An Application of Markov Models in Estimating Transition Probabilities for Postmenopausal Women with Osteoporosis
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An Application of Markov Models in Estimating Transition Probabilities for Postmenopausal Women with Osteoporosis
Zhengqing Li, PhD Section Head, Biometrics and Statistical Sciences, Procter 6.Gamble Pharmaceuticals, Mason,
Associate Director, New Drug Development. Europe,
Procter 6 Gamble Pharmaceuticals, United Kingdom
Key Words Bootstrap: Markov chains: Osteoporosis: Stochastic processes; Transition probabilities Correspondence Address Zhengqing Li, Biometrics and Statistical Sciences, The Procter 6 Gamble Company, 8700 Mason-Montgomery Rd., Box 2199, Mason, OH 45040 (e-mail: [email protected])
Vertebral fracture is a common consequence of osteoporosisfor postmenopausal women. Modeling the progression of the disease in vertebral fractures has been of great interest to physicians as well as patients. In this papert we propose a simple Markov chain model to study the progression of the vertebralfiactures for untreatedpostmenopausal women with osteoporosis. In this model, the state space consists of (0,1, 2, . . . Pi},which represents the possible number of vertebral fractures for a patient, as commonly assessed via spinal x-ray Based on the model, a
INTROOUCTlON Vertebral fractures, that is, severe deformation of spinal vertebra, are a well-recognized consequence of postmenopausal bone loss and are the most common osteoporotic fracture (1).All vertebral fractures are associated with increased mortality and morbidity, including back pain and decreased activity. Previous data have indicated that prevalent vertebral fractures substantially increase the risk of future vertebral fractures (2). However, the speed of progression of the disease has not been well characterized and is usually addressed via epidemiological studies. Data from clinical trials of osteoporosis therapies provide unique data that facilitate alternative modeling techniques. As a useful mathematical tool, Markov models have been well developed in the mathematical and statistical literature (3) and have been used in many areas (4). For example, for aggregated data (data that consist of the number of individuals in each state at specified observation times), statistical methods have been developed in estimating the transition probabilities (5). In the therapeutic area of postmenopausal osteoporosis, vertebral fractures have been assessed in clinical trials of up to three years dura-
simple estimatorof the oneyear transition probability matrix is proposed. An estimate for the m-year transition probability matrix is then den'vedfdlowing an application of the ChapmanKolmogorov equations. To estimate the confidence intervalsfor the m-year transition probabilities, a bootstrap procedure is proposed. The proposed bootstrap procedure can also be employed to construct confidence intervals for differences between two treatment groups. The methods are illustrated with data from clinical trials for an anti-resorptivetherapy
tion using annual spinal x-rays to see whether a patient sustains a fracture and the time period when the fracture occurs (6,7,8).It i
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