Abstract
Semi-Markov models are widely used for survival analysis and reliability analysis. In general, there are two competing parameterizations and each entails its own interpretation and inference properties. On the one hand, a semi-Markov process can be defined based on the distribution of sojourn times, often via hazard rates, together with transition probabilities of an embedded Markov chain. On the other hand, intensity transition functions may be used, often referred to as the hazard rates of the semi-Markov process. We summarize and contrast these two parameterizations both from a probabilistic and an inference perspective, and we highlight relationships between the two approaches. In general, the intensity transition based approach allows the likelihood to be split into likelihoods of two-state models having fewer parameters, allowing efficient computation and usage of many survival analysis tools. Nevertheless, in certain cases the sojourn time based approach is natural and has been exploited extensively in applications. In contrasting the two approaches and contemporary relevant R packages used for inference, we use two real datasets highlighting the probabilistic and inference properties of each approach. This analysis is accompanied by an Rvignette.
Highlights
In biostatistics, many models for survival and reliability analysis are two-state stochastic processes which lead to a particular event such as death, or the outcome of a particular drug treatment
A multi-state model is a continuous time stochastic process with values in a discrete set that is often applied for longitudinal medical studies, where the patients may experience several events and their related information is collected over time
Table : Akaike information criterion (AIC) and number of parameters for the models based on the sojourn times (Apporach I) and the intensity transition functions (Apporach II)
Summary
Many models for survival and reliability analysis are two-state stochastic processes which lead to a particular event such as death, or the outcome of a particular drug treatment. The risk of chronic diseases such as AIDS essentially depends on the time since infection, see [5] For these cases, applying the class of semi-Markov processes (SMP), as an extension of Markov processes, is fruitful. Especially those concerned with characterizing an individual’s progression through various stages of a disease, a three-state semi-Markov process known as the illness-death model is very popular (see for instance [5, 9]). For both datasets, we compare results of fully parametric models based on both approaches and highlight the implications of each modeling choice. All the numerical results (tables and figures) from the paper are freely reproducible via R code in a comprehensive detailed vignette available in [12]
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