Estimating Marginal Effects in Accelerated Failure Time Models for Serial Sojourn Times Among Repeated Events

Recurrent event data are commonly encountered in longitudinal studies when events occur repeatedly over time for each study subject. An accelerated failure time (AFT) model on the sojourn time between recurrent events is considered in this article. This model assumes that the covariate effect and the subject-specific frailty are additive on the logarithm of sojourn time, and the covariate effect maintains the same over distinct episodes, while the distributions of the frailty and the random error in the model are unspecified. With the ordinal nature of recurrent events, two scale transformations of the sojourn times are derived to construct semiparametric methods of log-rank type for estimating the marginal covariate effects in the model. The proposed estimation approaches/inference procedures also can be extended to the bivariate events, which alternate themselves over time. Examples and comparisons are presented to illustrate the performance of the proposed methods.     

Nonparametric inference and uniqueness for periodically observed progressive disease models

In many studies examining the progression of HIV and other chronic diseases, subjects are periodically monitored to assess their progression through disease states. This gives rise to a specific type of panel data which have been termed “chain-of-events data”; e.g. data that result from periodic observation of a progressive disease process whose states occur in a prescribed order and where state transitions are not observable. Using a discrete time semi-Markov model, we develop an algorithm for nonparametric estimation of the distribution functions of sojourn times in a J state progressive disease model. Issues of uniqueness for chain-of-events data are not well-understood. Thus, a main goal of this paper is to determine the uniqueness of the nonparametric estimators of the distribution functions of sojourn times within states. We develop sufficient conditions for uniqueness of the nonparametric maximum likelihood estimator, including situations where some but not all of its components are unique. We illustrate the methods with three examples.     

Estimation in a model for a semi-Markov process with covariates under right-censoring

A semi-Markov model with covariates is proposed for a multi-state process with a finite number of states such that the transition probabilities between the states and the distribution functions of the duration times between the occurrence of two states depend on a discrete covariate. The hazard rates for the time elapsed between two successive states depend on the covariate through a proportional hazards model involving a set of regression parameters, while the transition probabilities depend on the covariate in an unspecified way. We propose estimators for these parameters and for the cumulative hazard functions of the sojourn times. A difficulty comes from the fact that when a sojourn time in a state is right-censored, the next state is unknown. We prove that our estimators are consistent and asymptotically Gaussian under the model constraints.     

Modelling multi-stage processes through multivariate distributions

A new model combining parametric and semi-parametric approaches and following the lines of a semi-Markov model is developed for multi-stage processes. A Bivariate sojourn time distribution derived from the bivariate exponential distribution of Marshall & Olkin (1967) is adopted. The results compare favourably with the usual semi-parametric approaches that have been in use. Our approach also has several advantages over the models in use including its amenability to statistical inference. For example, the tests for symmetry and also for independence of the marginals of the sojourn time distributions, which were not available earlier, can now be conveniently derived and are enhanced in elegant forms. A unified Goodness-of-Fit test procedure for our proposed model is also presented. An application to the human resource planning involving real-life data from University of Nigeria is given.     

Nonparametric estimation of the sojourn time distributions for a multipath model

Summary. We use a multipath (multistate) model to describe data with multiple end points. Statistical inference based on the intermediate end point is challenging because of the problems of nonidentifiability and dependent censoring. We study nonparametric estimation for the path probability and the sojourn time distributions between the states. The methodology proposed can be applied to analyse cure models which account for the competing risk of death. Asymptotic properties of the estimators proposed are derived. Simulation shows that the methods proposed have good finite sample performance. The methodology is applied to two data sets.     

Semiparametric regression analysis of two‐sample current status data,with applications to tumorigenicity experiments

Current status data frequently occur in failure time studies, particularly in demographical studies and tumorigenicity experiments. Although commonly used in this context, proportional hazards and odds models are inadequate when survival functions cross. The authors consider a class of two‐sample models which is suitable for this situation and encompasses the proportional hazards and odds models. The estimating equations they propose lead to consistent and asymptotically Gaussian estimates of regression parameters in the extended model. Their approach is assessed through simulations and illustrated using data from a tumorigenicity experiment.     

Estimating parametric semi-Markov models from panel data using phase-type approximations

Inference for semi-Markov models under panel data presents considerable computational difficulties. In general the likelihood is intractable, but a tractable likelihood with the form of a hidden Markov model can be obtained if the sojourn times in each of the states are assumed to have phase-type distributions. However, using phase-type distributions directly may be undesirable as they require estimation of parameters which may be poorly identified. In this article, an approach to fitting semi-Markov models with standard parametric sojourn distributions is developed. The method involves establishing a family of Coxian phase-type distribution approximations to the parametric distribution and merging approximations for different states to obtain an approximate semi-Markov process with a tractable likelihood. Approximations are developed for Weibull and Gamma distributions and demonstrated on data relating to post-lung-transplantation patients.     

BAYESIAN HIDDEN MARKOV MODELS FOR LONGITUDINAL COUNTS

This paper describes a Bayesian approach to modelling carcinogenity in animal studies where the data consist of counts of the number of tumours present over time. It compares two autoregressive hidden Markov models. One of them models the transitions between three latent states: an inactive transient state, a multiplying state for increasing counts and a reducing state for decreasing counts. The second model introduces a fourth tied state to describe non‐zero observations that are neither increasing nor decreasing. Both these models can model the length of stay upon entry of a state. A discrete constant hazards waiting time distribution is used to model the time to onset of tumour growth. Our models describe between‐animal‐variability by a single hierarchy of random effects and the within‐animal variation by first‐order serial dependence. They can be extended to higher‐order serial dependence and multi‐level hierarchies. Analysis of data from animal experiments comparing the influence of two genes leads to conclusions that differ from those of Dunson (2000). The observed data likelihood defines an information criterion to assess the predictive properties of the three‐ and four‐state models. The deviance information criterion is appropriately defined for discrete parameters.     

Reliability of semi-Markov missions

We consider a device that is designed to perform missions consisting of a random sequence of phases or stages with random durations. The mission process is described by a Markov renewal process and the system is a complex one consisting of a number of components whose lifetimes depend on the phases of the mission. We discuss models and tools to compute system, mission, and phase reliabilities using Markov renewal theory. A simplified model involving a mission-based system with maximal repair is analyzed first, and the results are then extended to the case where there is no repair using intrinsic aging concepts. Our objective is to focus on computation of system reliability for these two possible extreme cases.     

Prediction of disease status: A regressive model approach for repeated measures

In this paper, regressive models are proposed for modeling a sequence of transitions in longitudinal data. These models are employed to predict the future status of the outcome variable of the individuals on the basis of their underlying background characteristics or risk factors. The estimation of parameters and also estimates of conditional and unconditional probabilities are shown for repeated measures. The goodness of fit tests are extended in this paper on the basis of the deviance and the Hosmer–Lemeshow procedures and generalized to repeated measures. In addition, to measure the suitability of the proposed models for predicting the disease status, we have extended the ROC curve approach to repeated measures. The procedure is shown for the conditional models for any order as well as for the unconditional model, to predict the outcome at the end of the study. The test procedures are also suggested. For testing the differences between areas under the ROC curves in subsequent follow-ups, two different test procedures are employed, one of which is based on permutation test. In this paper, an unconditional model is proposed on the basis of conditional models for the disease progression of depression among the elderly population in the USA on the basis of the Health and Retirement Survey data collected longitudinally. The illustration shows that the disease progression observed conditionally can be employed to predict the outcome and the role of selected variables and the previous outcomes can be utilized for predictive purposes. The results show that the percentage of correct predictions of a disease is quite high and the measures of sensitivity and specificity are also reasonably impressive. The extended measures of area under the ROC curve show that the models provide a reasonably good fit in terms of predicting the disease status during a long period of time. This procedure will have extensive applications in the field of longitudinal data analysis where the objective is to obtain estimates of unconditional probabilities on the basis of series of conditional transitional models.     

Censored regression with the multistate accelerated sojourn times model

Many disease processes are characterized by two or more successive health states, and it is often of interest and importance to assess state-specific covariate effects. However, with incomplete follow-up data such inference has not been satisfactorily addressed in the literature. We model the logarithm-transformed sojourn time in each state as linearly related to the covariates; however, neither the distributional form of the error term nor the dependence structure of the states needs to be specified. We propose a regression procedure to accommodate incomplete follow-up data. Asymptotic theory is presented, along with some tools for goodness-of-fit diagnostics. Simulation studies show that the proposal is reliable for practical use. We illustrate it by application to a cancer clinical trial.     

SOJOURN TIMES IN NON-HOMOGENEOUS QBD PROCESSES WITH PROCESSOR SHARING

We study sojourn times of customers in a processor sharing queue with a service rate that varies over time, depending on the number of customers and on the state of a random environment. An explicit expression is derived for the Laplace–Stieltjes transform of the sojourn time conditional on the state upon arrival and the amount of work brought into the system. Particular attention is paid to the conditional mean sojourn time of a customer as a function of his required amount of work, and we establish the existence of an asymptote as the amount of work tends to infinity. The method of random time change is then extended to include the possibility of a varying service rate. By means of this method, we explain the well-established proportionality between the conditional mean sojourn time and required amount of work in processor sharing queues without random environment. Based on numerical experiments, we propose an approximation for the conditional mean sojourn time. Although first presented for exponentially distributed service requirements, the analysis is shown to extend to phase-type services. The service discipline of discriminatory processor sharing is also shown to fall within the framework.     

Comparing waiting times in a multi-stage model: A log-rank approach

A proper log-rank test for comparing two waiting (i.e. sojourn, gap) times under right censored data has been absent in the survival literature. The classical log-rank test provides a biased comparison even under independent right censoring since the censoring induced on the time since state entry depends on the entry time unless the hazards are semi-Markov. We develop test statistics for comparing K waiting time distributions from a multi-stage model in which censoring and waiting times may be dependent upon the transition history in the multi-stage model. To account for such dependent censoring, the proposed test statistics utilize an inverse probability of censoring weighted (IPCW) approach previously employed to define estimators for the cumulative hazard and survival function for waiting times in multi-stage models. We develop the test statistics as analogues to K-sample log-rank statistics for failure time data, and weak convergence to a Gaussian limit is demonstrated. A simulation study demonstrates the appropriateness of the test statistics in designs that violate typical independence assumptions for multi-stage models, under which naive test statistics for failure time data perform poorly, and illustrates the superiority of the test under proportional hazards alternatives to a Mann–Whitney type test. We apply the test statistics to an existing data set of burn patients.     

Generalized bent-cable methodology for changepoint data: a Bayesian approach

The choice of the model framework in a regression setting depends on the nature of the data. The focus of this study is on changepoint data, exhibiting three phases: incoming and outgoing, both of which are linear, joined by a curved transition. Bent-cable regression is an appealing statistical tool to characterize such trajectories, quantifying the nature of the transition between the two linear phases by modeling the transition as a quadratic phase with unknown width. We demonstrate that a quadratic function may not be appropriate to adequately describe many changepoint data. We then propose a generalization of the bent-cable model by relaxing the assumption of the quadratic bend. The properties of the generalized model are discussed and a Bayesian approach for inference is proposed. The generalized model is demonstrated with applications to three data sets taken from environmental science and economics. We also consider a comparison among the quadratic bent-cable, generalized bent-cable and piecewise linear models in terms of goodness of fit in analyzing both real-world and simulated data. This study suggests that the proposed generalization of the bent-cable model can be valuable in adequately describing changepoint data that exhibit either an abrupt or gradual transition over time.     

Estimation of the Parameters of a Two-phase Tandem Queue with a Second Optional Service

In this article, we consider a two-phase tandem queueing model with a second optional service. In this model, the service is done by two phases. The first phase of service is essential for all customers and after the completion of the first phase of service, any customer receives the second phase of service with probability α, or leaves the system with probability 1 ? α. Also, there are two heterogeneous servers which work independently, one of them providing the first phase of service and the other a second phase of service. In this model, our main purpose is to estimate the parameters of the model, traffic intensity, and mean system size, in the steady state, via maximum likelihood and Bayesian methods. Furthermore, we find asymptotic confidence intervals for mean system size. Finally, by a simulation study, we compute the confidence levels and mean length for asymptotic confidence intervals of mean system size with a nominal level 0.95.     

The random-effects proportional hazards model with grouped survival data: a comparison between the grouped continuous and continuation ratio versions

Summary.  When analysing grouped time survival data having a hierarchical structure it is often appropriate to assume a random-effects proportional hazards model for the latent continuous time and then to derive the corresponding grouped time model. There are two formally equivalent grouped time versions of the proportional hazards model obtained from different perspec-tives, known as the continuation ratio and the grouped continuous models. However, the two models require distinct estimation procedures and, more importantly, they differ substantially when extended to time-dependent covariates and/or non-proportional effects. The paper discusses these issues in the context of random-effects models, illustrating the main points with an application to a complex data set on job opportunities for a cohort of graduates.     

Nonparametric estimation of quality adjusted lifetime distribution in a simple illness-death model

In this work, we consider the nonparametric estimation of quality adjusted lifetime (QAL) distribution in a simple illness-death model. We first derive the expression of QAL distribution in terms of the distribution of sojourn time in each health state. Next we substitute the estimate of sojourn time distributions in the expression of QAL distribution to obtain its estimate. Consistency and asymptotic normality of the proposed nonparametric estimator are established. Estimation in the presence of some missing data on the transition time to illness is also discussed. We conduct a simulation study to investigate the performance of the proposed estimator. For illustration, we analyse a data set of the Stanford Heart Transplant program. Extension to multistate progressive model is discussed along with an analysis of International Breast Cancer Study Group (IBCSG) Trial V data.     

On the functional central limit theorem for first passage time of nonlinear semi-Markov reward processes

Abstract

In this article we examine the functional central limit theorem for the first passage time of reward processes defined over a finite state space semi-Markov process. In order to apply this process for a wider range of real-world applications, the reward functions, considered in this work, are assumed to have general forms instead of the constant rates reported in the other studies. We benefit from the martingale theory and Poisson equations to prove and establish the convergence of the first passage time of reward processes to a zero mean Brownian motion. Necessary conditions to derive the results presented in this article are the existence of variances for sojourn times in each state and second order integrability of reward functions with respect to the distribution of sojourn times. We finally verify the presented methodology through a numerical illustration.     

Exploring Parameter Relations for Multi‐Stage Models in Stage‐Wise Constant and Time Dependent Hazard Rates

Single cohort stage‐frequency data are considered when assessing the stage reached by individuals through destructive sampling. For this type of data, when all hazard rates are assumed constant and equal, Laplace transform methods have been applied in the past to estimate the parameters in each stage‐duration distribution and the overall hazard rates. If hazard rates are not all equal, estimating stage‐duration parameters using Laplace transform methods becomes complex. In this paper, two new models are proposed to estimate stage‐dependent maturation parameters using Laplace transform methods where non‐trivial hazard rates apply. The first model encompasses hazard rates that are constant within each stage but vary between stages. The second model encompasses time‐dependent hazard rates within stages. Moreover, this paper introduces a method for estimating the hazard rate in each stage for the stage‐wise constant hazard rates model. This work presents methods that could be used in specific types of laboratory studies, but the main motivation is to explore the relationships between stage maturation parameters that, in future work, could be exploited in applying Bayesian approaches. The application of the methodology in each model is evaluated using simulated data in order to illustrate the structure of these models.     

Nonparametric estimator for the survival function of quality adjusted lifetime (QAL) in a three-state illness–death model

In this work, we consider nonparametric estimation of QAL distribution in a three-state illness–death model. In our approach, we first write down the expression for the distribution of QAL in terms of the joint distribution of the sojourn times in the three states. The estimate of the QAL distribution is obtained by substituting the estimates of sojourn time distributions in the expression of the QAL distribution. The proposed nonparametric estimate, assuming independence between time to illness and sojourn time in the state of illness, is uniformly consistent. Asymptotic normality has also been established. An estimate of asymptotic variance has been obtained. The performance of the proposed estimator is investigated by simulation. A data set of the Stanford Heart Transplant program has been analyzed for illustration.