Testing independence between two sequential gap times in the presence of covariates

In the risk analysis of sequential events, the successive gap times are often correlated, e.g. as a result of an individual heterogeneity. Correlation is usually accounted for by using a shared gamma‐frailty model, where the variance φ of the random individual effect quantifies the correlation between gap times. This method is known to yield satisfactory estimates of covariate effects, but underestimates φ, which could result in a lack of power of the test of independence. We propose a new test of independence between two sequential gap times where the first is the time elapsed from the origin. The test is based on an approximation of the hazard of the second event given the first gap time in a frailty model, with a frailty distribution belonging to the power variance function family. Simulation results show an increased power of the new test compared with the test derived from the gamma‐frailty model. In the realistic case where hazards are event specific, and using event‐specific approaches, the proposed estimation of the variance of the frailty is less biased than the gamma‐frailty based estimation for a wide range of values ( with the set of parameters considered), and similar for higher values. As an illustration, the methods are applied to a previously analysed asthma prevention trial with results showing a significant positive association between the successive times to asthmatic events. We also analyse data from a cohort of HIV‐seropositive patients in order to assess the effect of risk factors on the occurrence of two successive markers of progression of the HIV disease. The results demonstrate the ability of the proposed model to account for negative correlations between gap times.     

Semiparametric regression analysis on longitudinal pattern of recurrent gap times

In longitudinal studies, individual subject may experience recurrent events of the same type over a relatively long period of time. The longitudinal pattern of gaps between successive recurrent events is often of great research interest. In this article, the probability structure of the recurrent gap times is first explored in the presence of censoring. According to the discovered structure, we introduce the stratified proportional reverse-time hazards models with unspecified baseline functions to accommodate individual heterogeneity, when the longitudinal pattern parameter is of main interest. Inference procedures are proposed and studied by way of proper riskset construction. The proposed methodology is demonstrated by the Monte Carlo simulations and an application to a well-known Denmark schizophrenia cohort study data set.     

Estimating the extent of tracking in interval-censored chain-of-events data

This paper describes a method for determining whether the times between a chain of successive events (which all individuals experience in the same order) are correlated, for data in which the exact event times are not observed. Such data arise when individuals are only observed occasionally to determine which events have occurred. In such data, the (unknown) event times are interval censored. In addition, some individuals may have experienced some of the events before their first observation and may be lost to follow-up before experiencing the last event. Using a frailty model proposed by Aalen (1988, Mathematical Scientist 13, 90-103) but which has never been used to analyze real data, we examine whether individuals who develop early markers of HIV infection can also be expected to develop antibody and other indicators of HIV infection more rapidly.     

A semiparametric estimator of the bivariate distribution function for censored gap times

Let (T(1), T(2)) be gap times corresponding to two consecutive events, which are observed subject to random right-censoring. In this paper, a semiparametric estimator of the bivariate distribution function of (T(1), T(2)) and, more generally, of a functional E [φ(T(1),T(2))] is proposed. We assume that the probability of censoring for T(2) given the (possibly censored) gap times belongs to a parametric family of binary regression curves. We investigate the conditions under which the introduced estimator is consistent. We explore the finite sample behavior of the estimator and of its bootstrap standard error through simulations. The main conclusion of this paper is that the semiparametric estimator may be much more efficient than purely nonparametric methods. Real data illustration is included.     

Discrete-time nonparametric estimation for semi-Markov models of chain-of-events data subject to interval censoring and truncation

Chain-of-events data are longitudinal observations on a succession of events that can only occur in a prescribed order. One goal in an analysis of this type of data is to determine the distribution of times between the successive events. This is difficult when individuals are observed periodically rather than continuously because the event times are then interval censored. Chain-of-events data may also be subject to truncation when individuals can only be observed if a certain event in the chain (e.g., the final event) has occurred. We provide a nonparametric approach to estimate the distributions of times between successive events in discrete time for data such as these under the semi-Markov assumption that the times between events are independent. This method uses a self-consistency algorithm that extends Turnbull's algorithm (1976, Journal of the Royal Statistical Society, Series B 38, 290-295). The quantities required to carry out the algorithm can be calculated recursively for improved computational efficiency. Two examples using data from studies involving HIV disease are used to illustrate our methods.     

Nonparametric estimation of the bivariate recurrence time distribution

This article considers statistical models in which two different types of events, such as the diagnosis of a disease and the remission of the disease, occur alternately over time and are observed subject to right censoring. We propose nonparametric estimators for the joint distribution of bivariate recurrence times and the marginal distribution of the first recurrence time. In general, the marginal distribution of the second recurrence time cannot be estimated due to an identifiability problem, but a conditional distribution of the second recurrence time can be estimated non-parametrically. In the literature, statistical methods have been developed to estimate the joint distribution of bivariate recurrence times based on data on the first pair of censored bivariate recurrence times. These methods are inefficient in the model considered here because recurrence times of higher orders are not used. Asymptotic properties of the proposed estimators are established. Numerical studies demonstrate the estimators perform well with practical sample sizes. We apply the proposed method to the South Verona, Italy, psychiatric case register (PCR) data set for illustration of the methods and theory.     

Robust covariate-adjusted log-rank statistics and corresponding sample size formula for recurrent events data

Summary .   Recurrent events data are frequently encountered in clinical trials. This article develops robust covariate-adjusted log-rank statistics applied to recurrent events data with arbitrary numbers of events under independent censoring and the corresponding sample size formula. The proposed log-rank tests are robust with respect to different data-generating processes and are adjusted for predictive covariates. It reduces to the Kong and Slud (1997, Biometrika 84, 847–862) setting in the case of a single event. The sample size formula is derived based on the asymptotic normality of the covariate-adjusted log-rank statistics under certain local alternatives and a working model for baseline covariates in the recurrent event data context. When the effect size is small and the baseline covariates do not contain significant information about event times, it reduces to the same form as that of Schoenfeld (1983, Biometrics 39, 499–503) for cases of a single event or independent event times within a subject. We carry out simulations to study the control of type I error and the comparison of powers between several methods in finite samples. The proposed sample size formula is illustrated using data from an rhDNase study.     

Nonparametric and semiparametric trend analysis for stratified recurrence times

Recurrent event data are frequently encountered in longitudinal follow-up studies when the occurrences of multiple events are considered as the major outcomes. Suppose that the recurrent events are of the same type and the variable of interest is the recurrence time between successive events. In many applications, the distributional pattern of recurrence times can be used as an index for the progression of a disease. Such a distributional pattern is important for understanding the natural history of a disease or for confirming long-term treatment effect. In this article, we discuss and define the comparability of recurrence times. Nonparametric and semiparametric methods are developed for testing trend of recurrence time distributions and estimating trend parameters in regression models. The construction of the methods is based on comparable recurrence times from stratified data. A real data example is presented to illustrate the use of methodology.     

A two-sample comparison for multiple ordered event data

A longitudinal study is conducted to compare the process of particular disease between two groups. The process of the disease is monitored according to which of several ordered events occur. In the paper, the sojourn time between two successive events is considered as the outcome of interest. The group effects on the sojourn times of the multiple events are parameterized by scale changes in a semiparametric accelerated failure time model where the dependence structure among the multivariate sojourn times is unspecified. Suppose that the sojourn times are subject to dependent censoring and the censoring times are observed for all subjects. A log-rank-type estimating approach by rescaling the sojourn times and the dependent censoring times into the same distribution is constructed to estimate the group effects and the corresponding estimators are consistent and asymptotically normal. Without the dependent censoring, the independent censoring times in general are not available for the uncensored data. In order to complete the censoring information, pseudo-censoring times are generated from the corresponding nonparametrically estimated survival function in each group, and we can still obtained unbiased estimating functions for the group effects. A real application and a simulation study are conducted to illustrate the proposed methods.     

Hippocampal "time cells" bridge the gap in memory for discontiguous events

The hippocampus is critical to remembering the flow of events in distinct experiences and, in doing so, bridges temporal gaps between discontiguous events. Here, we report a robust hippocampal representation of sequence memories, highlighted by "time cells" that encode successive moments during an empty temporal gap between the key events, while also encoding location and ongoing behavior. Furthermore, just as most place cells "remap" when a salient spatial cue is altered, most time cells form qualitatively different representations ("retime") when the main temporal parameter is altered. Hippocampal neurons also differentially encode the key events and disambiguate different event sequences to compose unique, temporally organized representations of specific experiences. These findings suggest that hippocampal neural ensembles segment temporally organized memories much the same as they represent locations of important events in spatially defined environments.     

A Mixture Point Process for Repeated Failure Times,with an Application to a Recurrent Disease

We present a model that describes the distribution of recurring times of a disease in presence of covariate effects. After a first occurrence of the disease in an individual, the time intervals between successive cases are supposed to be independent and to be a mixture of two distributions according to the issue of the previous treatment. Both sub‐distributions of the model and the mixture proportion are allowed to involve covariates. Parametric inference is considered and we illustrate the methods with data of a recurrent disease and with simulations, using piecewise constant baseline hazard functions.     

Power calculation for case-cohort studies with nonrare events

Case-cohort design has been advocated in many epidemiologic studies when studying rare diseases or events. In this design, with a rare event, all the events are selected for risk-factor assessment. When the event is not rare, it is desirable to consider a generalized case-cohort design, where only a fraction of events are sampled. We provide a valid test statistic to compare hazards functions between two samples for this generalized design and give a method for calculating power. Our result generalizes the result in Cai and Zeng (2004, Biometrics60, 1015-1024), and it shows numerically that efficiency loss due to sampling only part of the events is very low under nonrare-events situation.     

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 dataset 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.     

A novel flexible model for lot sizing and scheduling with non-triangular,period overlapping and carryover setups in different machine configurations

This paper develops and tests an efficient mixed integer programming model for capacitated lot sizing and scheduling with non-triangular and sequence-dependent setup times and costs incorporating all necessary features of setup carryover and overlapping on different machine configurations. The model’s formulation is based on the asymmetric travelling salesman problem and allows multiple lots of a product within a period. The model conserves the setup state when no product is being processed over successive periods, allows starting a setup in a period and ending it in the next period, permits ending a setup in a period and starting production in the next period(s), and enforces a minimum lot size over multiple periods. This new comprehensive model thus relaxes all limitations of physical separation between the periods. The model is first developed for a single machine and then extended to other machine configurations, including parallel machines and flexible flow lines. Computational tests demonstrate the flexibility and comprehensiveness of the proposed models.     

A multistate model for bivariate interval-censored failure time data

SUMMARY: Interval-censored life-history data arise when the events of interest are only detectable at periodic assessments. When interest lies in the occurrence of two such events, bivariate-interval censored event time data are obtained. We describe how to fit a four-state Markov model useful for characterizing the association between two interval-censored event times when the assessment times for the two events may be generated by different inspection processes. The approach treats the two events symmetrically and enables one to fit multiplicative intensity models that give estimates of covariate effects as well as relative risks characterizing the association between the two events. An expectation-maximization (EM) algorithm is described for estimation in which the maximization step can be carried out with standard software. The method is illustrated by application to data from a trial of HIV patients where the events are the onset of viral shedding in the blood and urine among individuals infected with cytomegalovirus.     

Interdependence of Cell Cycle Events in Tetrahymena thermophila: Formation of Contractile Vacuole Pore and Fission Gap in Ciliary Meridians*

SYNOPSIS The formation of the contractile vacuole pore (CVP) in Tetrahymena thermophila. genotype mol^b/mol^b is temporally and spatially associated with the formation of the fission gap in the CVP meridian. New CVPs arise when fission gaps appear in CVP meridians, the new pores being found anterior to the gaps. When, however, CVP meridians are rotated 180°, the fission gaps develop late. In more than 1/3 of the 180°-rotated CVP meridians, the new CVPs are formed before the appearance of the fission gap. Evidently, the appearance of the gap is not a prerequisite for CVP formation. Nevertheless, mutants exist in which the absence of fission gap and CVP are correlated in some cases and in which the presence of supernumerary fission gap and CVP are correlated in other instances. It is suggested that the 2 developmental events, although not causally related to each other, may be controlled by a common morpho-genetic signal. This commits a certain site (mid-body) along a ciliary meridian to develop the fission gap as well as the CVP; however, after this step of commitment, the appearance of the fission gap is delayed in 180°-rotated CVP meridians.     

中国第四纪孢粉植物群事件的记录

通过总结中国不同区域现有的孢粉资料,发现和民孢粉群的演化序列中明显地存在与演化趋势和周期不符的孢粉植物群事伯,归纳起来有三种类型,时间约万年的孢粉群突变事件,延续时间约千年的孢粉群贫乏事件和延时间几百年的孢粉群颤动,突变事件有４起,Ｂｅｔｕｌａ－Ａｅｍｉｓｉａ事件（２．５ＭａＢ．Ｐ）;Ａｒｔｅｍｉｓｉａ－Ｇｒａ,ｉｎｅａｅ事件（１．４Ｍａ．Ｂ．Ｐ．）;Ｅｐｈｅｄｒａ＝Ｃｈｅｎｏｐｏｄｉａｃｅａ事件     

EXPERIMENTS WITH THE METHYLENE BLUE REDUCTION TEST FOR MILK

SUMMARY: The effects of shaking milk samples before the addition of methylene blue, delaying the time of addition of the dye, adding a second quantity of dye with a hypodermic syringe after the initial charge had been decolourized and of increasing the storage time or varying the testing time have been studied in five separate experiments. Vigorous shaking at the time of dye addition lengthened the reduction time. This effect was more pronounced when the milk was stored in a tube than in a bottle. Delayed addition of the dye resulted in a shorter reduction time than expected for inverted samples and longer than expected for samples which were not inverted between the time they were put into the 37° bath and the addition of the dye. Second additons of dye did not usually have reduction times exceeding 1 hr. Methylene blue reduction times did not necessarily decrease with successive tests carried out over a period of time on one sample: a time of 30 min. might persist over a series of tests for 4 hr. or more and small increase and decrease in reduction time with successive tests were also noted.     

Alternative responses to rare selection events are differentially vulnerable to changes in the frequency,scope, and intensity of environmental extremes

Extreme weather events are becoming more frequent, severe, and/or widespread as a consequence of anthropogenic climate change. While the economic and ecological implications of these changes have received considerable attention, the role of evolutionary processes in determining organismal responses to these critical challenges is currently unknown. Here we develop a novel theoretical framework that explores how alternative pathways for adaptation to rare selection events can influence population‐level vulnerabilities to future changes in the frequency, scope, and intensity of environmental extremes. We begin by showing that different life histories and trait expression profiles can shift the balance between additive and multiplicative properties of fitness accumulation, favoring different evolutionary responses to identical environmental phenomena. We then demonstrate that these different adaptive outcomes lead to predictable differences in population‐level vulnerabilities to rapid increases in the frequency, intensity, or scope of extreme weather events. Specifically, we show that when the primary mode of fitness accumulation is additive, evolution favors ignoring environmental extremes and lineages become highly vulnerable to extinction if the frequency or scope of extreme weather events suddenly increases. Conversely, when fitness accumulates primarily multiplicatively, evolution favors bet‐hedging phenotypes that cope well with historical extremes and are instead vulnerable to sudden increases in extreme event intensity. Our findings address a critical gap in our understanding of the potential consequences of rare selection events and provide a relatively simple rubric for assessing the vulnerabilities of any population of interest to changes in a wide variety of extreme environmental phenomena.     

Extreme climatic events and vegetation: the role of stabilizing processes

Current climatic trends involve both increasing temperatures and climatic variability, with extreme events becoming more frequent. Increasing concern on extreme climatic events has triggered research on vegetation shifts. However, evidences of vegetation shifts resulting from these events are still relatively rare. Empirical evidence supports the existence of stabilizing processes minimizing and counteracting the effects of these events, reinforcing community resilience. We propose a demographic framework to understand this inertia to change based on the balance between adult mortality induced by the event and enhanced recruitment or adult survival after the event. The stabilizing processes potentially contributing to this compensation include attenuation of the adult mortality caused by the event, due to site quality variability, to tolerance, phenotypic variability, and plasticity at population level, and to facilitative interactions. Mortality compensation may also occur by increasing future survival due to beneficial effect on growth and survival of the new conditions derived from global warming and increased climatic variability, to lowered competition resulting from reduced density in affected stands, or to antagonistic release when pathogens or predators are vulnerable to the event or the ongoing climatic conditions. Finally, mortality compensation may appear by enhanced recruitment due to release of competition with established vegetation, for instance as a consequence of gap openings after event‐caused mortality, or to the new conditions, which may be more favorable for seedling establishment, or to enhanced mutualistic interactions (pollination, dispersal). There are important challenges imposed by the need of long‐term studies, but a research agenda focused on potentially stabilizing processes is well suited to understand the variety of responses, including lack of sudden changes and community inertia that are frequently observed in vegetation under extreme events. This understanding is crucial for the establishment of sound management strategies and actions addressed to improve ecosystem resilience under climate change scenarios.