number of Monte Carlo simulations. We illustrate the proposed method using a real case example. 1. Introduction In survival studies, covariates or explanatory variables are usually employed to represent heterogeneity in a population. The main objective in such situations is to understand and exploit the relationship between lifetime and covariates. Regression models are useful in such contexts to assess the effect of covariates on lifetime. These models can be formulated in many ways and several types are in common use. Parametric regression models for lifetime involve specification for the distribution of a lifetime given a vector of covariates . The most commonly used parametric model is the Weibull regression model, which satisfies the proportional relationship between hazard rate functions of the lifetimes of two subjects. The maximum likelihood technique is usually employed to find estimates of the parameters of the model. For more properties and applications of parametric regression models, one should refer to Lawless [1]. In survival studies, there are many occasions where lifetime data are left censored. For example, baboons in the Amboseli Reserve, Kenya, sleep in the trees and descend for ageing at certain times of the day. Observers often arrive later in the day after this descent has occurred and on such days they can only ascertain that the descent took place before a particular time, so that the descent times are left censored (see [2]). On such occasions, a reversed hazard rate is more appropriate than a hazard rate to analyze lifetime data due to the fact that estimators of hazard rates are unstable when data are left censored. The reversed hazard rate of is defined as Introduced by Barlow et al. [3], the function has been used in various contexts such as the estimation of distribution function under left censoring [1], defining a new stochastic order [4], characterization of lifetime distributions [5–7], studying ageing behavior [8, 9], evolving new repair and maintenance strategies [10, 11], the mixed proportional hazards model