Use-level lifetime distribution estimation under dependent right censored test data

Accelerated life testing (ALT) is a practice for estimating unit reliability at normal use conditions using failure data obtained under more severe test conditions. We focus on life tests where a potential critical unit failure at X2 (unit lifetime) may be avoided by a degraded failure at some random time X1. Degraded and critical failures are linked through the degradation process,hence the situation under consideration is that of dependent competing risks. We apply the general result that if the copula C(.; .) of (X1; X2) is known, competing risks data uniquely determine the marginal distributions at each stress level. Interest here (and in life testing studies in general) is in unit lifetime. Accordingly, our target of estimation is to extrapolate a use-level lifetime distribution from which important reliability measures such as mean lifetime, warranty period among others are derived. The paper is based in part on a PhD thesis by Hove (2014)

Contributions to accelerated reliability testing

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, December 2014.Industrial units cannot operate without failure forever. When the operation of a unit deviates from industrial standards, it is considered to have failed. The time from the moment a unit enters service until it fails is its lifetime. Within reliability and often in life data analysis in general, lifetime is the event of interest. For highly reliable units, accelerated life testing is required to obtain lifetime data quickly. Accelerated tests where failure is not instantaneous, but the end point of an underlying degradation process are considered. Failure during testing occurs when the performance of the unit falls to some specified threshold value such that the unit fails to meet industrial specifications though it has some residual functionality (degraded failure) or decreases to a critical failure level so that the unit cannot perform its function to any degree (critical failure). This problem formulation satisfies the random signs property, a notable competing risks formulation originally developed in maintenance studies but extended to accelerated testing here. Since degraded and critical failures are linked through the degradation process, the open problem of modelling dependent competing risks is discussed. A copula model is assumed and expert opinion is used to estimate the copula. Observed occurrences of degraded and critical failure times are interpreted as times when the degradation process first crosses failure thresholds and are therefore postulated to be distributed as inverse Gaussian. Based on the estimated copula, a use-level unit lifetime distribution is extrapolated from test data. Reliability metrics from the extrapolated use-level unit lifetime distribution are found to differ slightly with respect to different degrees of stochastic dependence between the risks. Consequently, a degree of dependence between the risks that is believed to be realistic to admit is considered an important factor when estimating the use-level unit lifetime distribution from test data. Keywords: Lifetime; Accelerated testing; Competing risks; Copula; First passage time

Reliability Estimation of Rotary Lip Seal in Aircraft Utility System Based on Time-Varying Dependence Degradation Model and Its Experimental Validation

With several attractive properties, rotary lip seals are widely used in aircraft utility system, and their reliability estimation has drawn more and more attention. This work proposes a reliability estimation approach based on time-varying dependence analysis. The dependence between the two performance indicators of rotary lip seals, namely leakage rate and friction torque, is modeled by time-varying copula function with polynomial to denote the time-varying parameters, and an efficient copula selection approach is utilized to select the optimal copula function. Parameter estimation is carried out based on a Bayesian method and the reliability during the whole lifetime is calculated based on a Monte Carlo method. Degradation test for rotary lip seal is conducted and the proposed model is validated by test data. The optimal copula function and optimal order of polynomial are determined based on test data. Results show that this model is effective in estimating the reliability of rotary lip seals and can achieve a better goodness of fit

Joint modeling of bivariate time to event data with semi-competing risk

Indiana University-Purdue University Indianapolis (IUPUI)Survival analysis often encounters the situations of correlated multiple events including the same type of event observed from siblings or multiple events experienced by the same individual. In this dissertation, we focus on the joint modeling of bivariate time to event data with the estimation of the association parameters and also in the situation of a semi-competing risk. This dissertation contains three related topics on bivariate time to event mod els. The first topic is on estimating the cross ratio which is an association parameter between bivariate survival functions. One advantage of using cross-ratio as a depen dence measure is that it has an attractive hazard ratio interpretation by comparing two groups of interest. We compare the parametric, a two-stage semiparametric and a nonparametric approaches in simulation studies to evaluate the estimation perfor mance among the three estimation approaches. The second part is on semiparametric models of univariate time to event with a semi-competing risk. The third part is on semiparametric models of bivariate time to event with semi-competing risks. A frailty-based model framework was used to accommodate potential correlations among the multiple event times. We propose two estimation approaches. The first approach is a two stage semiparametric method where cumulative baseline hazards were estimated by nonparametric methods first and used in the likelihood function. The second approach is a penalized partial likelihood approach. Simulation studies were conducted to compare the estimation accuracy between the proposed approaches. Data from an elderly cohort were used to examine factors associated with times to multiple diseases and considering death as a semi-competing risk

Partial Identification and Inference in Censored Quantile Regression: A Sensitivity Analysis

In this paper we characterize the identified set and construct asymptotically valid and non-conservative confidence sets for the quantile regression coeffi cient in a linear quantile regression model, where the dependent variable is subject to possibly dependent censoring. The underlying censoring mechanism is characterized by an Archimedean copula for the dependent variable and the censoring variable. For a broad class of Archimedean copulas, we characterize an outer set of the corresponding identified set for the quantile regression coeffi cient via inequality constraints. For one-parameter ordered families of Archimedean copulas, we construct simple confidence sets by inverting asymptotically pivotal statistics related to kernel-based model specification testing. The methodology we develop in this paper allows practitioners to conduct sensitivity analysis of the robustness of conclusions on the quantile regression coeffi cient to the independent censoring mechanism. Bootstrap confidence sets are also constructed. Interpreting the dependent variable and the censoring variable in our censored quantile regression model as two competing risks, our methodology is useful in duration analysis with possibly dependent competin

Relaxing Assumptions on the Censoring Mechanism in Survival Link-Based Additive Models

Survival models are frequently encountered in applications. In these models, the response of interest, the time until a particular event occurs, is often right censored. Most estimation methods assume that the event time and the censoring time are stochastically independent and non-informative conditional on covariates. However, these assumptions may be questioned. The aim of this thesis is to relax these assumptions in a class of flexible parametric survival models, called survival link-based additive models. The assumption of non-informative censoring is relaxed by assuming that the marginal survival functions of the event and censoring times have parameters in common. In particular, we provide evidence on the efficiency gains produced by the newly introduced informative estimator when compared to its non-informative counterpart. The independence assumption is relaxed by modelling both the event time and the censoring time simultaneously using copula functions. We provide some preliminary arguments towards model identification although this topic is very challenging and requires more future work. In these survival link-based additive models, the baseline functions are estimated non-parametrically by monotonic P-splines, whereas covariate effects are flexibly determined using additive predictors that allow for a vast variety of effects. Parameter estimation is reliably carried out within a penalised maximum likelihood framework with integrated automatic multiple smoothing parameter selection. We derive the √n-consistency and asymptotic normality of the estimators proposed in this thesis. Their finite sample performance are investigated via Monte Carlo simulation studies, and the approaches illustrated using two cases study based on infants hospitalised for pneumonia as well as prostate cancer data. The R package GJRM has been extended to incorporate the developments discussed in this thesis to facilitate transparent and reproducible researc

Semi-parametric regression analysis of interval-censored failure time data

"July 2014."Dissertation Supervisor: Dr. (Tony) Jianguo Sun.Includes vita.By interval-censored data, we mean that the failure time of interest is known only to lie within an interval instead of being observed exactly. Many clinical trials and longitudinal studies may generate interval-censored data. One common example occurs in medical or health studies that entail periodic follow-ups. An important special case of interval-censored data is the so called current status data when each subject is observed only once for the status of the occurrence of the event of interest. That is, instead of observing the survival endpoint directly, we only know the observation time and whether or not the event of interest has occurred at that time. Such data may occur in many fields, for example, cross-sectional studies and tumorigenicity experiments. Sometimes we also refer current status data to as case I interval-censored data and the general case as case II interval-censored data. In the following, for simplicity, we will refer current status data and interval-censored data to case I and case II interval-censored data, respectively. The statistical analysis of both case I and case II interval-censored failure time data has recently attracted a great deal of attention and especially, many procedures have been proposed for their regression analysis under various models. However, due to the strict restrictions of existing regression analysis procedures and practical demands, new methodologies for regression analysis need to be developed. For regression analysis of interval-censored data, many approaches have been proposed and for most of them, the inference is carried out based on the asymptotic normality. It's well known that the symmetric property implied by the normal distribution may not be appropriate sometimes and could underestimate the variance of estimated parameters. In the first part of this dissertation, we adopt the linear transformation models for regression analysis of interval-censored data and propose an empirical likelihood-based procedure to address the underestimating problem fromIncludes bibliographical references (pages 127-134)

Interval-censored semi-competing risks data: a novel approach for modelling bladder cancer

Aquesta tesi tracta sobre tècniques d'anàlisi de supervivència en situacions amb múltiples esdeveniments i patrons complexes de censura. Proposem una nova metodologia per tractar la situació de riscos semi-competitius quan les dades estan censurades en un interval. La motivació del treball neix de la nostra col·laboració amb l'Estudi Espanyol del Càncer de Bufeta (SBC/EPICURO), el més gran estudi sobre càncer de bufeta realitzat fins ara a l'Estat Espanyol. La nostra contribució en el projecte es centra en la modelització i identificació de factors pronòstics de l'evolució de la malaltia.L'evolució de malalties complexes, com el càncer o la infecció VIH, es caracteritza per la ocurrència de múltiples esdeveniments en el mateix pacient: per exemple, la recaiguda de la malaltia o la mort. Aquests esdeveniments poden ser finals, quan el seguiment del pacient s'atura després de l'esdeveniment, o bé intermedis, quan l'individu continua sota observació. La presència d'esdeveniments finals complica l'anàlisi dels intermedis ja que n'impedeix la seva completa observació, induint una possible censura depenent.En aquest context, es requereixen metodologies apropiades. Els següents mètodes són emprats: riscos competitius, models multiestat i riscos semi-competitius. A resultes de l'aplicació de mètodes per riscos competitius i models multi-estat, proposem dues aportacions rellevants al coneixement de la malaltia: (1) la caracterització dels pacients amb un alt risc de progressió com a primer esdeveniment després de la diagnosi, i (2) la construcció d'un model pronòstic dinàmic per al risc de progressió.La situació de riscos competitius es dóna quan volem descriure el temps fins al primer entre K possibles esdeveniments, juntament amb un indicador del tipus d'esdeveniment observat. En l'estudi EPICURO, és rellevant estudiar el temps fins al primer entre recidiva, progressió o mort. La caracterització d'aquest primer esdeveniment permetria seleccionar el millor tractament d'acord amb el perfil de risc basal del pacient.Els models multi-estat descriuen les diferents evolucions que la malaltia pot seguir, establint relacions entre els esdeveniments d'interès: per exemple, un pacient pot experimentar una recidiva del tumor primari, i després morir, o bé pot morir sense haver tingut cap recaiguda de la malaltia. Una característica interessant d'aquests models és que permeten fer prediccions del risc de futurs esdeveniments per a un pacient, d'acord amb la història que hagi pogut tenir fins aquell moment. En el cas de càncer de bufeta podrem avaluar la influència que té en el risc de progressar haver patit o no una recidiva prèvia.Un cas especial de model multi-estat és aquell que conté un esdeveniment intermedi E1, i un esdeveniment final, E2. Siguin T1 i T2 els temps fins aquests esdeveniments, respectivament. Ni l'anàlisi de riscos competitius ni els models multi-estat permeten adreçar l'estudi de la distribució marginal de T1. En efecte, l'anàlisi de riscos competitius tracta amb la distribució del mínim entre els dostemps, T=min(T1,T2), mentre que els models multi-estat es centren en la distribució condicional de T2|T1, és a dir, en com la ocurrència de E1 modifica el risc de E2. En aquest cas, la distribució de T1 no és identificable a partir de les dades observades. La situació abans descrita, on la ocurrència d'un esdeveniment final impedeix l'observació de l'esdeveniment intermedi és coneguda com a riscos semi-competitius (Fine et al., 2001). L'estratègia d'aquests autors passà per assumir un model per a la distribució conjunta (T1, T2), i aleshores recuperar la distribució marginal de T1 derivada d'aquest model.Proposem una nova metodologia per tractar amb riscos semi-competitius quan el temps fins l'esdeveniment intermedi, T1, està censurat en un interval. En molts estudis mèdics longitudinals, la ocurrència de l'esdeveniment d'interès s'avalua en visites periòdiques del pacient, i per tant, T1 és desconegut, però es sap que pertany al interval comprès entre els temps de dues visites consecutives. Els mètodes per riscos semi-competitius en el context usual de censura per la dreta no són vàlids en aquest cas i és necessària una nova aproximació. En aquest treball ampliem la metodología semi-paramètrica proposada per Fine et al. (2001), que assumeix un model de còpula de Clayton (1978) per a descriure la dependència entre T1 i T2. Assumint el mateix model, desenvolupem un algoritme iteratiu que estima conjuntament el paràmetre d'associació del model de còpula, així com la funció de supervivència del temps intermedi T1.Fine, J. P.; Jiang, H. & Chappell, R. (2001), 'On Semi-Competing Risks Data', Biometrika 88(4), 907--919.Clayton, D. G. (1978), 'A Model for Association in Bivariate Life Tables and Its Application in Epidemiological Studies of Familial. Tendency in Chronic Disease Incidence', Biometrika 65(1), 141--151.La presente tesis trata sobre técnicas de análisis de supervivencia en situaciones con múltiples eventos y patrones complejos de censura. Proponemos una nueva metodología para tratar el problema de riesgos semi-competitivos cuando los datos están censurados en un intervalo. La motivación de este trabajo nace de nuestra colaboración con el estudio Español de Cáncer de Vejiga (SBC/EPICURO), el más grande estudio sobre cáncer de vejiga realizado en España hasta el momento. Nuestra participación en el mismo se centra en la modelización e identificación de factores pronósticos en el curso de la enfermedad.El curso de enfermedades complejas tales como el cáncer o la infección por VIH, se caracteriza por la ocurrencia de múltiples eventos en el mismo paciente, como por ejemplo la recaída o la muerte. Estos eventos pueden ser finales, cuando el seguimiento del paciente termina con el evento, o bien intermedios, cuando el individuo sigue bajo observación. La presencia de eventos finales complica el análisis de los eventos intermedios, ya que impiden su completa observación, induciendo una posible censura dependiente.En este contexto, se requieren metodologías apropiadas. Se utilizan los siguientes métodos: riesgos competitivos, modelos multiestado y riesgos semi-competitivos. De la aplicación de métodos para riesgos competitivos y modelos multi-estado resultan dos aportaciones relevantes sobre el conocimiento de la enfermedad: (1) la caracterización de los pacientes con un alto riesgo de progresión como primer evento después del diagnóstico, y (2) la construcción de un modelo pronóstico y dinámico para el riesgo de progresión.El problema de riesgos competitivos aparece cuando queremos describir el tiempo hasta el primero de K posibles eventos, junto con un indicador del tipo de evento observado. En el estudio SBC/EPICURO es relevante estudiar el tiempo hasta el primero entre recidiva, progresión o muerte. La caracterización de este primer evento permitiría seleccionar el tratamiento más adecuado de acuerdo con el perfil de riesgo basal del paciente.Los modelos multi-estado describen las diferentes tipologías que el curso de la enfermedad puede seguir, estableciendo relaciones entre los eventos de interés. Por ejemplo, un paciente puede experimentar una recidiva y después morir, o bien puede morir sin haber tenido recaída alguna. El potencial interesante de los modelos multi-estado es que permiten realizar predicciones sobre el riesgo de futuros eventos dada la historia del paciente hasta ese momento. En el caso del cáncer de vejiga, podremos evaluar la influencia que tiene en el riesgo de progresar el haber tenido o no una recidiva previa.Un caso especial de modelo multi-estado es el que contiene un evento intermedio E1 y uno final, E2. Sean T1 y T2 los tiempos hasta tales eventos, respectivamente. Ni el análisis de riesgos competitivos ni los modelos multi-estado permiten estudiar la distribución marginal de T1. En efecto, el análisis de riesgos competitivos trata con la distribución del mínimo entre los dos tiempos, T=min(T1,T2), mientras que los modelos multi-estado se centran en la distribución condicional de T2 dado T1, T2|T1, en cómo la ocurrencia de E1 modifica el riesgo de E2. En ambos casos, la distribución de T1 no es identificable a partir de los datos observados.La situación anteriormente descrita donde un evento final impide la observación de un evento intermedio se conoce como riesgos semi-competitivos (Fine et al. 2001). La estrategia de estos autores asume un modelo para la distribución conjunta (T1,T2) para así recuperar la distribución de T1 derivada de ese modelo.Proponemos una nueva metodología para tratar con riesgos semi-competitivos cuando el tiempo hasta el evento intermedio, T1, esta censurado en un intervalo. En muchos estudios médicos longitudinales, la ocurrencia del evento de interés se evalúa en visitas periódicas al paciente, por lo que T1 es desconocido, aunque se conoce que pertenece al intervalo comprendido entre los tiempos de dos visitas consecutivas. Los métodos para riesgos semi-competitivos en el contexto usual de censura por la derecha no son válidos en este caso y se requiere una nueva aproximación. En este trabajo ampliamos la metodología semi-paramétrica propuesta por Fine et al. (2001), que asume una cópula de Clayton (1978) para describir la dependencia entre T1 y T2. Bajo el mismo modelo de asociación, desarrollamos un algoritmo iterativo que estima conjuntamente el parámetro de asociación del modelo de cópula, así como la función de supervivencia del tiempo al evento intermedio T1.Fine, J. P.; Jiang, H. & Chappell, R. (2001), 'On Semi-Competing Risks Data', Biometrika 88(4), 907--919. Clayton, D. G. (1978), 'A Model for Association in Bivariate Life Tables and Its Application in Epidemiological Studies of Familial. Tendency in Chronic Disease Incidence', Biometrika 65(1), 141--151