An application of Markov models in estimating transition probabilities for postmenopausal women with osteoporosis

Vertebral fracture is a common consequence of osteoporosis for postmenopausal women. Modeling the progression of the disease in vertebral fractures has been of great interest to physicians as well as patients. In this paper, we propose a simple Markov chain model to study the progression of the vertebral fractures for untreated postmenopausal women with osteoporosis. In this model, the state space consists of {0, 1, 2,... 13}, which represents the possible number of vertebral fractures for a patient, as commonly assessed via spinal x-ray. Based on the model, a simple estimator of the one-year transition probability matrix is proposed. An estimate for the m-year transition probability matrix is then derived following an application of the Chapman-Kolmogorov equations. To estimate the confidence intervals for 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-resorptive therapy.