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Accessible to nonspecialists, this book explains the basic ideas in frailty modeling and statistical techniques, with a focus on real data application and interpretation of the results. It extensively explores how univariate frailty models can represent unobserved heterogeneity. It also emphasizes correlated frailty models as extensions of univariate and shared frailty models. The author analyzes similarities and differences between frailty and copula models, discusses problems related to frailty models, and describes parametric and semiparametric models using both frequentist and Bayesian approaches. He also shows how to apply the models to real data using R, SAS, and Stata.

Aging is a multifaceted and intricate physiological process characterized by a gradual decline in functional capacity, leading to increased susceptibility to diseases and mortality. While chronological age serves as a strong risk factor for age-related health conditions, considerable heterogeneity exists in the aging trajectories of individuals, suggesting that biological age may provide a more nuanced understanding of the aging process. However, the concept of biological age lacks a clear operationalization, leading to the development of various biological age predictors without a solid statistical foundation. This paper addresses these limitations by proposing a comprehensive operationalization of biological age, introducing the “AccelerAge” framework for predicting biological age, and introducing previously underutilized evaluation measures for assessing the performance of biological age predictors. The AccelerAge framework, based on Accelerated Failure Time (AFT) models, directly models the effect of candidate predictors of aging on an individual’s survival time, aligning with the prevalent metaphor of aging as a clock. We compare predictors based on the AccelerAge framework to a predictor based on the GrimAge predictor, which is considered one of the best-performing biological age predictors, using simulated data as well as data from the UK Biobank and the Leiden Longevity Study. Our approach seeks to establish a robust statistical foundation for biological age clocks, enabling a more accurate and interpretable assessment of an individual’s aging status.

Prediction and modeling of association football (soccer) outcomes has gained increasing interest in the scientific community in recent years, both due to betting concerns and the need for a deeper understanding of the factors influencing soccer events. We introduce and examine the validity of a Bayesian model, which belongs to the class of accelerated failure time (survival) models and is characterized by its straightforward structure. We implement MCMC methodology to estimate the posterior summaries of the model parameters and suggest a novel algorithm that can be used to transform simulated goal arrival times into predicted goals. The proposed model achieves exceptional in-sample and out-of-sample performance by replicating the entire league in a remarkably precise manner and by making accurate predictions on the second half of the league using the first half as a training dataset. The structure of the proposed model is extendable, allowing for the inclusion of in-play covariates that can be used to further map the complex dynamics of soccer matches.

Most reliability studies assume large samples or independence among components, but these assumptions often fail in practice, leading to imprecise inference. We address this issue by constructing confidence intervals (CIs) for the reliability of two-component systems with Weibull distributed failure times under a copula-frailty framework. Our construction integrates gamma-distributed frailties to capture unobserved heterogeneity and a copula-based dependence structure for correlated failures. The main contribution of this work is to derive adjusted CIs that explicitly incorporate the copula parameter in the variance-covariance matrix, achieving near-nominal coverage probabilities even in small samples or highly dependent settings. Through simulation studies, we show that, although traditional methods may suffice with moderate dependence and large samples, the proposed CIs offer notable benefits when dependence is strong or data are sparse. We further illustrate our construction with a synthetic example illustrating how penalized estimation can mitigate the issue of a degenerate Hessian matrix under high dependence and limited observations, so enabling uncertainty quantification despite deviations from nominal assumptions. Overall, our results fill a gap in reliability modeling for systems prone to correlated failures, and contribute to more robust inference in engineering, industrial, and biomedical applications.

Sustainable mining development requires structures on or within rock masses that can withstand deformation over a long period without compromising safety. Understanding of time-dependent behaviour of rocks is essential for such a purpose which is commonly investigated under sustained loading or so-called “creep” condition within the laboratory environment. A large number of experimental and analytical studies have examined creep behaviour of different rock types. However, some questions have still remained unanswered, particularly regarding the estimation of long-term strength of rocks and predicting their time-to-failure. This study proposes a novel method for prediction of time-to-failure of rock materials under creep loading governed by the secondary creep strain rate as well as estimation of their long-term strength through laboratory data. To do so, six different stress magnitudes ranging from 0.4 to 0.95 of the uniaxial compressive strength were selected for conventional creep compressive tests on Gosford sandstone. Throughout each experiment, the stress magnitude was kept constant until the sample reached failure. The results demonstrated that the secondary creep strain rate is strongly dependent on the magnitude of applied stress. A mere 10% reduction in the applied stress resulted in a decrease in the secondary creep strain rate of approximately three orders of magnitude. The proposed approach for time-to-failure prediction under creep loading included utilisation of secondary creep strain rates as a set of predictive indicators to overcome inherent variability or heterogeneity in rocks. Finally, the validation study was conducted based on the creep data obtained from various rock types to highlight consistent linear correlation between the secondary creep strain rate and the time-to-failure regardless of the magnitude of applied stress. Such an innovative approach can be a suitable tool for practitioners to better predict the stability of rock structures subjected to long-term loading leading to sustainable mining operation. Highlights Investigating the creep behaviour of a shaly sandstone at different stress magnitudes and examination of its long-term deformation responses; Introducing a novel and versatile predictive model for long-term strength estimation and time-to-failure prediction of rock materials under creep loading; Development of a universal correlation between the creep failure time and the secondary creep strain rate for different rock types.

The prediction of time to slope failure (TOF) is one of the most pivotal concerns for both geological risk researchers and practitioners. Conventional inverse velocity method (IVM), based on the analysis of displacement monitoring data, has become an effective method to solve this problem because it is easy to perform and the prediction results are generally acceptable. Practically, some limitations like random instrumental noise, environmental noise, and measurement error are ubiquitous factors hampered the reliability of the prediction. In this work, traditional IVM method and modified IVM with three different filters are respectively detected on velocity time series from an landslide event in an open-pit coal mine with the propose of improving, in retrospect, the accuracy of failure predictions. Simultaneously, the effects of noise on the appraisal of IVM graphics are also assessed and explanation. The results demonstrate that the sliding process of landslides can be divided into three signature stages based on the IVM. Noteworthily, the slope failure critical point occurs at the end of the progressive stage and generally coincides with a major acceleration event in which almost integrity of the slope is lost, transitioning to a linear trend ever since. Additionally, the short-term smoothing filter (SSF) and long-term smoothing filter (LSF) models can provide more accuracy and useful information about the probable failure time. Finally, with the intention of enhancing the feasible use of the method and supporting pre-determined response plans, two-level alert procedures combing SSF and LSF are proposed.

This paper deals with a reliability system hit by three types of shocks ranked as harmless, critical, or extreme, depending on their magnitudes, being below H1, between H1 and H2, and above H2, respectively. The system’s failure is caused by a single extreme shock or by a total of N critical shocks. In addition, the system fails under occurrences of M pairs of shocks with lags less than some δ (δ-shocks) in any order. Thus, the system fails when one of the three named cumulative damages occurs first. Thus, it fails due to the competition of the three associated shock processes. We obtain a closed-form joint distribution of the time-to-failure, shock count upon failure, δ-shock count, and cumulative damage to the system on failure, to name a few. In particular, the reliability function directly follows from the marginal distribution of the failure time. In a modified system, we restrict δ-shocks to those with small lags between consecutive harmful shocks. We treat the system as a generalized random walk process and use an embellished variant of discrete operational calculus developed in our earlier work. We demonstrate analytical tractability of our formulas which are also validated, through Monte Carlo simulation.

This study presents an integrated experimental–analytical–AI approach to evaluate the long-term hydrostatic performance of PE-RT/Al/PE-RT multi-layer pipes. Hydrostatic pressure tests were conducted on four pipe sizes (16, 20, 25, and 32 mm) at temperatures of 20 °C, 60 °C, 95 °C, and 110 °C, with failure times ranging from 1 h to over 10,000 h. The stress distribution within each layer was determined using stress equilibrium in pipe layers, revealing that the aluminum layer carries 81% to 84% of the internal pressure, while the inner and outer PE-RT layers contribute 12% to 16% and < 2%, respectively. A novel K -coefficient, ranging from 0.76 to 0.89, was introduced to relate aluminum hoop stress to its yield strength, simplifying the structural design process. To complement the analytical framework, an artificial neural network model was developed using 6 input features (diameter, layer thicknesses, temperature, failure time), a single hidden layer with 25 neurons, and trained on about 100 experimental samples. The ANN achieved a correlation coefficient R 2  > 0.99, RMSE = 0.93 bar, enabling accurate prediction of rupture pressure, stress profile, and K -coefficient for unseen configurations. This study provides a robust and time-efficient predictive tool to estimate long-term pressure capacity without requiring extensive physical testing. The proposed model supports more efficient and cost-effective design of PE-RT/Al/PE-RT pipes Additionally, a comparative analysis between PE-RT/Al/PE-RT and PE-X/Al/PE-X pipes was conducted to evaluate differences in material performance and long-term durability.

Recurrent event data arise frequently in various fields such as biomedical sciences, public health, engineering, and social sciences. In many instances, the observation of the recurrent event process can be stopped by the occurrence of a correlated failure event, such as treatment failure and death. In this article, we propose a joint scale-change model for the recurrent event process and the failure time, where a shared frailty variable is used to model the association between the two types of outcomes. In contrast to the popular Cox-type joint modeling approaches, the regression parameters in the proposed joint scale-change model have marginal interpretations. The proposed approach is robust in the sense that no parametric assumption is imposed on the distribution of the unobserved frailty and that we do not need the strong Poisson-type assumption for the recurrent event process. We establish consistency and asymptotic normality of the proposed semiparametric estimators under suitable regularity conditions. To estimate the corresponding variances of the estimators, we develop a computationally efficient resampling-based procedure. Simulation studies and an analysis of hospitalization data from the Danish Psychiatric Central Register illustrate the performance of the proposed method. Supplementary materials for this article are available online.

Broken adaptive ridge (BAR) is a penalized regression method that performs variable selection via a computationally scalable surrogate to L0 regularization. The BAR regression has many appealing features; it converges to selection with L0 penalties as a result of reweighting L2 penalties, and satisfies the oracle property with grouping effect for highly correlated covariates. In this paper, we investigate the BAR procedure for variable selection in a semiparametric accelerated failure time model with complex high-dimensional censored data. Coupled with Buckley-James-type responses, BAR-based variable selection procedures can be performed when event times are censored in complex ways, such as right-censored, left-censored, or double-censored. Our approach utilizes a two-stage cyclic coordinate descent algorithm to minimize the objective function by iteratively estimating the pseudo survival response and regression coefficients along the direction of coordinates. Under some weak regularity conditions, we establish both the oracle property and the grouping effect of the proposed BAR estimator. Numerical studies are conducted to investigate the finite-sample performance of the proposed algorithm and an application to real data is provided as a data example.