A Semi-Markov Model Based on Generalized Weibull Distribution with an Illustration for HIV Disease

• Published in: Biometrical journal Vol. 47; no. 6; pp. 825 - 833
• Main Authors: Foucher, Yohann, Mathieu, Eve, Saint-Pierre, Philippe, Durand, Jean-François, Daurès, Jean-Pierre
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag 01-12-2005; WILEY‐VCH Verlag; Wiley-VCH; Wiley-VCH Verlag
• Subjects: Adult; Age Factors; Aged; Aged, 80 and over; Algorithms; Applications; Biology, psychology, social sciences; Disease Progression; Disease Transmission, Infectious - statistics & numerical data; Distribution theory; Exact sciences and technology; Female; Generalized Weibull distribution; Hazard function; Hepatitis - complications; Heterosexuality; HIV; HIV Infections - etiology; HIV Infections - transmission; Homosexuality; Human immunodeficiency virus; Humans; Likelihood Functions; longitudinal analysis; Longitudinal Studies; Male; Markov Chains; Mathematics; Medical sciences; Methodology; Middle Aged; Models, Statistical; Multi-state model; Multivariate Analysis; Probability; Probability and statistics; Probability theory and stochastic processes; Proportional Hazards Models; Sciences and techniques of general use; Semi-Markov process; Sex Factors; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistical Distributions; Statistics; Substance-Related Disorders; Biometrics; Homogeneous model; Stochastic model; Semimarkovian process; Hazard function; Exponential distribution; Chronic disease; Life science; longitudinal analysis; Continuous time; Semi-Markov process; Covariate; Waiting time; Generalized Weibull distribution; Multi-state model; Parametric method; Statistical method; Multi state model; Time distribution; HIV; Weibull distribution
• ISSN: 0323-3847; 1521-4036
• DOI: 10.1002/bimj.200410170

Multi‐state stochastic models are useful tools for studying complex dynamics such as chronic diseases. Semi‐Markov models explicitly define distributions of waiting times, giving an extension of continuous time and homogeneous Markov models based implicitly on exponential distributions. This paper develops a parametric model adapted to complex medical processes. (i) We introduced a hazard function of waiting times with a U or inverse U shape. (ii) These distributions were specifically selected for each transition. (iii) The vector of covariates was also selected for each transition. We applied this method to the evolution of HIV infected patients. We used a sample of 1244 patients followed up at the hospital in Nice, France. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)