Estimation of life expectancy in the Middle Ages

Summary. In the Middle Ages data on landholders were recorded. We use a selection of these data for estimating the life expectancy of adult males from the higher ranks of English society. The data consist of the age at which the landholder inherited land, a sequence of ages at which he was known to be alive and possibly the age at death. We explain that the date of death is left truncated and right censored at an unknown censoring time. We propose a model to fit (a selection of) the data, we give estimators of the unknown parameters in the model and we present estimates of adult life expectancy.     

The analysis of incomplete data using stochastic covariates

In the development of many diseases there are often associated random variables which continuously reflect the progress of a subject towards the final expression of the disease (failure). At any given time these processes, which we call stochastic covariates, may provide information about the current hazard and the remaining time to failure. Likewise, in situations when the specific times of key prior events are not known, such as the time of onset of an occult tumour or the time of infection with HIV-1, it may be possible to identify a stochastic covariate which reveals, indirectly, when the event of interest occurred. The analysis of carcinogenicity trials which involve occult tumours is usually based on the time of death or sacrifice and an indicator of tumour presence for each animal in the experiment. However, the size of an occult tumour observed at the endpoint represents data concerning tumour development which may convey additional information concerning both the tumour incidence rate and the rate of death to which tumour-bearing animals are subject. We develop a stochastic model for tumour growth and suggest different ways in which the effect of this growth on the hazard of failure might be modelled. Using a combined model for tumour growth and additive competing risks of death, we show that if this tumour size information is used, assumptions concerning tumour lethality, the context of observation or multiple sacrifice times are no longer necessary in order to estimate the tumour incidence rate. Parametric estimation based on the method of maximum likelihood is outlined and is applied to simulated data from the combined model. The results of this limited study confirm that use of the stochastic covariate tumour size results in more precise estimation of the incidence rate for occult tumours.     

A joint model of cancer incidence,metastasis, and mortality

Many diseases, especially cancer, are not static, but rather can be summarized by a series of events or stages (e.g. diagnosis, remission, recurrence, metastasis, death). Most available methods to analyze multi-stage data ignore intermediate events and focus on the terminal event or consider (time to) multiple events as independent. Competing-risk or semi-competing-risk models are often deficient in describing the complex relationship between disease progression events which are driven by a shared progression stochastic process. A multi-stage model can only examine two stages at a time and thus fails to capture the effect of one stage on the time spent between other stages. Moreover, most models do not account for latent stages. We propose a semi-parametric joint model of diagnosis, latent metastasis, and cancer death and use nonparametric maximum likelihood to estimate covariate effects on the risks of intermediate events and death and the dependence between them. We illustrate the model with Monte Carlo simulations and analysis of real data on prostate cancer from the SEER database.     

Estimating progression-free survival in paediatric brain tumour patients when some progression statuses are unknown

In oncology, progression-free survival time, which is defined as the minimum of the times to disease progression or death, often is used to characterize treatment and covariate effects. We are motivated by the desire to estimate the progression time distribution on the basis of data from 780 paediatric patients with choroid plexus tumours, which are a rare brain cancer where disease progression always precedes death. In retrospective data on 674 patients, the times to death or censoring were recorded but progression times were missing. In a prospective study of 106 patients, both times were recorded but there were only 20 non-censored progression times and 10 non-censored survival times. Consequently, estimating the progression time distribution is complicated by the problems that, for most of the patients, either the survival time is known but the progression time is not known, or the survival time is right censored and it is not known whether the patient's disease progressed before censoring. For data with these missingness structures, we formulate a family of Bayesian parametric likelihoods and present methods for estimating the progression time distribution. The underlying idea is that estimating the association between the time to progression and subsequent survival time from patients having complete data provides a basis for utilizing covariates and partial event time data of other patients to infer their missing progression times. We illustrate the methodology by analysing the brain tumour data, and we also present a simulation study.     

Bayesian Modeling of Recurrent Event Data with Dependent Censoring

The recurrent-event setting, where the subjects experience multiple occurrences of the event of interest, are encountered in many biomedical applications. In analyzing recurrent event data, non informative censoring is often assumed for the implementation of statistical methods. However, when a terminating event such as death serves as part of the censoring mechanism, validity of the censoring assumption may be violated because recurrence can be a powerful risk factor for death. We consider joint modeling of recurrent event process and terminating event under a Bayesian framework in which a shared frailty is used to model the association between the intensity of the recurrent event process and the hazard of the terminating event. Our proposed model is implemented on data from a well-known cancer study.     

Simultaneous inference of a misclassified outcome and competing risks failure time data

Ipsilateral breast tumor relapse (IBTR) often occurs in breast cancer patients after their breast conservation therapy. The IBTR status' classification (true local recurrence versus new ipsilateral primary tumor) is subject to error and there is no widely accepted gold standard. Time to IBTR is likely informative for IBTR classification because new primary tumor tends to have a longer mean time to IBTR and is associated with improved survival as compared with the true local recurrence tumor. Moreover, some patients may die from breast cancer or other causes in a competing risk scenario during the follow-up period. Because the time to death can be correlated to the unobserved true IBTR status and time to IBTR (if relapse occurs), this terminal mechanism is non-ignorable. In this paper, we propose a unified framework that addresses these issues simultaneously by modeling the misclassified binary outcome without a gold standard and the correlated time to IBTR, subject to dependent competing terminal events. We evaluate the proposed framework by a simulation study and apply it to a real data set consisting of 4477 breast cancer patients. The adaptive Gaussian quadrature tools in SAS procedure NLMIXED can be conveniently used to fit the proposed model. We expect to see broad applications of our model in other studies with a similar data structure.     

Isotonic estimation of survival under a misattribution of cause of death

Several authors have indicated that incorrectly classified cause of death for prostate cancer survivors may have played a role in the observed recent peak and decline of prostate cancer mortality. Motivated by the suggestion we studied a competing risks model where other cause of death may be misattributed as a death of interest. We first consider a na?ve approach using unconstrained nonparametric maximum likelihood estimation (NPMLE), and then present the constrained NPMLE where the survival function is forced to be monotonic. Surprising observations were made as we studied their small-sample and asymptotic properties in continuous and discrete situations. Contrary to the common belief that the non-monotonicity of a survival function NPMLE is a small-sample problem, the constrained NPMLE is asymptotically biased in the continuous setting. Other isotonic approaches, the supremum (SUP) method and the Pooled-Adjacent-Violators (PAV) algorithm, and the EM algorithm are also considered. We found that the EM algorithm is equivalent to the constrained NPMLE. Both SUP method and PAV algorithm deliver consistent and asymptotically unbiased estimator. All methods behave well asymptotically in the discrete time setting. Data from the Surveillance, Epidemiology and End Results (SEER) database are used to illustrate the proposed estimators.     

Nonparametric Survival Estimation when Death is Reported with Delay

In disease registries there can be a delay between death of asubject and the reporting of this death to the data analyst.If researchers use the Kaplan-Meier estimator and implicitlyassumed that subjects who have yet to have death reported arestill alive, i.e. are censored at the time of analysis, the Kaplan-Meierestimator is typically inconsistent. Assuming censoring is independentof failure, we provide a simple estimator that is consistentand asymptotically efficient. We also provide estimates of theasymptotic variance of our estimator and simulations that demonstratethe favorable performance of these estimators. Finally, we demonstrateour methods by analyzing AIDS survival data. This analysis underscoresthe pitfalls of not accounting for delay when estimating thesurvival distribution and suggests a significant reduction inbias by using our estimator.     

Modeling of semi-competing risks by means of first passage times of a stochastic process

In semi-competing risks one considers a terminal event, such as death of a person, and a non-terminal event, such as disease recurrence. We present a model where the time to the terminal event is the first passage time to a fixed level c in a stochastic process, while the time to the non-terminal event is represented by the first passage time of the same process to a stochastic threshold S, assumed to be independent of the stochastic process. In order to be explicit, we let the stochastic process be a gamma process, but other processes with independent increments may alternatively be used. For semi-competing risks this appears to be a new modeling approach, being an alternative to traditional approaches based on illness-death models and copula models. In this paper we consider a fully parametric approach. The likelihood function is derived and statistical inference in the model is illustrated on both simulated and real data.     

Bayesian inference for a partially observed birth–death process using data on proportions

Stochastic kinetic models are often used to describe complex biological processes. Typically these models are analytically intractable and have unknown parameters which need to be estimated from observed data. Ideally we would have measurements on all interacting chemical species in the process, observed continuously in time. However, in practice, measurements are taken only at a relatively few time‐points. In some situations, only very limited observation of the process is available, for example settings in which experimenters can only observe noisy observations on the proportion of cells that are alive. This makes the inference task even more problematic. We consider a range of data‐poor scenarios and investigate the performance of various computationally intensive Bayesian algorithms in determining the posterior distribution using data on proportions from a simple birth‐death process.     

Estimating heritability for cause specific mortality based on twin studies

There has been considerable interest in studying the magnitude and type of inheritance of specific diseases. This is typically derived from family or twin studies, where the basic idea is to compare the correlation for different pairs that share different amount of genes. We here consider data from the Danish twin registry and discuss how to define heritability for cancer occurrence. The key point is that this should be done taking censoring as well as competing risks due to e.g.  death into account. We describe the dependence between twins on the probability scale and show that various models can be used to achieve sensible estimates of the dependence within monozygotic and dizygotic twin pairs that may vary over time. These dependence measures can subsequently be decomposed into a genetic and environmental component using random effects models. We here present several novel models that in essence describe the association in terms of the concordance probability, i.e., the probability that both twins experience the event, in the competing risks setting. We also discuss how to deal with the left truncation present in the Nordic twin registries, due to sampling only of twin pairs where both twins are alive at the initiation of the registries.     

A comparison of models for dynamic life tables. Application to mortality data from the Valencia Region (Spain)

The graduation of mortality data aims to estimate the probabilities of death at age x, q ( x ), by means of an age-dependent function, whose parameters are adjusted from the crude probabilities that are directly obtainable from the data. However, current life tables have a problem, the need for periodic updates due to changes in mortality over short periods of time. The table containing mortality rates for different ages in different years, q ( xt ), is called a dynamic life table, which captures mortality variation over time. This paper proposes a review of the most commonly used dynamic models and compares the results obtained by each of them when applied to mortality data from the Valencia Region (Spain). The result of the comparison leads us to the conclusion that the Lee-Carter method offers the best results for both sexes, while that based on Heligman and Pollard functions provides the best fit for men alone. Our working method is of additional interest as it may be applied to mortality data for a wide range of ages in any geographical location, allowing the most appropriate dynamic life table to be selected for the case at hand.     

Non-parametric Tests for Recurrent Events under Competing Risks

Abstract.  We consider a data set on nosocomial infections of patients hospitalized in a French intensive care facility. Patients may suffer from recurrent infections of different types and they also have a high risk of death. To deal with such situations, a model of recurrent events with competing risks and a terminal event is introduced. Our aim was to compare the occurrence rates of two types of events. For this purpose, we propose two tests: one to detect if the occurrence rate of a given type of event increases with time; a second to detect if the instantaneous probability of experiencing an event of a given type is always greater than the one of another type. The asymptotic properties of the test statistics are derived and Monte Carlo methods are used to study the power of the tests. Finally, the procedures developed are applied to the French nosocomial infections data set.     

Maximum Likelihood Estimation of Life-Span Based on Censored and Passively Registered Historical Data

We consider the estimation of life length of people who were born in the seventeenth or eighteenth century in England. The data consist of a sequence of times of life events that is either ended by a time of death or is right-censored by an unobserved time of migration. We propose a semi parametric model for the data and use a maximum likelihood method to estimate the unknown parameters in this model. We prove the consistency of the maximum likelihood estimators and describe an algorithm to obtain the estimates numerically. We have applied the algorithm to data and the estimates found are presented.     

Regression Analysis of Bivariate Current Status Data Using a Multistate Model

We consider bivariate current status data with death which often occur in animal tumorigenicity experiments. Instead of observing exact tumor onset time, the existence of tumor is known at death time or sacrifice time. Such an incomplete data structure makes it difficult to investigate the effect of treatment on tumor onset times. Furthermore, when tumor onsets occur at two sites, information for the order of their onsets is unknown. A multistate model is applied to incorporate the sequential occurrence of events. For the inference of parameters, an EM algorithm is applied and a real NTP (National Toxicology Program) dataset is analyzed as an illustrative example.     

Refusal to Answer Specific Questions in a Survey: A Case Study

It is well known that non ignorable item non response may occur when the cause of the non response is the value of the latent variable of interest. In these cases, a refusal by a respondent to answer specific questions in a survey should be treated sometimes as a non ignorable item non response. The Rasch-Rasch model (RRM) is a new two-dimensional item response theory model for addressing non ignorable non response. This article demonstrates the use of the RRM on data from an Italian survey focused on assessment of healthcare workers’ knowledge about sudden infant death syndrome (that is, a context in which non response is presumed to be more likely among individuals with a low level of competence). We compare the performance of the RRM with other models within the Rasch model family that assume the unidimensionality of the latent trait. We conclude that this assumption should be considered unreliable for the data at hand, whereas the RRM provides a better fit of the data.     

Estimation of river and stream temperature trends under haphazard sampling

Long-term temporal trends in water temperature in rivers and streams are typically estimated under the assumption of evenly-spaced space-time measurements. However, sampling times and dates associated with historical water temperature datasets and some sampling designs may be haphazard. As a result, trends in temperature may be confounded with trends in time or space of sampling which, in turn, may yield biased trend estimators and thus unreliable conclusions. We address this concern using multilevel (hierarchical) linear models, where time effects are allowed to vary randomly by day and date effects by year. We evaluate the proposed approach by Monte Carlo simulations with imbalance, sparse data and confounding by trend in time and date of sampling. Simulation results indicate unbiased trend estimators while results from a case study of temperature data from the Illinois River, USA conform to river thermal assumptions. We also propose a new nonparametric bootstrap inference on multilevel models that allows for a relatively flexible and distribution-free quantification of uncertainties. The proposed multilevel modeling approach may be elaborated to accommodate nonlinearities within days and years when sampling times or dates typically span temperature extremes.     

Efficient estimation of the distribution of time to composite endpoint when some endpoints are only partially observed

Two common features of clinical trials, and other longitudinal studies, are (1) a primary interest in composite endpoints, and (2) the problem of subjects withdrawing prematurely from the study. In some settings, withdrawal may only affect observation of some components of the composite endpoint, for example when another component is death, information on which may be available from a national registry. In this paper, we use the theory of augmented inverse probability weighted estimating equations to show how such partial information on the composite endpoint for subjects who withdraw from the study can be incorporated in a principled way into the estimation of the distribution of time to composite endpoint, typically leading to increased efficiency without relying on additional assumptions above those that would be made by standard approaches. We describe our proposed approach theoretically, and demonstrate its properties in a simulation study.     

CAUSAL EFFECTS OF TREATMENTS FOR INFORMATIVE MISSING DATA DUE TO PROGRESSION/DEATH

In longitudinal clinical trials, when outcome variables at later time points are only defined for patients who survive to those times, the evaluation of the causal effect of treatment is complicated. In this paper, we describe an approach that can be used to obtain the causal effect of three treatment arms with ordinal outcomes in the presence of death using a principal stratification approach. We introduce a set of flexible assumptions to identify the causal effect and implement a sensitivity analysis for non-identifiable assumptions which we parameterize parsimoniously. Methods are illustrated on quality of life data from a recent colorectal cancer clinical trial.     

Accelerated failure time models for the analysis of competing risks

Competing risks are common in clinical cancer research, as patients are subject to multiple potential failure outcomes, such as death from the cancer itself or from complications arising from the disease. In the analysis of competing risks, several regression methods are available for the evaluation of the relationship between covariates and cause-specific failures, many of which are based on Cox’s proportional hazards model. Although a great deal of research has been conducted on estimating competing risks, less attention has been devoted to linear regression modeling, which is often referred to as the accelerated failure time (AFT) model in survival literature. In this article, we address the use and interpretation of linear regression analysis with regard to the competing risks problem. We introduce two types of AFT modeling framework, where the influence of a covariate can be evaluated in relation to either a cause-specific hazard function, referred to as cause-specific AFT (CS-AFT) modeling in this study, or the cumulative incidence function of a particular failure type, referred to as crude-risk AFT (CR-AFT) modeling. Simulation studies illustrate that, as in hazard-based competing risks analysis, these two models can produce substantially different effects, depending on the relationship between the covariates and both the failure type of principal interest and competing failure types. We apply the AFT methods to data from non-Hodgkin lymphoma patients, where the dataset is characterized by two competing events, disease relapse and death without relapse, and non-proportionality. We demonstrate how the data can be analyzed and interpreted, using linear competing risks regression models.