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, 34 (4), 652-63

A Model for Time to Fracture With a Shock Stream Superimposed on Progressive Degradation: The Study of Osteoporotic Fractures

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A Model for Time to Fracture With a Shock Stream Superimposed on Progressive Degradation: The Study of Osteoporotic Fractures

Xin He et al. Stat Med.

Abstract

Osteoporotic hip fractures in the elderly are associated with a high mortality in the first year following fracture and a high incidence of disability among survivors. We study first and second fractures of elderly women using data from the Study of Osteoporotic Fractures. We present a new conceptual framework, stochastic model, and statistical methodology for time to fracture. Our approach gives additional insights into the patterns for first and second fractures and the concomitant risk factors. Our modeling perspective involves a novel time-to-event methodology called threshold regression, which is based on the plausible idea that many events occur when an underlying process describing the health or condition of a person or system encounters a critical boundary or threshold for the first time. In the parlance of stochastic processes, this time to event is a first hitting time of the threshold. The underlying process in our model is a composite of a chronic degradation process for skeletal health combined with a random stream of shocks from external traumas, which taken together trigger fracture events.

Keywords: disease progression; elderly; first hitting time; health process; threshold regression; time-to-event data.

Figures

Figure 1
Figure 1
A conceptual stochastic process model for skeletal strength, an external shock process and time to fracture.
Figure 2
Figure 2
Pictorial representation of the time to skeletal fracture as a first hitting time for a composite stochastic process consisting of a stream of random shocks superimposed on a progressive degradation process for skeletal strength.
Figure 3
Figure 3
Plots of the probability density and cumulative distribution functions for the shock distribution with α = 1 and β = 1: (a) g(υ) = (1/υ2) exp(−1/υ), (b) G(υ) = exp(−1/υ).
Figure 4
Figure 4
Plot of the survival function (s) = exp[(1 − e0.035s)/1.225], corresponding to parameter values α = 1 and β = 1 for the shock distribution and a degradation curve starting at initial strength y0 = 35 with rate parameter λ = −0.035.
Figure 5
Figure 5
Plots of the Kaplan-Meier and fitted survival functions, as well as the 95% confidence bands for the Kaplan-Meier function for (a) first and (b) second fracture times based on the TR degradation-shock model without covariates. The fitted model for second fractures has a large proportion of subjects who will not experience a second fracture (a large cure rate).
Figure 6
Figure 6
A plot showing the calibration and predictive power of the model for first fracture. The curves compare the cumulative sum of actual and expected fractures, based on martingale residuals, when patients are ranked by their estimated skeletal strength y0.

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