Statistical inference for dependent competing risks data under adaptive Type-II progressive hybrid censoring

In this article, we consider statistical inference based on dependent competing risks data from Marshall-Olkin bivariate Weibull distribution. The maximum likelihood estimates of the unknown model parameters have been computed by using the Newton-Raphson method under adaptive Type II progressive hybrid censoring with partially observed failure causes. The existence and uniqueness of maximum likelihood estimates are derived. Approximate confidence intervals have been constructed via the observed Fisher information matrix using the asymptotic normality property of the maximum likelihood estimates. Bayes estimates and highest posterior density credible intervals have been calculated under gamma-Dirichlet prior distribution by using the Markov chain Monte Carlo technique. Convergence of Markov chain Monte Carlo samples is tested. In addition, a Monte Carlo simulation is carried out to compare the effectiveness of the proposed methods. Further, three different optimality criteria have been taken into account to obtain the most effective censoring plans. Finally, a real-life data set has been analyzed to illustrate the operability and applicability of the proposed methods

Forecasting Remission Time of a Treatment Method for Leukemia as an Application to Statistical Inference Approach

In this paper, Weibull-Linear Exponential distribution (WLED) has been investigated whether being it is a well - fit distribution to a clinical real data. These data represent the duration of remission achieved by a certain drug used in the treatment of leukemia for a group of patients. The statistical inference approach is used to estimate the parameters of the WLED through the set of the fitted data. The estimated parameters are utilized to evaluate the survival and hazard functions and hence assessing the treatment method through forecasting the duration of remission times of patients. A two-sample prediction approach has been applied to obtain a predictive sample based on the Bayes estimates of the parameters. The statistical inference approach is applied to the case of censored data namely Type-II hybrid censoring scheme, which is common in clinical studies

Stress-strength reliability estimation for the inverted exponentiated Rayleigh distribution under unified progressive hybrid censoring with application

In this paper, we studied the estimation of a stress-strength reliability model (R = P(X>Y)) based on inverted exponentiated Rayleigh distribution under the unified progressive hybrid censoring scheme (unified PHCS). The maximum likelihood estimates of the unknown parameters were obtained using the stochastic expectation-maximization algorithm (stochastic EMA). The asymptotic confidence intervals were also created. Under squared error and Linex and generalized entropy loss functions, the Gibbs sampler together with Metropolis-Hastings algorithm was provided to compute the Bayes estimates and the credible intervals. Extensive simulations were performed to see the effectiveness of the proposed estimation methods. Also, parallel to the development of reliability studies, it is necessary to study its application in different sciences such as engineering. Therefore, droplet splashing data under two nozzle pressures were proposed to exemplify the theoretical outcomes

Regression methods for survival and multistate models.

A common research interest in medical, biological, and engineering research is determining whether certain independent variables are correlated with the survival or failure times. Standard statistical techniques cannot usually be applied for failure-time data due to the lack of complete data or in other word, due to censoring. From a statistical perspective, the study of time to event data is even more challenging when further complexities such as high dimensionality or multivariablity is added to the model. In this dissertation, we consider the predicating patient survival from proteomic profile of patient serum using matrix-assisted laser desorption/ionization time-of-flight (MALDI-TOF) data of non-small cell lung cancer patients. Due to much larger dimension of features in a mass spectrum compared to the study sample size, traditional linear regression modeling of survival times with high number of proteomic features is not feasible. Hence, we consider latent factor and regularized/penalized methods for fitting such models in order to predict patient survival from the mass spectrometry features. Extensive numerical studies involving both simulated as well as real mass spectrometry data are used to compare four popular regression methods, namely, partial least squares (PLS), sparse partial least square (SPLS), least absolute shrinkage and selection operator (LASSO) and elastic net regularization, on processed spectra. Right censoring is handled through a residual based multiple imputation. Overall, more complex methods such as the elastic net and SPLS result in better performances provided the operational parameters are chosen carefully via cross validation. For survival time prediction, we recommend using the elastic net based on a selected set of features. As a type of multivariate survival data, multistate models have a wide range of applications. Most of the existing regression approaches to analyze such data are based on parametric and semi-parametric procedures in which one should rely on specific model structures. In this dissertation, we construct non-parametric regression estimators of a number of temporal functions in a multistate system based on a univariate continuous baseline covariate. These estimators include state occupation probabilities, state entry, exit and waiting (sojourn) times distribution functions of a general progressive (e.g. acyclic) multistate model. The data are subject to right censoring and the censoring mechanism is explainable by observable covariates that could be time dependent. The resulting estimators are valid even if the multistate process is non-Markov. The performance of the estimators is studied using a detailed simulation. We illustrate our estimators using a data set on bone marrow transplant patients. Finally, some extension of the proposed methods to more general case with multivariate covariates are presented along with plans for future developments

Modified weibull distributions in reliability engineering

Ph.DDOCTOR OF PHILOSOPH

Planning and inference of sequential accelerated life tests

Ph.DDOCTOR OF PHILOSOPH

Survival analysis of random censoring with inverse Maxwell distribution: an application to guinea pigs data

In real-life situations, performing an experiment up to a certain period of time or getting the desired number of failures is time-consuming and costly. Many of the available observations remain censored and only give the survival information of testing units up to a noted time and not about the exact failure times. In this article, the inverse Maxwell distribution having an upside-down hazard rate is considered a survival lifetime model. The censoring time is also assumed to follow the inverse Maxwell distribution with a different parameter. The probability of failure of an item before censoring and expected and observed time on the test is derived from a random censoring scheme. The maximum likelihood estimators with their confidence intervals for the parameters are obtained for a randomly censored setup. The Bayes estimators are also obtained by taking the inverted gamma distribution as a prior under squared error loss function. In Bayesian analysis, the two techniques i.e. Markov Chain Monte Carlo and Tierney-Kadane approximation methods are used for estimation purposes. For checking the performances of proposed estimators, we performed an extensive simulation study. A real data, guinea pigs, is analyzed to support the proposed study

Bayesian Inference for Cure Rate Models

Η ανάλυση επιβίωσης αποτελείται από ένα σύνολο στατιστικών μεθόδων που στοχεύει στη μελέτη του χρόνου μέχρι την εμφάνιση ενός συγκεκριμένου γεγονότος όπως ο θάνατος. Για την πλειονότητα των μεθόδων αυτών, θεωρείται πως όλα τα άτομα που συμμετέχουν υπόκεινται στο γεγονός που μας ενδιαφέρει. Ωστόσο, υπάρχουν περιπτώσεις όπου η υπόθεση αυτή δεν είναι ρεαλιστική, καθώς υπάρχουν ασθενείς που δεν θα βιώσουν το γεγονός αυτό στη διάρκεια της μελέτης. Για αυτό το λόγο, έχουν αναπτυχθεί ορισμένα μοντέλα επιβίωσης που επιτρέπουν την ύπαρξη ασθενών οι οποίοι δε βιώνουν το συμβάν και ονομάζονται μακροχρόνια επιζώντες. Τα μοντέλα αυτά ονομάζονται μοντέλα ρυθμού θεραπείας και υποθέτουν ότι, καθώς ο χρόνος αυξάνεται, η συνάρτηση επιβίωσης τείνει σε μια τιμή p ∈ (0,1), που αντιπροσωπεύει το ποσοστό των μακροχρόνια επιζώντων, αντί να τείνει στο μηδέν όπως στην κλασική ανάλυση επιβίωσης. Πρόσφατα, ο Rocha (2016) πρότεινεμία νέα προσέγγισητωνπροβλημάτωνεπιβίωσης μεμακροχρόνια επιζώντες. Η μεθοδολογία του για τη μοντελοποίηση του ποσοστού των μακροχρόνια επιζώντων βασίστηκε στη χρήση των «ελαττωματικών» (defective) κατανομών, οι οποίες χαρακτηρίζονται από το γεγονός ότι το ολοκλήρωμα της συνάρτησης πιθανότητάς τους δεν ισούται με τη μονάδα για ορισμένες επιλογές του πεδίου ορισμού κάποιων παραμέτρων τους. Σκοπός της παρούσας διπλωματικής εργασίας, είναι να παράσχει νέους Μπεϋζιανούς εκτιμητές των παραμέτρων των «ελαττωματικών» μοντέλων κάτω από την υπόθέση της δεξιάς λογοκρισίας. Επίσης, θα αναπτυχθούν αλγόριθμοι Markov chain Monte Carlo (MCMC) για τη συμπερασματολογία σχετικά με τις παραμέτρους μιας ευρείας κατηγορίας μοντέλων ρυθμού θεραπείας βασισμένων στις «ελαττωματικές» αυτές κατανομές, ενώ οι Μπεϋζιανοί εκτιμητές και τα αντίστοιχα διαστήματα αξιοπιστίας θα ληφθούν από τα δείγματα της από κοινού εκ των υστέρων κατανομής. Επιπλέον, η συμπεριφορά των Μπεϋζιανών εκτιμητών θα αξιολογηθεί και θα συγκριθεί με αυτή των εκτιμητών μεγίστης πιθανοφάνειας του Rocha (2016) μέσω πειραμάτων προσομοίωσης. Ακόμη, τα προτεινόμενα αυτά μοντέλα-κατανομές θα εφαρμοσθούν σε πραγματικά σετ δεδομένων, όπου και θα συγκριθούν μεταξύ τους μέσω κατάλληλων στατιστικών μεγεθών. Τέλος, αξίζει να σημειωθεί πως η παρούσα διπλωματική εργασία αποτελεί την πρώτη προσπάθεια διερεύνησης των πλεονεκτημάτων της Μπεϋζιανής προσέγγισης στη συμπερασματολογία για τις παραμέτρους αρκετών μοντέλων ρυθμού θεραπείας, κάτω από την υπόθεση της δεξιάς λογοκρισίας, καθώς και της απόκτησης νέων Μπεϋζιανών εκτιμητών, χωρίς όμως τη συμπερίληψη της πληροφορίας από συν μεταβλητές.Survival analysis consists of a set of statistical methods in the field of biostatistics, whose main aim is to study the time until the occurrence of a specified event, such as death. For the majority of these methods it is assumed that all the individuals taking part in the study are subject to the event of interest. However, there are situations where this assumption is unrealistic, since there are observations not susceptible to the event of interest or cured. For this reason, there have been developed some survival models which allow for patients that may never experience the event, usually called long-term survivors. These models, called Cure Rate Models, assume that, as time increases, the survival function tends to a value p ∈ (0,1), representing the cure rate, instead of tending to zero as in standard survival analysis. Recently, Rocha (2016) proposed a new approach to modelling the situations in which there are long-term survivors in survival studies. His methodology was based on the use of defective distributions to model cure rates. In contrast to the standard distributions, the defective ones are characterized by having probability density functions which integrate to values less than one for certain choices of the domain of some of their parameters. The aim of the present thesis is to provide new Bayesian estimates for the parameters of the defective models used for cure rate modelling under the assumption of right censoring. We will develop Markov chain Monte Carlo (MCMC) algorithms for inferring the parameters of a broad class of defective models, both for the baseline distributions (Gompertz & Inverse Gaussian), as well as, for their extension under the Marshall-Olkin family of distributions. The Bayesian estimates of the distributions’ parameters, as well as their associated credible intervals, will be obtained from the samples drawn from their joint posterior distribution. In addition, Bayesian estimates’ behaviour will be evaluated and compared with the maximum likelihood estimates obtained by Rocha (2016) through simulation experiments. Finally, we will apply the competing models and approaches to real datasets and we will compare them through various statistical measures. This work will be the first attempt to explore the advantages of the Bayesian approach to inference for defective cure rate models under the assumption of right censoring mechanism, as well as the first presentation of new Bayesian estimates for several defective distributions, but without incorporating covariate information

Bayesian Models for Joint Longitudinal and Multi-State Survival Data

Biomedical data commonly include repeated measures of biomarkers and disease states over time. When the processes determining the biomarker levels and disease states are related, a joint longitudinal and survival model is needed to properly handle the data. In a recent study of adrenal cancer patients at the University of Michigan, their tumors were monitored with repeated radiography scans. Other body measurements, called morphomics, were also measured from these scans. At each scan, it was noted whether the patient's disease was stable, progressing or regressing. In addition, the data include time to death or end of follow-up. Motivated by this data we explore joint models for longitudinal and survival data of several types. In Chapter 2 we compare computational approaches to joint longitudinal and survival models with a single type of event. We examine different joint model formulations especially those most often implemented in software available to statisticians and clinicians. We apply and compare several models to the adrenal data and perform a simulation study to further evaluate each model and software. In Chapter 3 we examine the relationship between a morphomic variable and time to first disease state change which can be either cancer progression or regression, in the adrenal cancer data. We develop Bayesian joint models for longitudinal and competing risks survival data. A seldom considered aspect of competing risk joint models is the relationship between the two competing outcomes. This cannot be examined when using the most common technique, cause-specific hazards models. With that motivation for our future projects, we work under the assumption that each risk has a latent failure time for each individual. We begin with the simple case of conditionally independent risks and model the survival times using parametric distributions. We apply our models to the adrenal data and examine the performance via simulations. In Chapter 4 we extend our joint longitudinal and competing risks models for dependent competing risks. We begin with a discussion of survival copulas and the general joint survival function we will use which is based on an Archimedean copula model. We prove that dependent variables with this joint survival function can be written in terms of independent variables which is useful for simulating data. We develop the model with Weibull marginals. We fit this model to the adrenal data and examine the models using a simulation study. We discuss interpretations of the model and how it can be used to learn about the dependence between risks. Finally, in Chapter 5 we will develop a joint model that incorporates multiple longitudinal outcomes and multistate survival data. We will develop an appropriate model and apply it to the adrenal cancer data.PHDBiostatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169659/1/acullen_1.pd

Statistical inference for the Nadarajah-Haghighi distribution based on ranked set sampling with applications

In this article, the maximum likelihood and Bayes inference methods are discussed for determining the two unknown parameters and specific lifetime parameters of the Nadarajah-Haghighi distribution, such as the survival and hazard rate functions, with the inclusion of ranked set sampling and simple random sampling. The estimated confidence intervals for the two parameters and any function of them are developed based on the Fisher-information matrix. Metropolis-Hastings algorithm and Lindley-approximation are used for generating the Bayes estimates and related highest posterior density credible ranges for the unknown parameters and reliability parameters under the presumption of conjugate gamma priors. A Monte-Carlo simulation study and a real-life data set have been used to assess the efficacy of the proposed methods