Nonparametric estimation of the survival function for ordered multivariate failure time data: a comparative study

In longitudinal studies of disease, patients may experience several events through a follow-up period. In these studies, the sequentially ordered events are often of interest and lead to problems that have received much attention recently. Issues of interest include the estimation of bivariate survival, marginal distributions and the conditional distribution of gap times. In this work we consider the estimation of the survival function conditional to a previous event. Different nonparametric approaches will be considered for estimating these quantities, all based on the Kaplan-Meier estimator of the survival function. We explore the finite sample behavior of the estimators through simulations. The different methods proposed in this article are applied to a data set from a German Breast Cancer Study. The methods are used to obtain predictors for the conditional survival probabilities as well as to study the influence of recurrence in overall survival.

Bayesian Analysis for the Generalized Lognormal Distribution Applied to Failure Time Analysis

There are several versions of the lognormal distribution in the statistical literature, one is based in the exponential transformation of generalized normal distribution (GN). This paper presents the Bayesian analysis for the generalized lognormal distribution (logGN) considering independent non-informative Jeffreys distributions for the parameters as well as the procedure for implementing the Gibbs sampler to obtain the posterior distributions of parameters. The results are used to analyze failure time models with right-censored and uncensored data. The proposed method is illustrated using actual failure time data of computers.

On the Accelerated Failure Time Model for Current Status and Interval Censored Data

This paper introduces a novel approach to making inference about the regression parameters in the accelerated failure time (AFT) model for current status and interval censored data. The estimator is constructed by inverting a Wald type test for testing a null proportional hazards model. A numerically efficient Markov chain Monte Carlo (MCMC) based resampling method is proposed to simultaneously obtain the point estimator and a consistent estimator of its variance-covariance matrix. We illustrate our approach with interval censored data sets from two clinical studies. Extensive numerical studies are conducted to evaluate the finite sample performance of the new estimators.

Robust accelerated failure time regression

Robust estimators for accelerated failure time models with asymmetric (or symmetric) error distribution and censored observations are proposed. It is assumed that the error model belongs to a log-location-scale family of distributions and that the mean response is the parameter of interest. Since scale is a main component of mean, scale is not treated as a nuisance parameter. A three steps procedure is proposed. In the first step, an initial high breakdown point S estimate is computed. In the second step, observations that are unlikely under the estimated model are rejected or down weighted. Finally, a weighted maximum likelihood estimate is computed. To define the estimates, functions of censored residuals are replaced by their estimated conditional expectation given that the response is larger than the observed censored value. The rejection rule in the second step is based on an adaptive cut-off that, asymptotically, does not reject any observation when the data are generat ed according to the model. Therefore, the final estimate attains full efficiency at the model, with respect to the maximum likelihood estimate, while maintaining the breakdown point of the initial estimator. Asymptotic results are provided. The new procedure is evaluated with the help of Monte Carlo simulations. Two examples with real data are discussed.

Some concepts and models usefulninthe analysis of discrete life time data

This thesis entitled Reliability Modelling and Analysis in Discrete time Some Concepts and Models Useful in the Analysis of discrete life time data.The present study consists of five chapters. In Chapter II we take up the derivation of some general results useful in reliability modelling that involves two component mixtures. Expression for the failure rate, mean residual life and second moment of residual life of the mixture distributions in terms of the corresponding quantities in the component distributions are investigated. Some applications of these results are also pointed out. The role of the geometric,Waring and negative hypergeometric distributions as models of life lengths in the discrete time domain has been discussed already. While describing various reliability characteristics, it was found that they can be often considered as a class. The applicability of these models in single populations naturally extends to the case of populations composed of sub-populations making mixtures of these distributions worth investigating. Accordingly the general properties, various reliability characteristics and characterizations of these models are discussed in chapter III. Inference of parameters in mixture distribution is usually a difficult problem because the mass function of the mixture is a linear function of the component masses that makes manipulation of the likelihood equations, leastsquare function etc and the resulting computations.very difficult. We show that one of our characterizations help in inferring the parameters of the geometric mixture without involving computational hazards. As mentioned in the review of results in the previous sections, partial moments were not studied extensively in literature especially in the case of discrete distributions. Chapters IV and V deal with descending and ascending partial factorial moments. Apart from studying their properties, we prove characterizations of distributions by functional forms of partial moments and establish recurrence relations between successive moments for some well known families. It is further demonstrated that partial moments are equally efficient and convenient compared to many of the conventional tools to resolve practical problems in reliability modelling and analysis. The study concludes by indicating some new problems that surfaced during the course of the present investigation which could be the subject for a future work in this area.

Parametric accelerated failure time models with random effects and an application to kidney transplant survival

Accelerated failure time models with a shared random component are described, and are used to evaluate the effect of explanatory factors and different transplant centres on survival times following kidney transplantation. Different combinations of the distribution of the random effects and baseline hazard function are considered and the fit of such models to the transplant data is critically assessed. A mixture model that combines short- and long-term components of a hazard function is then developed, which provides a more flexible model for the hazard function. The model can incorporate different explanatory variables and random effects in each component. The model is straightforward to fit using standard statistical software, and is shown to be a good fit to the transplant data. Copyright (C) 2004 John Wiley Sons, Ltd.

Quantifying and Comparing the Accuracy of Binary Biomarkers When Predicting a Failure Time Outcome

The positive and negative predictive value are standard measures used to quantify the predictive accuracy of binary biomarkers when the outcome being predicted is also binary. When the biomarkers are instead being used to predict a failure time outcome, there is no standard way of quantifying predictive accuracy. We propose a natural extension of the traditional predictive values to accommodate censored survival data. We discuss not only quantifying predictive accuracy using these extended predictive values, but also rigorously comparing the accuracy of two biomarkers in terms of their predictive values. Using a marginal regression framework, we describe how to estimate differences in predictive accuracy and how to test whether the observed difference is statistically significant.

Sensitivity and out-of-sample error in continuous time data assimilation

Data assimilation refers to the problem of finding trajectories of a prescribed dynamical model in such a way that the output of the model (usually some function of the model states) follows a given time series of observations. Typically though, these two requirements cannot both be met at the same time–tracking the observations is not possible without the trajectory deviating from the proposed model equations, while adherence to the model requires deviations from the observations. Thus, data assimilation faces a trade-off. In this contribution, the sensitivity of the data assimilation with respect to perturbations in the observations is identified as the parameter which controls the trade-off. A relation between the sensitivity and the out-of-sample error is established, which allows the latter to be calculated under operational conditions. A minimum out-of-sample error is proposed as a criterion to set an appropriate sensitivity and to settle the discussed trade-off. Two approaches to data assimilation are considered, namely variational data assimilation and Newtonian nudging, also known as synchronization. Numerical examples demonstrate the feasibility of the approach.

Using Just-in-Time Data to Intervene with At-Risk Students

With increasing state and federal policy pressure to increase retention and graduation rates, academic affairs leaders seek metrics to track that predict student success. This report analyzes behavioral, academic, and financial indicators of student risk and suggests actions administrators can take to intervene with students who cross risk thresholds.

Using Just in Time Data to Intervene with At-Risk Students

With increasing state and federal policy pressure to increase retention and graduation rates, academic affairs leaders seek metrics to track that predict student success. This report analyzes behavioral, academic, and financial indicators of student risk and suggests actions administrators can take to intervene with students who cross risk thresholds.

Bayesian space-time data fusion for real-time forecasting and map uncertainty

Environmental computer models are deterministic models devoted to predict several environmental phenomena such as air pollution or meteorological events. Numerical model output is given in terms of averages over grid cells, usually at high spatial and temporal resolution. However, these outputs are often biased with unknown calibration and not equipped with any information about the associated uncertainty. Conversely, data collected at monitoring stations is more accurate since they essentially provide the true levels. Due the leading role played by numerical models, it now important to compare model output with observations. Statistical methods developed to combine numerical model output and station data are usually referred to as data fusion. In this work, we first combine ozone monitoring data with ozone predictions from the Eta-CMAQ air quality model in order to forecast real-time current 8-hour average ozone level defined as the average of the previous four hours, current hour, and predictions for the next three hours. We propose a Bayesian downscaler model based on first differences with a flexible coefficient structure and an efficient computational strategy to fit model parameters. Model validation for the eastern United States shows consequential improvement of our fully inferential approach compared with the current real-time forecasting system. Furthermore, we consider the introduction of temperature data from a weather forecast model into the downscaler, showing improved real-time ozone predictions. Finally, we introduce a hierarchical model to obtain spatially varying uncertainty associated with numerical model output. We show how we can learn about such uncertainty through suitable stochastic data fusion modeling using some external validation data. We illustrate our Bayesian model by providing the uncertainty map associated with a temperature output over the northeastern United States.

Early Warning For Large Earthquakes: Observations, Models and Real-Time Data Analysis

This thesis is a collection of works focused on the topic of Earthquake Early Warning, with a special attention to large magnitude events. The topic is addressed from different points of view and the structure of the thesis reflects the variety of the aspects which have been analyzed. The first part is dedicated to the giant, 2011 Tohoku-Oki earthquake. The main features of the rupture process are first discussed. The earthquake is then used as a case study to test the feasibility Early Warning methodologies for very large events. Limitations of the standard approaches for large events arise in this chapter. The difficulties are related to the real-time magnitude estimate from the first few seconds of recorded signal. An evolutionary strategy for the real-time magnitude estimate is proposed and applied to the single Tohoku-Oki earthquake. In the second part of the thesis a larger number of earthquakes is analyzed, including small, moderate and large events. Starting from the measurement of two Early Warning parameters, the behavior of small and large earthquakes in the initial portion of recorded signals is investigated. The aim is to understand whether small and large earthquakes can be distinguished from the initial stage of their rupture process. A physical model and a plausible interpretation to justify the observations are proposed. The third part of the thesis is focused on practical, real-time approaches for the rapid identification of the potentially damaged zone during a seismic event. Two different approaches for the rapid prediction of the damage area are proposed and tested. The first one is a threshold-based method which uses traditional seismic data. Then an innovative approach using continuous, GPS data is explored. Both strategies improve the prediction of large scale effects of strong earthquakes.