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n an infectious disease cohort study, individuals who have been infected with apathogenare often recruited for follow up. The period between infection and the onset of symptomatic disease, referred to as the incubation period, is of interest because of its importance on disease surveillance and control. However, the incubation period is often difficult to ascertain due to the uncertainty associated with asymptomatic infection onset time. An additional complication is that the observed infected sub-jects are likely to have longer incubation periods due to the prevalent sampling. In this paper, we demonstrate how to estimate the distribution of the incubation period with the uncertain infection onset, subject to left-truncation and right-censoring. We employ a family of sufficiently general parametric models, the generalized odds-rate class of regression models, for the underlying incubation period and its correlation with covariates. In simulation studies, we assess the finite sample performance of the model fitting and hazard function estimation. The proposed method is illustrated on data from the HIV/AIDS study on injection drug users admitted to a detoxification program in Badalona, Spain, Ministerio de Ciencia e Innovación, PID2019‐104830RB‐I00; Departament d'Economia i Coneixement de la Generalitat de Catalunya, 2017 SGR 622 (GRBIO); Ministerio de Economía y Competitividad, MTM2015‐64465‐C2‐1‐R (MINECO/FEDER); National Cancer Institute, CA016672; CA193878, Peer Reviewed, Postprint (author's final draft)

Cross-sectional sampling is often used when investigating inter-event times, resulting in left-truncated and right-censored data. In this paper, we consider a semiparametric truncation model in which the truncating variable is assumed to belong to a certain parametric family. We examine two methods of estimating both the truncation and the lifetime distributions. We obtain asymptotic representations of the estimators for the lifetime distribution and establish their weak convergence. Both of the proposed estimators perform better than Wang’s (1991) nonparametric maximum likelihood estimator in terms of the integrated mean squared error, when the parametric family for the truncation is sufficiently close to its true distribution. The full likelihood approach is preferable to the conditional likelihood approach in estimating the lifetime distribution, though not necessarily the truncation distribution. In an application to Alzheimer’s disease data, hypothesis tests reject the uniform truncation distribution, but several other parametric models lead to similar behaviour of the truncation and lifetime distributions after disease onset.

Cross-sectional sampling is often used when investigating inter-event times, resulting in left-truncated and right-censored data. In this paper, we consider a semiparametric truncation model in which the truncating variable is assumed to belong to a certain parametric family. We examine two methods of estimating both the truncation and the lifetime distributions. We obtain asymptotic representations of the estimators for the lifetime distribution and establish their weak convergence. Both of the proposed estimators perform better than Wang’s (1991) nonparametric maximum likelihood estimator in terms of the integrated mean squared error, when the parametric family for the truncation is sufficiently close to its true distribution. The full likelihood approach is preferable to the conditional likelihood approach in estimating the lifetime distribution, though not necessarily the truncation distribution. In an application to Alzheimer’s disease data, hypothesis tests reject the uniform truncation distribution, but several other parametric models lead to similar behaviour of the truncation and lifetime distributions after disease onset.

Evaluating and understanding the risk and safety of using medications for autoimmune disease in a woman during her pregnancy will help both clinicians and pregnant women to make better treatment decisions. However, utilizing spontaneous abortion (SAB) data collected in observational studies of pregnancy to derive valid inference poses two major challenges. First, the data from the observational cohort are not random samples of the target population due to the sampling mechanism. Pregnant women with early SAB are more likely to be excluded from the cohort, and there may be substantial differences between the observed SAB time and those in the target population. Second, the observed data are heterogeneous and contain a "cured" proportion. In this article, we consider semiparametric models to simultaneously estimate the probability of being cured and the distribution of time to SAB for the uncured subgroup. To derive the maximum likelihood estimators, we appropriately adjust the sampling bias in the likelihood function and develop an expectation-maximization algorithm to overcome the computational challenge. We apply the empirical process theory to prove the consistency and asymptotic normality of the estimators. We examine the finite sample performance of the proposed estimators in simulation studies and illustrate the proposed method through an application to SAB data from pregnant women.

In forestry, many processes of interest are binary and they can be modeled using lifetime analysis. However, available data are often incomplete, being interval- and right-censored as well as lefttruncated, which may lead to biased parameter estimates. While censoring can be easily considered in lifetime analysis, left truncation is more complicated when individual age at selection is unknown. In this study, we designed and tested a maximum likelihood estimator that deals with left truncation by taking advantage of prior knowledge about the time when the individuals enter the experiment. Whenever a model is available for predicting the time of selection, the distribution of the delayed entries can be obtained using Bayes’ theorem. It is then possible to marginalize the likelihood function over the distribution of the delayed entries in the experiment to assess the joint distribution of time of selection and time to event. This estimator was tested with continuous and discrete Gompertz-distributed lifetimes. It was then compared with two other estimators: a standard one in which left truncation was not considered and a second estimator that implemented an analytical correction. Our new estimator yielded unbiased parameter estimates with empirical coverage of confidence intervals close to their nominal value. The standard estimator leaded to an overestimation of the long-term probability of survival

Aging societies have given rise to important challenges in the field of health insurance. Elderly policyholders need to be provided with fair premiums based on their individual health status, whereas insurance companies want to plan for the potential costs of tackling lifetimes above mean expectations. In this article, we focus on a large cohort of policyholders in Barcelona (Spain), aged 65 years and over. A shared-parameter joint model is proposed to analyse the relationship between annual demand for emergency claims and time until death outcomes, which are subject to left truncation. We compare different functional forms of the association between both processes, and, furthermore, we illustrate how the fitted model provides time-dynamic predictions of survival probabilities. The parameter estimation is performed under the Bayesian framework using Markov chain Monte Carlo methods.

Peer Reviewed

Background When patients are admitted to an intensive care unit (ICU) their risk of getting an infection will be highly depend on the length of stay at-risk in the ICU. In addition, risk of infection is likely to vary over calendar time as a result of fluctuations in the prevalence of the pathogen on the ward. Hence risk of infection is expected to depend on two time scales (time in ICU and calendar time) as well as competing events (discharge or death) and their spatial location. The purpose of this paper is to develop and apply appropriate statistical models for the risk of ICU-acquired infection accounting for multiple time scales, competing risks and the spatial clustering of the data. Methods A multi-center data base from a Spanish surveillance network was used to study the occurrence of an infection due to Methicillin-resistant Staphylococcus aureus (MRSA). The analysis included 84,843 patient admissions between January 2006 and December 2011 from 81 ICUs. Stratified Cox models were used to study multiple time scales while accounting for spatial clustering of the data (patients within ICUs) and for death or discharge as competing events for MRSA infection. Results Both time scales, time in ICU and calendar time, are highly associated with the MRSA hazard rate and cumulative risk. When using only one basic time scale, the interpretation and magnitude of several patient-individual risk factors differed. Risk factors concerning the severity of illness were more pronounced when using only calendar time. These differences disappeared when using both time scales simultaneously. Conclusions The time-dependent dynamics of infections is complex and should be studied with models allowing for multiple time scales. For patient individual risk-factors we recommend stratified Cox regression models for competing events with ICU time as the basic time scale and calendar time as a covariate. The inclusion of calendar time and stratification by ICU allow to indirectly account f

For a linear regression model subject to left-truncation and right-censoring where the truncation and censoring points are known constants (or always observed if random), Karlsson and Laitila (Stat Probab Lett 78:2567–2571,2008) proposed a semiparametric estimator which deals with left-truncation by trimming and right-censoring by ‘winsorizing’. The estimator was motivated by a zero moment condition where a transformed error term appears with trimmed and winsorized tails. This paper takes the semiparametric estimator further by deriving the asymptotic distribution that was not shown in Karlsson and Laitila (Stat Probab Lett 78:2567–2571,2008) and discusses its implementation aspects in practice, albeit brief.

In a prevalent cohort study with follow-up subjects identified as prevalent cases are followed until failure (defined suitably) or censoring. When the dates of the initiating events of these prevalent cases are ascertainable, each observed datum point consists of a backward recurrence time and a possibly censored forward recurrence time. Their sum is well known to be the left truncated lifetime. It is common to term these left truncated lifetimes "length biased" if the initiating event times of all the incident cases (including those not observed through the prevalent sampling scheme) follow a stationary Poisson process. Statistical inference is then said to be carried out under stationarity. Whether or not stationarity holds, a further assumption needed for estimation of the incident survivor function is the independence of the lifetimes and their accompanying truncation times. That is, it must be assumed that survival does not depend on the calendar date of the initiating event. We show how this assumption may be checked under stationarity, even though only the backward recurrence times and their associated (possibly censored) forward recurrence times are observed. We prove that independence of the lifetimes and truncation times is equivalent to equality in distribution of the backward and forward recurrence times, and exploit this equivalence as a means of testing the former hypothesis. A simulation study is conducted to investigate the power and Type 1 error rate of our proposed tests, which include a bootstrap procedure that takes into account the pairwise dependence between the forward and backward recurrence times, as well as the potential censoring of only one of the members of each pair. We illustrate our methods using data from the Canadian Study of Health and Aging. We also point out an equivalence of the problem presented here to a non-standard changepoint problem.

The product-limit estimator under left truncation and right censoring was first proposed and shown to be asymptotically normal by Tsai et al. (1987). In this article, we shall develop an Edgeworth expansion for the studentized product-limit estimator. This is achieved by representing the studentized product-limit estimator as a U -statistic plus some negligible remainder terms. The bootstrap is employed to approximate the distribution of the studentized product-limit estimator and shown to be second-order accurate.

The product-limit estimator under left truncation and right censoring was first proposed and shown to be asymptotically normal by Tsai et al. (1987). In this article, we shall develop an Edgeworth expansion for the studentized product-limit estimator. This is achieved by representing the studentized product-limit estimator as a U -statistic plus some negligible remainder terms. The bootstrap is employed to approximate the distribution of the studentized product-limit estimator and shown to be second-order accurate.

The product-limit estimator under left truncation and right censoring was first proposed and shown to be asymptotically normal by Tsai et al. (1987). In this article, we shall develop an Edgeworth expansion for the studentized product-limit estimator. This is achieved by representing the studentized product-limit estimator as a U -statistic plus some negligible remainder terms. The bootstrap is employed to approximate the distribution of the studentized product-limit estimator and shown to be second-order accurate.

We consider a model where elements of a single type are life tested. All elements are observed up to the time of their failures or censorings. Three types of events are possible to observe during life testing for each element: failure, censoring, and warning, where a warning can only be observed before a failure or before censoring has occurred. It is essential to know if warnings influence subsequent failures. Two subsets of data are simultaneously considered: the first consisting of only the times of the first occurrences of failure, censoring, or warning, and the second consisting of the times for those elements where warnings occurred before failures or censorings. The first subset belongs to the competing risks model, and the second consists of left-truncated data. Estimators of the cumulative hazard function before and after warnings are derived and proved to be consistent, with asymptotic normal distributions. A null hypothesis where the cumulative hazard functions before and after warnings are proportional and a corresponding alternative hypothesis that they are not proportional are defined. Under this null hypothesis an estimator for the constant of proportionality is derived and showed to be strongly consistent. Martingale techniques are used and numerical examples are provided.

Retrospectively ascertained data are common in many areas, including demography, epidemiology, and actuarial science. The main objective of this article is to study the effect of retrospective ascertainment on inference regarding recurrent events of time to pregnancy (TTP) data. For the particular TTP dataset that we consider, couples are included retrospectively based on their first pregnancy and then followed prospectively to a second pregnancy or to end of study. We consider a conditional model for the recurrent events data where the second TTP is included only if it is observed and a full model where the nonobserved second TTPs are included as suitably right censored. We furthermore consider two different approaches to modeling the dependencies of the recurrent events. A traditional frailty model, where the frailty enters the model as an unobserved covariate, and a marginal frailty model are applied. We find that efficiency is gained from including the second TTPs, with the full model being the most efficient. Further, the marginal frailty model is preferred over the traditional frailty model because estimates of covariate effects are easier to interpret and are ore robust to changes in the frailty distribution.

Retrospectively ascertained data are common in many areas, including demography, epidemiology, and actuarial science. The main objective of this article is to study the effect of retrospective ascertainment on inference regarding recurrent events of time to pregnancy (TTP) data. For the particular TTP dataset that we consider, couples are included retrospectively based on their first pregnancy and then followed prospectively to a second pregnancy or to end of study. We consider a conditional model for the recurrent events data where the second TTP is included only if it is observed and a full model where the nonobserved second TTPs are included as suitably right censored. We furthermore consider two different approaches to modeling the dependencies of the recurrent events. A traditional frailty model, where the frailty enters the model as an unobserved covariate, and a marginal frailty model are applied. We find that efficiency is gained from including the second TTPs, with the full model being the most efficient. Further, the marginal frailty model is preferred over the traditional frailty model because estimates of covariate effects are easier to interpret and are ore robust to changes in the frailty distribution.

Retrospectively ascertained data are common in many areas, including demography, epidemiology, and actuarial science. The main objective of this article is to study the effect of retrospective ascertainment on inference regarding recurrent events of time to pregnancy (TTP) data. For the particular TTP dataset that we consider, couples are included retrospectively based on their first pregnancy and then followed prospectively to a second pregnancy or to end of study. We consider a conditional model for the recurrent events data where the second TTP is included only if it is observed and a full model where the nonobserved second TTPs are included as suitably right censored. We furthermore consider two different approaches to modeling the dependencies of the recurrent events. A traditional frailty model, where the frailty enters the model as an unobserved covariate, and a marginal frailty model are applied. We find that efficiency is gained from including the second TTPs, with the full model being the most efficient. Further, the marginal frailty model is preferred over the traditional frailty model because estimates of covariate effects are easier to interpret and are ore robust to changes in the frailty distribution.

In survival analysis, as in other branches of applied statistics, confirmatory analysis is often requested before an initial finding can be cleared of suspicion of being due just to mass significance. This paper points out that the special temporal structure of the classical clinical trial with staggered entry allows re‐use of the initial survivors by left‐truncating their remaining survival times. This procedure can often save a considerable amount of calendar time. The procedure is illustrated by data from the Danish Breast Cancer Cooperative Group.

In survival analysis, as in other branches of applied statistics, confirmatory analysis is often requested before an initial finding can be cleared of suspicion of being due just to mass significance. This paper points out that the special temporal structure of the classical clinical trial with staggered entry allows re‐use of the initial survivors by left‐truncating their remaining survival times. This procedure can often save a considerable amount of calendar time. The procedure is illustrated by data from the Danish Breast Cancer Cooperative Group.

In a prevalent cohort study with follow-up subjects identified as prevalent cases are followed until failure (defined suitably) or censoring. When the dates of the initiating events of these prevalent cases are ascertainable, each observed datum point consists of a backward recurrence time and a possibly censored forward recurrence time. Their sum is well known to be the left truncated lifetime. It is common to term these left truncated lifetimes "length biased" if the initiating event times of all the incident cases (including those not observed through the prevalent sampling scheme) follow a stationary Poisson process. Statistical inference is then said to be carried out under stationarity. Whether or not stationarity holds, a further assumption needed for estimation of the incident survivor function is the independence of the lifetimes and their accompanying truncation times. That is, it must be assumed that survival does not depend on the calendar date of the initiating event. We show how this assumption may be checked under stationarity, even though only the backward recurrence times and their associated (possibly censored) forward recurrence times are observed. We prove that independence of the lifetimes and truncation times is equivalent to equality in distribution of the backward and forward recurrence times, and exploit this equivalence as a means of testing the former hypothesis. A simulation study is conducted to investigate the power and Type 1 error rate of our proposed tests, which include a bootstrap procedure that takes into account the pairwise dependence between the forward and backward recurrence times, as well as the potential censoring of only one of the members of each pair. We illustrate our methods using data from the Canadian Study of Health and Aging. We also point out an equivalence of the problem presented here to a non-standard changepoint problem.