A general threshold stress hybrid hazard model for lifetime data

In this paper we propose a hybrid hazard regression model with threshold stress which includes the proportional hazards and the accelerated failure time models as particular cases. To express the behavior of lifetimes the generalized-gamma distribution is assumed and an inverse power law model with a threshold stress is considered. For parameter estimation we develop a sampling-based posterior inference procedure based on Markov Chain Monte Carlo techniques. We assume proper but vague priors for the parameters of interest. A simulation study investigates the frequentist properties of the proposed estimators obtained under the assumption of vague priors. Further, some discussions on model selection criteria are given. The methodology is illustrated on simulated and real lifetime data set.     

Inverse gaussian accelerated test models based on cumulative damage

The failure of a system under environmental stress often can be described by an accelerated test model which incorporates the environmental variable L. Here, the failure of such a system at environmental level L is modeled as the first passage of accumulated damage to a critical threshold value. Assuming a discrete additive damage model leads to a Birnbaum–Saunders-type distribution for the failure time which can be closely approximated by an inverse Gaussian-type model. However, if a continuous damage model based on a Gaussian process is assumed, a more general family of inverse Gaussian accelerated test models is obtained. Three sets of failure data are discussed to illustrate the usefulness of this general family.     

Bivariate Constant-Stress Accelerated Degradation Model and Inference

To assess the reliability of highly reliable products that have two or more performance characteristics (PCs) in an accurate manner, relations between the PCs should be taken duly into account. If they are not independent, it would then become important to describe the dependence of the PCs. For many products, the constant-stress degradation test cannot provide sufficient data for reliability evaluation and for this reason, accelerated degradation test is usually performed. In this article, we assume that a product has two PCs and that the PCs are governed by a Wiener process with a time scale transformation, and the relationship between the PCs is described by the Frank copula function. The copula parameter is dependent on stress and assumed to be a function of stress level that can be described by a logistic function. Based on these assumptions, a bivariate constant-stress accelerated degradation model is proposed here. The direct likelihood estimation of parameters of such a model becomes analytically intractable, and so the Bayesian Markov chain Monte Carlo (MCMC) method is developed here for this model for obtaining the maximum likelihood estimates (MLEs) efficiently. For an illustration of the proposed model and the method of inference, a simulated example is presented along with the associated computational results.     

Asymptotic Comparison Between Constant-stress Testing and Step-stress Testing for Type-I Censored Data from Exponential Distribution

By running the life tests at higher stress levels than normal operating conditions, accelerated life testing quickly yields information on the lifetime distribution of a test unit. The lifetime at the design stress is then estimated through extrapolation using a regression model. In constant-stress testing, a unit is tested at a fixed stress level until failure or the termination time point of the test, while step-stress testing allows the experimenter to gradually increase the stress levels at some pre-fixed time points during the test. In this article, the optimal k-level constant-stress and step-stress accelerated life tests are compared for the exponential failure data under Type-I censoring. The objective is to quantify the advantage of using the step-stress testing relative to the constant-stress one. A log-linear relationship between the mean lifetime parameter and stress level is assumed and the cumulative exposure model holds for the effect of changing stress in step-stress testing. The optimal design point is then determined under C-optimality, D-optimality, and A-optimality criteria. The efficiency of step-stress testing compared to constant-stress testing is discussed in terms of the ratio of optimal objective functions based on the information matrix.     

Optimal simple step stress accelerated life test design for reliability prediction

A step stress accelerated life testing model is presented to obtain the optimal hold time at which the stress level is changed. The experimental test is designed to minimize the asymptotic variance of reliability estimate at time ζ$\zeta$. A Weibull distribution is assumed for the failure time at any constant stress level. The scale parameter of the Weibull failure time distribution at constant stress levels is assumed to be a log-linear function of the stress level. The maximum likelihood function is given for the step stress accelerated life testing model with Type I censoring, from which the asymptotic variance and the Fisher information matrix are obtained. An optimal test plan with the minimum asymptotic variance of reliability estimate at time ζ$\zeta$ is determined.     

Interlitter Response Variability in a Threshold Dose-Response Model

In this article, we extend a previously formulated threshold dose-response model with random litter effects that was applied to a data set from a developmental toxicity study. The dose-response pattern of the data indicates that a threshold dose level may exist. Additionally, there is noticeable variation between the responses across the dose levels. With threshold estimation being critical, the assumed variability structure should adequately model the variation while not taking away from the estimation of the threshold as well as the other parameters directly involved in the dose-response relationship. In the prior formulation, the random effect was modeled assuming identical variation in the interlitter response probabilities across all dose levels, that is, the model had a single parameter to account for the interlitter variability. In this new model, the random effect is modeled as having different response variability across dose levels, that is, multiple interlitter variability parameters. We performed the likelihood ratio test (LRT) to compare our extended model to the previous model. We conducted a simulation study to compare the bias of each model when fit to data generated with the underlying parametric structure of the opposing model. The extended threshold dose-response model with multiple response variation was less biased.     

Weibull accelerated life tests when there are competing causes of failure

Accelerated life testing of a product under more severe than normal conditions is commonly used to reduce test time and costs. Data collected at such accelerated conditions are used to obtain estimates of the parameters of a stress translation function. This function is then used to make inference about the product's life under normal operating conditions. We consider the problem of accelerated life tests when the product of interest is a p component series system. Each of the components is assumed to have an independent Weibull time to failure distribution with different shape parameters and different scale parameters which are increasing functions stress. A general model i s used for the scale parameter includes the standard engineering models as special This model also has an appealing biological interpretation     

Exponential progressive step-stress life-testing with link function based on Box–Cox transformation

In order to quickly extract information on the life of a product, accelerated life-tests are usually employed. In this article, we discuss a k-stage step-stress accelerated life-test with M-stress variables when the underlying data are progressively Type-I group censored. The life-testing model assumed is an exponential distribution with a link function that relates the failure rate and the stress variables in a linear way under the Box–Cox transformation, and a cumulative exposure model for modelling the effect of stress changes. The classical maximum likelihood method as well as a fully Bayesian method based on the Markov chain Monte Carlo (MCMC) technique is developed for inference on all the parameters of this model. Numerical examples are presented to illustrate all the methods of inference developed here, and a comparison of the ML and Bayesian methods is also carried out.     

Random effects models for estimation of the probability and time to progression of a continuous biomarker

Biomarkers play a key role in the monitoring of disease progression. The time taken for an individual to reach a biomarker exceeding or lower than a meaningful threshold is often of interest. Due to the inherent variability of biomarkers, persistence criteria are sometimes included in the definitions of progression, such that only two consecutive measurements above or below the relevant threshold signal that “true” progression has occurred. In previous work, a novel approach was developed, which allowed estimation of the time to threshold using the parameters from a linear mixed model where the residual variance was assumed to be pure measurement error. In this paper, we extend this methodology so that serial correlation can be accommodated. Assuming that the Markov property holds and applying the chain rule of probabilities, we found that the probability of progression at each timepoint can be expressed simply as the product of conditional probabilities. The methodology is applied to a cohort of HIV positive individuals, where the time to reach a CD4 count threshold is estimated. The second application we present is based on a study on abdominal aortic aneurysms, where the time taken for an individual to reach a diameter exceeding 55 mm is studied. We observed that erroneously ignoring the residual correlation when it is strong may result in substantial overestimation of the time to threshold. The estimated probability of the biomarker reaching a threshold of interest, expected time to threshold, and confidence intervals are presented for selected patients in both applications.     

Analysis of accelerated life tests with weibull failure distribution via generalized linear models

Efficient industrial experiments for reliability analysis of manufactured goods may consist in subjecting the units to higher stress levels than those of the usual working conditions. This results in the so called "accelerated life tests" where, for each pre-fixed stress level, the experiment ends after the failure of a certain pre-fixed proportion of units or a certain test time is reached. The aim of this paper is to determine estimates of the mean lifetime of the units under usual working conditions from censored failure data obtained under stress conditions. This problem is approached through generalized linear modelling and related inferential techniques, considering a Weibull failure distribution and a log-linear stress-response relationship. The general framework considered has as particular cases, the Inverse Power Law model, the Eyring model, the Arrhenius model and the generalized Eyring model. In order to illustrate the proposed methodology, a numerical example is provided.     

Guaranteed Local Maximum Likelihood Detection of a Change Point in Nonparametric Logistic Regression

We consider nonparametric logistic regression and propose a generalized likelihood test for detecting a threshold effect that indicates a relationship between some risk factor and a defined outcome above the threshold but none below it. One important field of application is occupational medicine and in particular, epidemiological studies. In epidemiological studies, segmented fully parametric logistic regression models are often threshold models, where it is assumed that the exposure has no influence on a response up to a possible unknown threshold, and has an effect beyond that threshold. Finding efficient methods for detection and estimation of a threshold is a very important task in these studies. This article proposes such methods in a context of nonparametric logistic regression. We use a local version of unknown likelihood functions and show that under rather common assumptions the asymptotic power of our test is one. We present a guaranteed non asymptotic upper bound for the significance level of the proposed test. If applying the test yields the acceptance of the conclusion that there was a change point (and hence a threshold limit value), we suggest using the local maximum likelihood estimator of the change point and consider the asymptotic properties of this estimator.     

Design of accelerated life test plans under progressive Type II interval censoring with random removals

This paper investigates the design of accelerated life test (ALT) plans under progressive Type II interval censoring with random removals. Units’ lifetimes are assumed to follow a Weibull distribution, and the number of random removals at each inspection is assumed to follow a binomial distribution. The optimal ALT plans, which minimize the asymptotic variance of an estimated quantile at use condition, are determined. The expected duration of the test and the expected number of inspections on each stress level are calculated. A numerical study is conducted to investigate the properties of the derived ALT plans under different parameter values. For illustration purpose, a numerical example is also given.     

Modelling Accelerated Degradation Data Using Wiener Diffusion With A Time Scale Transformation

Engineering degradation tests allow industry to assess the potential life span of long-life products that do not fail readily under accelerated conditions in life tests. A general statistical model is presented here for performance degradation of an item of equipment. The degradation process in the model is taken to be a Wiener diffusion process with a time scale transformation. The model incorporates Arrhenius extrapolation for high stress testing. The lifetime of an item is defined as the time until performance deteriorates to a specified failure threshold. The model can be used to predict the lifetime of an item or the extent of degradation of an item at a specified future time. Inference methods for the model parameters, based on accelerated degradation test data, are presented. The model and inference methods are illustrated with a case application involving self-regulating heating cables. The paper also discusses a number of practical issues encountered in applications. This revised version was published online in July 2006 with corrections to the Cover Date.     

Marginal Projected Multivariate Linear Models for Clustered Angular Data

Among the diverse frameworks that have been proposed for regression analysis of angular data, the projected multivariate linear model provides a particularly appealing and tractable methodology. In this model, the observed directional responses are assumed to correspond to the angles formed by latent bivariate normal random vectors that are assumed to depend upon covariates through a linear model. This implies an angular normal distribution for the observed angles, and incorporates a regression structure through a familiar and convenient relationship. In this paper we extend this methodology to accommodate clustered data (e.g., longitudinal or repeated measures data) by formulating a marginal version of the model and basing estimation on an EM‐like algorithm in which correlation among within‐cluster responses is taken into account by incorporating a working correlation matrix into the M step. A sandwich estimator is used for the parameter estimates’ covariance matrix. The methodology is motivated and illustrated using an example involving clustered measurements of microbril angle on loblolly pine (Pinus taeda L.) Simulation studies are presented that evaluate the finite sample properties of the proposed fitting method. In addition, the relationship between within‐cluster correlation on the latent Euclidean vectors and the corresponding correlation structure for the observed angles is explored.     

Accelerated life tests for log-normal series system with dependent masked data under Type-I progressive hybrid censoring

This article considers the constant stress accelerated life test for series system products, where independent log-normal distributed lifetimes are assumed for the components. Based on Type-I progressive hybrid censored and masked data, the expectation-maximization algorithm is applied to obtain the estimation for the unknown parameters, and the parametric bootstrap method is used for the standard deviation estimation. In addition, Bayesian approach combining latent variable with Gibbs sampling is developed. Further, the reliability functions of the system and components are estimated at use stress level. The proposed method is illustrated through a numerical example under different masking probabilities and censoring schemes.     

Multivariate Latent Growth Models for Mixed Data with Covariate Effects

The paper presents an extension of a new class of multivariate latent growth models (Bianconcini and Cagnone, 2012) to allow for covariate effects on manifest, latent variables and random effects. The new class of models combines: (i) multivariate latent curves that describe the temporal behavior of the responses, and (ii) a factor model that specifies the relationship between manifest and latent variables. Based on the Generalized Linear and Latent Variable Model framework (Bartholomew and Knott, 1999), the response variables are assumed to follow different distributions of the exponential family, with item-specific linear predictors depending on both latent variables and measurement errors. A full maximum likelihood method is used to estimate all the model parameters simultaneously. Data coming from the Data WareHouse of the University of Bologna are used to illustrate the methodology.     

Do Flow Rates Respond Asymmetrically to Water Level? Evidence from the Edwards Aquifer

This research examines the time series relationship between the Comal Springs flow rate and the water level in the Edwards Aquifer (Well J-17). The empirical methodology utilizes threshold autoregression (TAR) and momentum-TAR models that allow for asymmetry in responses and adjustments to a disequilibrium in the long-run cointegrating relationship. Based on the results, an asymmetric error-correction model (AECM) is proposed to characterize the short-run and long-run dynamic relationship between spring flow and water level. The results have implications for the management of water resources, water demand, and ecosystems.     

Optimal bivariate step-stress accelerated life test for Type-I hybrid censored data

This paper presents a step-stress accelerated life test for two stress variables to obtain optimal hold times under a Type-I hybrid censoring scheme. An exponentially distributed life and a cumulative exposure model are assumed. The maximum-likelihood estimates are given, from which the asymptotic variance and the Fisher information matrix are obtained. The optimal test plan is determined for each combination of stress levels by minimizing the asymptotic variance of reliability estimate at a typical operating condition. Finally, simulation results are discussed to illustrate the proposed criteria. Simulation results show that the proposed optimum plan is robust, and the initial estimates have a small effect on optimal values.     

Weibull lagged effect step-stress model with competing risks

In many survival analysis studies, failure can come from one of several competing risks. Additionally, where survival times are lengthy, researchers can increase stress levels to cause units to fail faster. One type of accelerated testing is a step-stress test where the increase is presented in quantum jumps at predetermined time points. If the impact of the increase is not immediately attained, an interim lag period is modeled. In this article, we propose a two-competing risk step-stress model with a lag period where each independent risk follows a Weibull lifetime distribution, the interim lag period is linear, and the attainment point is assumed known. We obtain the maximum likelihood estimators and the observed information matrix; we construct confidence intervals and provide estimates of coverage probabilities using large sample theory, percentile bootstrap, and bias-corrected accelerated (BCa) bootstrap methods.     

Optimum constant-stress life test plans for Pareto distribution under type-I censoring

The paper considers the case of constant-stress partially accelerated life testing (CSPALT) when two stress levels are involved under type-I censoring. The lifetimes of test items are assumed to follow a two-parameter Pareto lifetime distribution. Maximum-likelihood method is used to estimate the parameters of CSPALT model. Confidence intervals for the model parameters are constructed. Optimum CSPALT plans that determine the best choice of the proportion of test units allocated to each stress are developed. Such optimum test plans minimize the generalized asymptotic variance of the maximum-likelihood estimators of the model parameters. For illustration, Monte Carlo simulation studies are presented.