Parametric counting process models

In biostatistics it has become a tradition to use non- and semiparametric methods, like those considered in the previous two chapters, to analyze censored survival data, while parametric methods are more common in reliability studies of failure times of technical equipment. In our opinion biostatistics would gain from the use of a wider range of statistical methods, including parametric methods, than is the current practice. In this chapter, we discuss the basic modeling and inferential issues for parametric counting process models. More advanced models and methods are discussed in Chapters 6, 7, 10, and 11. As in Chapter 4, we consider counting processes N 1,N 2, …,N n , where N i (t) counts the number of occurrences of the event of interest for individual i in [0, t]; i=1,2, …,n. For survival data the event can occur at most once for each individual, and N i (t) can only take the values 0 or 1. In this chapter we consider situations where the intensity processes of the N i (t) may be specified by a q-dimensional parameter θ =(θ 1, …,θ q ) T . We write the intensity process of N i (t) as λ i (t;θ) to highlight this parametric specification.