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Analyzing Toxicity Data Using Statistical Models for Time-To-Death: An Introduction
Published in Michael C. Newman, Alan W. McIntosh, Metal Ecotoxicology, 2020
Philip M. Dixon, Michael C. Newman
Although the Weibull distribution is quite flexible, its hazard function is monotonic: always increasing if β > 1 and always decreasing if β < 1. The log-normal and log-logistic distributions are similar distributions with hazard functions that may monotonically increase, monotonically decrease, or change directions over time (Figure 6). The hazard curves for both distributions are different for an accelerated time model and a proportional hazards model (Figure 7). For the proportional hazards model, the hazard of a treatment group is some constant proportion of the baseline hazard at all times. The hazard for the accelerated time model increases more quickly than the hazard under a proportional hazards mode, and then declines closer to the baseline level (Figure 7). Although the effect of the treatment is the same in the proportional hazard and accelerated failure models, the median time of death and the distributions of times-to-death will be different.
Marginally and Conditionally Specified Multivariate Survival Models: A Survey
Published in Donald B. Owen, Subir Ghosh, William R. Schucany, William B. Smith, Statistics of Quality, 2020
A very popular survival model is the proportional hazards model. In this case we set μ = 0, σ = 1, γ = 1 in Eq. (5) to obtain what might be called a “Lehmann alternatives” model: () F¯X(x)=[F¯0(x)]δ
Condition-based maintenance for a degradation-shock dependence system under warranty
Published in International Journal of Production Research, 2023
We use a proportional-hazards model (Cox 1992) to estimate the extra effects of the degradation-rate acceleration. A regression-based model that is widely used is the proportional-hazards model. The time-dependent proportional-hazards model is generally expressed as (Elsayed 2012; Shyur, Elsayed, and Luxhøj 1999) where and denote the regression coefficient and stress function, respectively; is modified multiplicatively by covariates and is referred to as the baseline failure-rate function (Elsayed 2012; Shyur, Elsayed, and Luxhøj 1999). The degradation path is any unknown function, which can be either monotonically decreasing or monotonically increasing. The function for degradation is considered to endure item-to-item deviation by including the random variable X, and it is given by
Modeling lane-transgressing behavior of e-bike riders on road sections with marked bike lanes: A survival analysis approach
Published in Traffic Injury Prevention, 2020
Sirui Nan, Li Yan, Ran Tu, Tiezhu Li
The logarithm partial likelihood function (Eq. (7)) is used to estimate the parameters of Cox proportional hazards model. The regression coefficient [β1, β2,…, βq] is estimated by maximizing the value of the lnLP using the Newton-Raphson algorithm.