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A non-homogeneous Semi-Markov model for Interval Censoring

M. N. M. van Lieshout, R. L. Markwitz

Abstract

Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on semi-Markov processes in which interval generation is assumed to be time-dependent and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to homogeneous approaches by way of a parametric example.

A non-homogeneous Semi-Markov model for Interval Censoring

Abstract

Previous approaches to modelling interval-censored data have often relied on assumptions of homogeneity in the sense that the censoring mechanism, the underlying distribution of occurrence times, or both, are assumed to be time-invariant. In this work, we introduce a model which allows for non-homogeneous behaviour in both cases. In particular, we outline a censoring mechanism based on semi-Markov processes in which interval generation is assumed to be time-dependent and we propose a Markov point process model for the underlying occurrence time distribution. We prove the existence of this process and derive the conditional distribution of the occurrence times given the intervals. We provide a framework within which the process can be accurately modelled, and subsequently compare our model to homogeneous approaches by way of a parametric example.
Paper Structure (18 sections, 9 theorems, 65 equations, 4 figures)

This paper contains 18 sections, 9 theorems, 65 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The non-homogeneous semi-Markov process
  3. Definition and notation
  4. Conditional intensities and non-explosion conditions
  5. Renewal function: existence and boundedness
  6. Age and excess distributions
  7. A model for non-homogeneous interval-censoring
  8. Model formulation
  9. Conditional distribution
  10. Modelling considerations
  11. Non-homogeneous point process densities
  12. Parametric modelling of the mark kernel
  13. Statistical aspects
  14. Illustrations in practice
  15. Model mis-specification
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $(S_n, X_n)_{n=1}^{\infty}$ be an alternating non-homogeneous semi-Markov process with values in $\{ 0, 1\} \times \mathbb{R}^+$ with $S_0=1$, $X_0=0$ and semi-Markov kernels $G_Y(x,\cdot)$, $G_Z(x, \cdot)$ that follow Gamma distributions with shape and rate parameters $\theta_Y(x) = (k_Y(x), \l for some $c > 0$. Write $X_\infty = \lim_{n\to \infty} X_n$ for the time of explosion. Then $\mathb

Figures (4)

  • Figure 1: A visualisation of a semi-Markov process with initial values $S_0 = 1$ and $X_0 = 0$. At the dotted line, one cycle has passed - i.e. the process has taken both possible state values. The jump times correspond to a change of state. For a given time $t$ in which the process is in state $1$, a non-zero age $A(t)$ and excess $B(t)$ are recorded.
  • Figure 2: The solid line is the actual probability density of interval length for $k=1$ and $\lambda(0.6;1) = 1$. The broken line is the estimated survival time density.
  • Figure 3: Probability density function of the starting time $f_x(\cdot)$ with $x=1$ for various choices of $g_Y$ and $m$.
  • Figure 4: A comparison between a regular and clustered model with a 'peak time' added by changing the intensity function within a critical range.

Theorems & Definitions (18)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Corollary 2.2.1
  • proof
  • Corollary 2.2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 8 more