The thesis presents fundamental characteristics of survival analysis in the case of competing risks and their relationships. In the case without regression, basic nonparametric estimates and a logarithmic likelihood function for parameter estimates is given. The main focus is on Cox's proportional hazards model (PH), a model with accelerated time (AFT) and a flexible regression model (FG) are also mentioned. The identifiability of the associated survival function is solved using copulas. Basics of copula theory and the measurement of dependence by correlation coefficients (Pearson, Spearman and Kendal) are described in a separate chapter. A substantial part of the theory is practically used in a generated case without regression.

Los modelos de sobrevivencia o duración son útiles para modelar la distribución subyacente del periodo en el que ocurre el evento específico. El presente artículo analiza la duración de la tasa de referencia del Banco Central de Reserva del Perú (BCRP) y sus determinantes, haciendo uso de los modelos de sobrevivencia para un análisis que incluye los riesgos competitivos. En presencia de riesgos competitivos, el enfoque convencional de la duración puede obtener resultados sesgados y no interpretables. Por lo que, siguiendo la propuesta inicial de Gutiérrez y Lozano (2010), se recurre al análisis de riesgos competitivos a fin de analizar la duración entre los cambios de tasa de política monetaria en Perú, tomando en cuenta los dos escenarios posibles (incrementos y recortes) y magnitudes (25 pb y más de 25 pb); así como las variables que inciden en su comportamiento. Las regresiones bajo riesgos competitivos sugieren un comportamiento asimétrico en lo que respecta a las variables que definen los cambios de la tasa de referencia (incrementos o recortes). Variables como la inflación, producto y la tasa de referencia del periodo afectan al riesgo de ambos estados; sin embargo, un recorte en la tasa de referencia es también determinado por la brecha de la inflación local respecto a la extranjera y la duración de la tasa de referencia previa. En particular, los resultados son consistentes con una economía regida bajo el marco de metas de inflación. Se extrae que, el BCRP puede mantener la tasa de referencia en un nivel constante hasta que las variables de interés, como la inflación, se encuentren en condiciones críticas. Los resultados de las pruebas también confirman que la duración de tasas con cambios pequeños y grandes no son estadísticamente diferentes en las subidas de tasas. / Survival or duration models are useful for modeling the underlying distribution of the period in which the specific event occurs. This article analyzes the duration of the monetary policy rate of Peru and its determinants, in base of survival models including competing risks. In the presence of competing risks, the conventional duration method could get biased and uninterpretable results. Therefore, following the initial proposal of Gutierrez and Lozano (2010), this article includes competitive risks in order to analyze the duration between changes in the monetary policy rate of Peru, taking into account two possible scenarios, rate hikes and rate cuts, and magnitudes (25 bp and more than 25 bp); as well as the variables that affect their behavior. The regressions under competing risks suggest an asymmetric behavior between the variables that define the specific event of the monetary policy rate (increases or decreases). The models for rate hikes and rate cuts agree in finding the influences of variables, in the risk of both specific events: inflation, domestic product and the monetary policy rate. However, a cut in the monetary policy rate is also determined by the gap between local and US inflation and the duration of the previous rate. The results are consistent with an economy under the inflation targeting framework. As an inference, the Central Reserve Bank of Peru can maintain the reference rate at a constant level until the variables of interest, such as inflation, are in critical conditions. Test results also confirm that the duration of rates with small and large changes are not statistically different in rate hikes. / Trabajo de investigación

Indiana University-Purdue University Indianapolis (IUPUI) / Observational studies and clinical trials with time-to-event data frequently involve multiple event types, known as competing risks. The cumulative incidence function (CIF) is a particularly useful parameter as it explicitly quantifies clinical prognosis. Common issues in competing risks data analysis on the CIF include interval censoring, missing event types, and left truncation. Interval censoring occurs when the event time is not observed but is only known to lie between two observation times, such as clinic visits. Left truncation, also known as delayed entry, is the phenomenon where certain participants enter the study after the onset of disease under study. These individuals with an event prior to their potential study entry time are not included in the analysis and this can induce selection bias. In order to address unmet needs in appropriate methods and software for competing risks data analysis, this thesis focuses the following development of application and methods. First, we develop a convenient and exible tool, the R package intccr, that performs semiparametric regression analysis on the CIF for interval-censored competing risks data. Second, we adopt the augmented inverse probability weighting method to deal with both interval censoring and missing event types. We show that the resulting estimates are consistent and double robust. We illustrate this method using data from the East-African International Epidemiology Databases to Evaluate AIDS (IeDEA EA) where a significant portion of the event types is missing. Last, we develop an estimation method for semiparametric analysis on the CIF for competing risks data subject to both interval censoring and left truncation. This method is applied to the Indianapolis-Ibadan Dementia Project to identify prognostic factors of dementia in elder adults. Overall, the methods developed here are incorporated in the R package intccr. / 2021-05-06

Indiana University-Purdue University Indianapolis (IUPUI) / It is generally of interest to explore if the risk of death would be modified by medical conditions (e.g., illness) that have occurred prior. This situation gives rise to semicompeting risks data, which are a mixture of competing risks and progressive state data. This type of data occurs when a non-terminal event can be censored by a well-defined terminal event, but not vice versa. In the first part of this dissertation, the shared gamma-frailty conditional Markov model (GFCMM) is adopted because it bridges the copula models and illness-death models. Maximum likelihood estimation methodology has been proposed in the literature. However, we found through numerical experiments that the unrestricted model sometimes yields nonparametric biased estimation. Hence a practical guideline is provided for using the GFCMM that includes (i) a score test to assess whether the restricted model, which does not exhibit estimation problems, is reasonable under a proportional hazards assumption, and (ii) a graphical illustration to evaluate whether the unrestricted model yields nonparametric estimation with substantial bias for cases where the test provides a statistical significant result against the restricted model. However, the scientific question of interest that whether the status of non-terminal event alters the risk to terminal event can only be partially addressed based on the aforementioned approach. Therefore in the second part of this dissertation, we adopt a Markov illness-death model, whose transition intensities are essentially equivalent to the marginal hazards defined in GFCMM, but with different interpretations; we develop three nonparametric tests, including a linear test, a Kolmogorov-Smirnov-type test, and a L2-distance-type test, to directly compare the two transition intensities under consideration. The asymptotic properties of the proposed test statistics are established using empirical process theory. The performance of these tests in nite samples is numerically evaluated through extensive simulation studies. All three tests provide similar power levels with non-crossing curves of cumulative transition intensities, while the linear test is suboptimal when the curves cross. Eventually, the proposed tests successfully address the scientific question of interest. This research is applied to Indianapolis-Ibadan Dementia Project (IIDP) to explore whether dementia occurrence changes mortality risk. / 2022-05-06

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Understanding the lifetime behavior of their products is crucial to the success of any company in the manufacturing and engineering industries. Statistical methods for lifetime data are a key component to achieving this level of understanding. Sometimes a statistical procedure must be updated to be adequate for modeling specific data as is discussed in Chapter 2. However, there are cases in which the methods used in industrial standards are themselves inadequate. This is distressing as more appropriate statistical methods are available but remain unused. The research in Chapter 4 deals with such a situation. The research in Chapter 3 serves as a combination of both scenarios and represents how both statisticians and engineers from the industry can join together to yield beautiful results. After introducing basic concepts and notation in Chapter 1, Chapter 2 focuses on lifetime prediction for a product consisting of multiple components. During the production period, some components may be upgraded or replaced, resulting in a new ``generation" of component. Incorporating this information into a competing risks model can greatly improve the accuracy of lifetime prediction. A generalized competing risks model is proposed and simulation is used to assess its performance. In Chapter 3, optimal and compromise test plans are proposed for constant amplitude fatigue testing. These test plans are based on a nonlinear physical model from the fatigue literature that is able to better capture the nonlinear behavior of fatigue life and account for effects from the testing environment. Sensitivity to the design parameters and modeling assumptions are investigated and suggestions for planning strategies are proposed. Chapter 4 considers the analysis of ADDT data for the purposes of estimating a thermal index. The current industry standards use a two-step procedure involving least squares regression in each step. The methodology preferred in the statistical literature is the maximum likelihood procedure. A comparison of the procedures is performed and two published datasets are used as motivating examples. The maximum likelihood procedure is presented as a more viable alternative to the two-step procedure due to its ability to quantify uncertainty in data inference and modeling flexibility. / Ph. D.

2024 / Contemporary randomized trials frequently assess treatment effects across multiple time-to-event outcomes. In scenarios involving competing risks, prioritized outcomes, or informative censoring, alternatives to conventional methods to estimate and test for treatment effects are needed. For competing risks data, we proposed a doubly robust estimator for the difference in the restricted mean times lost to a specific cause. The estimator relies on non-parametric pseudo-observations of the cumulative incidence function, and therefore does not rely on the proportional hazard assumption. We evaluated the performance of the estimator in different scenarios of model misspecification. We applied the estimator to compare the event-free time lost to disease progression in the POPLAR and OAK studies for non-small-cell lung cancer. For prioritized time-to-event outcomes, we compared the performance of novel tests that prioritize events with higher clinical importance to traditional tests that do not. None of the tests was uniformly best when component-wise treatment effects varied. As these tests differ in how they characterize the treatment effect over the entire disease course, we proposed a generalizable framework to quantify the information used and ignored by each test. Under the Gumbel survival copula model, we also derived analytically the true value of the treatment effect corresponding to each test. We illustrated these methods using a five-component prioritized outcome in the SPRINT randomized trial. For informative censoring, we considered the issue of differential censoring between randomization groups in oncology trials. We assessed the impact of informative censoring on the treatment effect estimation, as well as on the performance of generalized log-rank tests under a delayed effect setting. We showed how to generate informative censoring data from survival copulas with piece-wise exponential marginals. We also derived the relationship between the copula rank correlation and the probability of informative censoring. We showed how to use this relationship to guide the choice of an adequate copula model to analyze informative censoring data. / 2027-02-12T00:00:00Z

The typical research of Alzheimer's disease includes a series of cognitive states. Multi-state models are often used to describe the history of disease evolvement. Competing risks models are a sub-category of multi-state models with one starting state and several absorbing states. Analyses for competing risks data in medical papers frequently assume independent risks and evaluate covariate effects on these events by modeling distinct proportional hazards regression models for each event. Jeong and Fine (2007) proposed a parametric proportional sub-distribution hazard (SH) model for cumulative incidence functions (CIF) without assumptions about the dependence among the risks. We modified their model to assure that the sum of the underlying CIFs never exceeds one, by assuming a proportional SH model for dementia only in the Nun study. To accommodate left censored data, we computed non-parametric MLE of CIF based on Expectation-Maximization algorithm. Our proposed parametric model was applied to the Nun Study to investigate the effect of genetics and education on the occurrence of dementia. After including left censored dementia subjects, the incidence rate of dementia becomes larger than that of death for age < 90, education becomes significant factor for incidence of dementia and standard errors for estimates are smaller. Multi-state Markov model is often used to analyze the evolution of cognitive states by assuming time independent transition intensities. We consider both constant and duration time dependent transition intensities in BRAiNS data, leading to a mixture of Markov and semi-Markov processes. The joint probability of observing a sequence of same state until transition in a semi-Markov process was expressed as a product of the overall transition probability and survival probability, which were simultaneously modeled. Such modeling leads to different interpretations in BRAiNS study, i.e., family history, APOE4, and sex by head injury interaction are significant factors for transition intensities in traditional Markov model. While in our semi-Markov model, these factors are significant in predicting the overall transition probabilities, but none of these factors are significant for duration time distribution.