Dynamic Modeling and Statistical Analysis of Event Times

This review article provides an overview of recent work in the modeling and analysis of recurrent events arising in engineering, reliability, public health, biomedicine and other areas. Recurrent event modeling possesses unique facets making it different and more difficult to handle than single event settings. For instance, the impact of an increasing number of event occurrences needs to be taken into account, the effects of covariates should be considered, potential association among the interevent times within a unit cannot be ignored, and the effects of performed interventions after each event occurrence need to be factored in. A recent general class of models for recurrent events which simultaneously accommodates these aspects is described. Statistical inference methods for this class of models are presented and illustrated through applications to real data sets. Some existing open research problems are described.Comment: Published at http://dx.doi.org/10.1214/088342306000000349 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adjusting for bias introduced by instrumental variable estimation in the Cox Proportional Hazards Model

Instrumental variable (IV) methods are widely used for estimating average treatment effects in the presence of unmeasured confounders. However, the capability of existing IV procedures, and most notably the two-stage residual inclusion (2SRI) procedure recommended for use in nonlinear contexts, to account for unmeasured confounders in the Cox proportional hazard model is unclear. We show that instrumenting an endogenous treatment induces an unmeasured covariate, referred to as an individual frailty in survival analysis parlance, which if not accounted for leads to bias. We propose a new procedure that augments 2SRI with an individual frailty and prove that it is consistent under certain conditions. The finite sample-size behavior is studied across a broad set of conditions via Monte Carlo simulations. Finally, the proposed methodology is used to estimate the average effect of carotid endarterectomy versus carotid artery stenting on the mortality of patients suffering from carotid artery disease. Results suggest that the 2SRI-frailty estimator generally reduces the bias of both point and interval estimators compared to traditional 2SRI.Comment: 27 pages, 8 figures, 4 table

Bayesian analysis of default and credit migration : latent factor models for event count and time-to-event data

This thesis develops Bayesian models to explain credit default and migration risk. Credit risk models used in practice are based on an assumption of conditionally independent events given a realization of systematic risk factors. The systematic risk can be modelled with both observed and unobserved factors. On the one hand we consider generalised linear mixed models (GLMMs) for default count data where random e ects account for unobserved factor risk. On the other hand we consider survival models with shared frailties to model unobserved factors in time-to-default and time-to-rating-transition data. The latter models are developed in the Anderson-Gill counting process framework for the Cox proportional hazards model to allow multiple events and time-dependent covariates. Using Standard and Poor's data on default and rating transitions we control for observed macroeconomic factors in the xed e ect parts of the models. We allow the latent factors to have autoregressive time series structure. The results from both kinds of model show clear evidence of heterogeneity between industry sectors/countries and time period suggesting that di erent latent factor effects are present in di erent sectors. This is an important message that should be accounted for in risk analyses. We implement Bayesian inference for all our models and use the MCMC approach (Gibbs sampling). We show some tractable model formulations that capture the main sources and implement Bayesian model choice procedures to select the most explanatory models. There are couple of contributions in this thesis: First, this is an analysis of industry e ects on default and migration rates using vector-valued random e ects in default count models and vector-valued dynamic frailties in time-to-event/survival models. While this has been done before in models for default counts (McNeil-Wendin) it is quite novel for time-to-event models. Koopman, Lucas and Schwaab (2012) which has some similarities but the estimation is by Monte Carlo maximum likelihood, not by Bayesian methods. Second, estimation of rating transition model with shared dynamic frailties for di erent industry sectors and macroeconomic covariates using Bayesian techniques (MCMC). This is a new model which is based on a simpler model used in medical statistics (Manda & Mayer(2005)) that has been adapted and extended for the credit risk application. We show how to estimate the new model using a Bayesian approach. Finally, we use the model to compute point-in-time dynamic estimates of rating transition probabilities for di erent industry sectors and forecast these into the future, while taking into account macroeconomic factors. This can be very useful for risk management applications and economic scenario generation

Partial Likelihood Estimation of a Cox Model with Random Effects: an EM Algorithm based on Penalized Likelihood.

The aim of this paper is to present a general EM algorithm to estimate Mixed Proportional Hazard models including more than one random effect, through partial likelihood. We assume only that the mixing distributions admit Laplace transforms. We show how to transform inference in a single complicated model in the estimation of MPH models involving only a single frailty, which are easily manageable. We then face on gamma unobserved heterogeneity. This choice is a weak assumption as the heterogeneity distribution among survivors converges to a gamma distribution, often quickly, for many types of unobserved heterogeneity distributions. The proposed approach can thus be used to estimate a wide class of models. We describe how to use the penalized partial likelihood within the EM algorithm, to improve speed and stability. The behaviour of the estimator on different clusterings and sample sizes is assessed through a Monte Carlo study. We also provide an application on the ratiffcation of ILO conventions by developing countries over the period 1975-1995. Both the simulations and the empirical results indicate an important decrease in computing time. Furthermore, our procedure converges in settings where a standard EM algorithm does not.Random Effects, Duration analysis, Dynamic model

A Comparison of Parameter Estimation Methods for Shared Frailty Models

This paper compares six different parameter estimation methods for shared frailty models via a series of simulation studies. A shared frailty model is a survival model that incorporates a random effect term, where the frailties are common or shared among individuals within specific groups. Several parameter estimation methods are available for fitting shared frailty models, such as penalized partial likelihood (PPL), expectation-maximization (EM), pseudo full likelihood (PFL), hierarchical likelihood (HL), maximum marginal likelihood (MML), and maximization penalized likelihood (MPL) algorithms. These estimation methods are implemented in various R packages, providing researchers with various options for analyzing clustered survival data using shared frailty models. However, there is a limited amount of research comparing the performance of these parameter estimation methods for fitting shared frailty models. Consequently, it can be challenging for users to determine the most appropriate method for analyzing clustered survival data. To address this gap, this paper aims to conduct a series of simulation studies to compare the performance of different parameter estimation methods implemented in R packages. We will evaluate several key aspects, including parameter estimation, bias and variance of the parameter estimates, rate of convergence, and computational time required by each package. Through this systematic evaluation, our goal is to provide a comprehensive understanding of the advantages and limitations associated with each estimation method

A Modification Of The EM Algorithm To Estimate An Andersen-Gill Gamma Frailty Model For Multivariate Failure Time Data

A modification of the Andersen-Gill gamma shared frailty model is presented. The variance of the frailty is directly modeled by means of a generalized linear model, the EM algorithm is modified in order to simultaneously estimate a semiparametric model for the failure times and a model for the variance of the frailty. A simulation study is conducted to evaluate the performance of the proposed algorithm (EMB algorithm) and compared with other methods, a marginal model, and a conditional model. Multivariate data from a nosocomial infection study is used to illustrate the methods. The EMB fit turned out to be better than the fit obtained from a marginal model or from a conditional model. The EMB provided the best fit (being the least over-dispersed and having the highest AIC and the highest pseudo-R square) and estimated the parameters most efficiently. The proposed method is able to capture and to take into account unobservable random effects in semiparametric models

Frailty multi-state models for the analysis of survival data from multicenter clinical trials

Proportional hazards models are among the most popular regression models in survival analysis. Multi-state models generalise them in the sense of jointly considering different types of events along with their interrelations, whereas frailty models introduce random effects to account for unobserved risk factors, possibly shared by groups of subjects. The integration of frailty and multi-state methodology is interesting to control for unobserved heterogeneity in presence of complex event history structures, particularly appealing in multicenter clinical trials applications. In the present thesis we propose the incorporation of nested frailties in the transition-specific hazard function; then, we develop and evaluate both parametric and semi-parametric inference. Simulation studies, performed thanks to an innovative method for generating dependent multi-state survival data, show that parametric inference is correct but extremely imprecise, whilst semiparametric methods are very competitive to evaluate the effect of covariates. Two case studies are presented, relative to cancer multicenter clinical trials. The multi-state nature of the models allows to study the treatment effect taking into account intermediate events, while the presence of frailties reduces the attenuation effect due to clustering. Finally, we present two new software tools, one to fit parametric frailty models with up to twenty possible combinations of baseline and frailty distributions, and one implementing semiparametric inference for multilevel frailty models, essential to fit the new nested frailty multi-state models

Growth Collapses

We study episodes where economic growth decelerates to negative rates. While the majority of these episodes are of short duration, a substantial fraction last for a longer period of time than can be explained as the result of business-cycle dynamics. The duration, depth and associated output loss of these episodes differs dramatically across regions. We investigate the factors associated with the entry of countries into these episodes as well as their duration. We find that while countries fall into crises for multiple reasons, including wars, export collapses, sudden stops and political transitions, most of these variables do not help predict the duration of crises episodes. In contrast, we find that a measure of the density of a country’s export product space is significantly associated with lower crisis duration. We also find that unconditional and conditional hazard rates are decreasing in time, a fact that is consistent with either strong shocks to fundamentals or with models of poverty traps.