Bivariate Frailty Model and Association Measure
In this paper, we present a bivariate frailty model and the association measure. The relationship between the conditional and the unconditional hazard gradients are derived and some examples are provided. A correlated frailty model is presented and its ap
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Bivariate Frailty Model and Association Measure Ramesh C. Gupta
Abstract In this paper, we present a bivariate frailty model and the association measure. The relationship between the conditional and the unconditional hazard gradients are derived and some examples are provided. A correlated frailty model is presented and its application in the competing risk theory is given. Some applications to real data sets are also pointed out. Keywords Hazard gradient · Bivariate proportional hazard model Bivariate additive model · Association measure · Correlated frailty model Competing risk
10.1 Introduction Frailty models are random effect models, where the randomness is introduced as an unobserved frailty. The proportional hazard frailty model is a classical frailty model due to Vaupel et al. [32] in which the baseline failure rate is multiplied by the unknown frailty variable and is given by, λ(t|v) = vλ0 (t), t > 0,
(10.1.1)
where λ0 (t) is the baseline hazard independent of v. In addition to the above classical frailty model, various other models, including the additive frailty model and the accelerated failure time frailty model, have also been studied; see [5, 11, 26]. In modeling survival data by frailty models, the choice of the frailty distribution has been of interest. It has been observed that the choice of frailty distribution strongly affects the estimate of the baseline hazard as well as the conditional probabilities, see [1, 15–19]. In the shared frailty models, the assumptions about the frailty distributions play an important role in the model’s interpretation since the frailty distribution links R. C. Gupta (B) University of Maine, Orono, ME, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 A. Adhikari et al. (eds.), Mathematical and Statistical Applications in Life Sciences and Engineering, https://doi.org/10.1007/978-981-10-5370-2_10
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the two processes of interest. For more discussion see [27] and [28]. In order to compare different frailties, in the univariate case, Gupta and Kirmani [10] investigated as to how the well-known stochastic orderings between distributions of two frailties translate into orderings between the corresponding survival functions. More recently, Gupta and Gupta [7, 8] studied a similar problem for a general frailty model which includes the classical frailty model (10.1.1) as well as the additive frailty model. For some applications of the model (10.1.1) and related models, the reader is referred to Price and Manatunga [31] and Kersey et al. [20], who applied cure models, frailty models and frailty mixture models to analyze survival data. Xue and Ding [36] applied the bivariate frailty model to inpatients mental health data. Hens et al. [12] applied the bivariate correlated gamma frailty model for type I interval censored data. Wienke et al. [33] applied the correlated gamma frailty model to fit bivariate time to event (occurence of breast cancer) data. Wienke et al. [34] used three correlated frailty models to analyze bivariate sur
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