Computational Methods for Multiplicative Intensity Models Using Weighted Gamma Processes: Proportional Hazards, Marked Point Processes, and Panel Count Data

Article excerpt

1. INTRODUCTION

Aalen (1975, 1978) developed a unified theory for nonparametric inference in multiplicative intensity models from a frequentist perspective. This treatment included, for example, the life-testing model, the multiple decrement model, birth and death processes, and branching processes. A Bayesian treatment for the real line was given by Lo and Weng (1989), who modeled hazard rates in the multiplicative intensity model as mixtures of a known kernel k with a finite measure [mu] modeled as a weighted gamma measure on the real line. That is, a hazard r is modeled as

r(x|[mu]) = [[integral].sub.R] k(x, v) [mu](dv). (1)

Lo and Weng (1989) showed that using arbitrary kernels provides the user with a great deal of flexibility; for instance, the choice of the kernel k(x, v) = I{v [less than or equal to] x} gives monotoneincreasing hazards as considered by Dykstra and Laud (1981), whereas kernels k(x, v) = I{|x - a| [greater than or equal to] v} and k(x, v) = I{|x - a| [less than or equal to] v} give U-shaped hazards (with minimum and maximum at a) similar to those of Glaser (1980), and normal density kernels k(x, v) = exp(-.5(x - v)[.sup.2]/[[tau].sup.2])/square root of (2[pi][[tau].sup.2]) can be used to estimate hazards without shape restriction.

In this article we develop a general approach to Bayesian inference for hazard (intensity) rates in nonparametric and semiparametric multiplicative intensity models by incorporating kernel mixtures of spatial weighted gamma process priors. This approach extends the work of Lo and Weng (1989) and Dykstra and Laud (1981) from a nonparametric setting on the real line to the nonparametric and semiparametric settings over general spaces and applies to the nonparametric multiplicative intensity models considered by Aalen (1975, 1978) and their semiparametric extensions developed by Andersen, Borgan, Gill, and Keiding (1993, chap. III). Models that fall within this framework that have been considered using gamma and weighted gamma processes from a Bayesian perspective include Markov models used in survival analysis subject to certain types of censoring, filtering, and truncation (Arjas and Gasbarra 1994; Laud, Smith, and Damien 1996; Gasbarra and Karia 2000), as well as Poisson point process models used in reliability (Kuo and Ghosh 1997), forest ecology (Wolpert and Ickstadt 1998a), and health exposure analysis (Best, Ickstadt, and Wolpert 2000). Another related application was given by Ibrahim, Chen, and MacEachern (1999) who used a weighted gamma process to select variables in Cox proportional hazards models.

A major contribution of this article is to develop a unified computational treatment of these problems from a Bayesian perspective. As was shown by Lo and Weng (1989) (see also Lo, Brunner, and Chan 1996) the posterior for multiplicative intensity models under weighted gamma processes share common structural features with posterior distributions for models subject to the Dirichlet process (i.e., Dirichlet process mixture models). (See Lo 1984 for background and posterior descriptions of Dirichlet process mixture models.) Recently, James (2003) extended these results to an abstract semiparametric setting (see Sec. 3), thus providing explicit calculus for relating posteriors for spatial semiparametric intensity models to posteriors for Dirichlet process mixture models. This equivalence, in combination with our use of a weighted gamma process approximation (Sec. 3), allows us to use efficient Dirichlet process computational procedures and to approximate the laws for general functionals of interest. An important aspect is that we avoid ad hoc methods used to approximate likelihoods. For example, in survival analysis problems we do not discretize time, as is often done to simplify computations. Another nice benefit of using Dirichlet process methods is that they are well understood and have a rich literature that can be drawn on. …