Left Truncation Research Articles

A generalization of the Kaplan–Meier estimator for analyzing bivariate mortality under right-censoring and left-truncation with applications in model-checking for survival copula models

In this paper we provide a new nonparametric estimator of the joint distribution of two lifetimes under random right censoring and left truncation which can be seen as a bivariate extension of the Kaplan–Meier estimator. We derive asymptotic results for this estimator, including uniform n1/2-consistency, and develop a general methodology for bivariate lifetime modeling, a critical issue in studying reversion conditions that are commonplace in defined benefit pensions and private annuity contracts. An application to goodness-of-fit for survival copula models is discussed. We show that the procedures that we use are consistent, and propose a bootstrap procedure based on our estimator to compute the critical values. The new technique that we propose is tested on the Canadian dataset initially studied by Frees et al. (1996).

Jointly modeling the relationship between longitudinal and survival data subject to left truncation with applications to cystic fibrosis.

Numerous methods for joint analysis of longitudinal measures of a continuous outcome y and a time to event outcome T have recently been developed either to focus on the longitudinal data y while correcting for nonignorable dropout, to predict the survival outcome T using the longitudinal data y, or to examine the relationship between y and T. The motivating problem for our work is in joint modeling of the serial measurements of pulmonary function (FEV1% predicted) and survival in cystic fibrosis (CF) patients using registry data. Within the CF registry data, an additional complexity is that not all patients have been followed from birth; therefore, some patients have delayed entry into the study while others may have been missed completely, giving rise to a left truncated distribution. This paper shows in joint modeling situations where y and T are not independent, that it is necessary to account for this left truncation to obtain valid parameter estimates related to both survival and the longitudinal marker. We assume a linear random effects model for FEV1% predicted, where the random intercept and slope of FEV1% predicted, along with a specified transformation of the age at death follow a trivariate normal distribution. We develop an expectation-maximization algorithm for maximum likelihood estimation of parameters, which takes left truncation and right censoring of survival times into account. The methods are illustrated using simulation studies and using data from CF patients in a registry followed at Rainbow Babies and Children's Hospital, Cleveland, OH.

Selection, Reallocation, and Spillover: Identifying the Sources of Gains from Multinational Production

Assessing the productivity gains from multinational production has been a vital topic of economic research. Positive aggregate productivity gains are often attributed to within-firm productivity improvement; however, an alternative, less emphasized explanation is between-firm selection and market reallocation, whereby competition from multinationals leads to factor reallocation and the survival of only the most productive domestic firms. We investigate the roles of the two different mechanisms in determining the aggregate productivity gains by exploring their distinct predictions on the distributions of domestic firms: within-firm productivity improvement shifts the productivity and the revenue distributions rightward while between-fim selection and market reallocation raise the left truncation of the distributions and shift revenue leftward. Using a rich cross-country firm-level panel dataset, we find significant evidence of both mechanisms, but between-firm selection and market reallocation accounts for the majority of aggregate productivity gains, suggesting that ignoring this channel could lead to substantial bias in understanding the nature of gains from multinational production.

Selection and Market Reallocation: Productivity Gains from Multinational Production

Assessing productivity gains from multinational production has been a vital topic of economic research and policy debate. Positive productivity gains are often attributed to productivity spillovers; however, an alternative, much less emphasized channel is selection and market reallocation, whereby competition leads to factor and revenue reallocation within and between domestic firms and exits of the least productive firms. We investigate the roles of these different mechanisms in determining aggregate-productivity gains using a unifying framework that explores the mechanisms' distinct predictions on the distributions of domestic firms: within-firm productivity improvement shifts rightward or reshapes the productivity distribution, while selection and market reallocation move the revenue and employment distributions leftward and raise left truncations. Using a rich cross-country firm-level panel dataset, we find significant evidence of both mechanisms and effects of competition in product, technology, and labor space. However, selection and market reallocation account for the majority of aggregate-productivity gains, suggesting ignoring this channel could lead to substantial bias in understanding the nature of productivity gains from multinational production. (JEL D22, D24, F14, F23, G32, O47)

Investigating hospital heterogeneity with a multi-state frailty model: application to nosocomial pneumonia disease in intensive care units

BackgroundMultistate models have become increasingly useful to study the evolution of a patient’s state over time in intensive care units ICU (e.g. admission, infections, alive discharge or death in ICU). In addition, in critically-ill patients, data come from different ICUs, and because observations are clustered into groups (or units), the observed outcomes cannot be considered as independent. Thus a flexible multi-state model with random effects is needed to obtain valid outcome estimates.MethodsWe show how a simple multi-state frailty model can be used to study semi-competing risks while fully taking into account the clustering (in ICU) of the data and the longitudinal aspects of the data, including left truncation and right censoring. We suggest the use of independent frailty models or joint frailty models for the analysis of transition intensities. Two distinct models which differ in the definition of time t in the transition functions have been studied: semi-Markov models where the transitions depend on the waiting times and nonhomogenous Markov models where the transitions depend on the time since inclusion in the study. The parameters in the proposed multi-state model may conveniently be computed using a semi-parametric or parametric approach with an existing R package FrailtyPack for frailty models. The likelihood cross-validation criterion is proposed to guide the choice of a better fitting model.ResultsWe illustrate the use of our approach though the analysis of nosocomial infections (ventilator-associated pneumonia infections: VAP) in ICU, with “alive discharge” and “death” in ICU as other endpoints. We show that the analysis of dependent survival data using a multi-state model without frailty terms may underestimate the variance of regression coefficients specific to each group, leading to incorrect inferences. Some factors are wrongly significantly associated based on the model without frailty terms. This result is confirmed by a short simulation study. We also present individual predictions of VAP underlining the usefulness of dynamic prognostic tools that can take into account the clustering of observations.ConclusionsThe use of multistate frailty models allows the analysis of very complex data. Such models could help improve the estimation of the impact of proposed prognostic features on each transition in a multi-centre study. We suggest a method and software that give accurate estimates and enables inference for any parameter or predictive quantity of interest.

New Results Concerning Probability Distributions with Increasing Generalized Failure Rates

Lariviere and Porteus (2001) introduce the concept of generalized failure rate of a continuous random variable and demonstrate its importance. If the valuation distribution of a product has an increasing generalized failure rate (that is, the distribution is IGFR), then the associated revenue function is unimodal, and when the generalized failure rate is strictly increasing, the global maximum is uniquely specified. Assuming that the distribution is IGFR is thus useful and frequently-held in recent pricing, revenue and supply chain management literature. This note extends the IGFR concept in three ways. First, it investigates left and right truncations of valuation distributions — truncations arise when price floors or price ceilings are imposed. Second, it generalizes the IGFR definition to situations where the buyer’s payment is different from the seller’s receipt due to, say, taxes or subsidies. Third, we extend the IGFR notion to discrete distributions and contrast it with the continuous distribution case. The note also addresses two errors in the previous IGFR literature. Finally, we analyze all common (continuous and discrete) distributions for the prevalence of the IGFR property, and derive and tabulate their generalized failure rates, for future reference.

Assessing the effect of vaccine on spontaneous abortion using time‐dependent covariates Cox models

In studying the safety of vaccine for influenza A (H1N1) given during pregnancy, spontaneous abortion (SAB) is one of the important points to consider. Because women may receive the vaccine any time during their pregnancy, evaluation of the effect of the vaccine on SAB should only take place after vaccination and in the risk window for SAB, that is, the first 20 weeks of gestation. In addition, when such studies are conducted through pregnancy registries where recruitment occurs after pregnancy recognition, the accrued subjects are left truncated in the sense that they are not followed from the start of pregnancy. As previously reported, left truncation needs to be properly handled using survival analysis methods to avoid bias. In the context of time-dependent vaccine exposure, a time-dependent covariate Cox model can be used to simultaneously take into account the left truncation and the vaccine exposure timing. In this communication, we illustrate the approach using the Vaccine and Medication in Pregnancy Surveillance System data. We explain in details how the model is fitted using different software. We recommend survival analysis methods together with collection of necessary data to study the effects of vaccine on SAB.

Modeling left-truncated and right-censored survival data with longitudinal covariates

There is a surge in medical follow-up studies that include longitudinal covariates in the modeling of survival data. So far, the focus has been largely on right censored survival data. We consider survival data that are subject to both left truncation and right censoring. Left truncation is well known to produce biased sample. The sampling bias issue has been resolved in the literature for the case which involves baseline or time-varying covariates that are observable. The problem remains open however for the important case where longitudinal covariates are present in survival models. A joint likelihood approach has been shown in the literature to provide an effective way to overcome those difficulties for right censored data, but this approach faces substantial additional challenges in the presence of left truncation. Here we thus propose an alternative likelihood to overcome these difficulties and show that the regression coefficient in the survival component can be estimated unbiasedly and efficiently. Issues about the bias for the longitudinal component are discussed. The new approach is illustrated numerically through simulations and data from a multi-center AIDS cohort study.

Left truncated and right censored Weibull data and likelihood inference with an illustration

The Weibull distribution is a very popular distribution for modeling lifetime data. Left truncation and right censoring are often observed in lifetime data. Here, the EM algorithm is applied to estimate the model parameters of the Weibull distribution fitted to data containing left truncation and right censoring. The maximization part of the EM algorithm is carried out using the EM gradient algorithm (Lange, 1995). The Weibull distribution is also fitted using the Newton–Raphson (NR) method. The two methods of estimation are then compared through an extensive Monte Carlo simulation study. The asymptotic variance–covariance matrix of the MLEs under the EM framework is obtained through the missing information principle (Louis, 1982), and asymptotic confidence intervals for the parameters are then constructed. The asymptotic confidence intervals corresponding to the missing information principle and the observed information matrix are compared in terms of coverage probabilities, through a simulation study. Finally, all the methods of inference discussed here are illustrated through some numerical examples.

A Parametric Approach to Estimate Survival Time of Diabetic Nephropathy with Left Truncated and Right Censored Data

Diabetic Nephropathy (DN) is the deleterious effect of diabetes mellitus on renal structure, or a function caused by it. The rate of rise in, serum creatinine is observed for the progression of DN. Retrospective data is collected from 132 patients. As 60 patients form uncensored cases, remaining 72 patients are censored cases. In this paper we have developed three models. The first model is based only on uncensored cases with left truncation. The second model also includes censored data. Fisher's information matrix has been applied to find variances and asymptotic confidence interval for the estimated parameters. The method of maximum likelihood is used to estimate unknown parameters in both the models. In the last model, onset times are arranged as to estimate mean onset time of DN for all the patients by using application of order statistic. We aim to predict onset time of DN for new diabetic patients.

Erratum

Family RelationsVolume 61, Issue 2 p. fm6-fm6 Free Access Erratum This article corrects the following: Financial Issues and Relationship Outcomes Among Cohabiting Individuals Jeffrey Dew, Volume 60Issue 2Family Relations pages: 178-190 First Published online: March 9, 2011 First published: 13 March 2012 https://doi.org/10.1111/j.1741-3729.2012.00714.xCitations: 1AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Correction to “Financial Issues and Relationship Outcomes Among Cohabiting Individuals” published in the April 2011 issue of Family Relations. Specifically the paragraph on page 183 that states, “The Cox regressions were also adjusted for the data's left truncation…” The models were not corrected for left truncation. The Cox regressions still estimate the hazard of cohabiting couples splitting up during the period of the first two waves of the NSFH based on their conflict types and the control covariates. These estimates are not, however, corrected for the left truncation. I apologize for the error. References Dew, J. (2011). Financial Issues and Relationship Outcomes Among Cohabiting Individuals. Family Relations, 60: 178– 190. doi: 10.1111/j.1741-3729.2010.00641.x Wiley Online LibraryWeb of Science®Google Scholar Citing Literature Volume61, Issue2April 2012Pages fm6-fm6 ReferencesRelatedInformation

Comparison of estimators in nested case–control studies with multiple outcomes

Reuse of controls in a nested case-control (NCC) study has not been considered feasible since the controls are matched to their respective cases. However, in the last decade or so, methods have been developed that break the matching and allow for analyses where the controls are no longer tied to their cases. These methods can be divided into two groups; weighted partial likelihood (WPL) methods and full maximum likelihood methods. The weights in the WPL can be estimated in different ways and four estimation procedures are discussed. In addition, we address modifications needed to accommodate left truncation. A full likelihood approach is also presented and we suggest an aggregation technique to decrease the computation time. Furthermore, we generalize calibration for case-cohort designs to NCC studies. We consider a competing risks situation and compare WPL, full likelihood and calibration through simulations and analyses on a real data example.

Clarification and correction

We would like to clarify a few points and report an erratum to our paper entitled “Parametric mixture models to evaluate and summarize hazard ratios in the presence of competing risks with time-dependent hazards and delayed entry” 1. Finally, it has come to our attention that stata 11 (and more recent versions) can account for left truncation in the Fine and Gray subdistribution proportional hazards model. In closing, we would like to thank Professor Alvaro Muñoz for his careful reading of our paper.

Estimating survival of dental fillings on the basis of interval‐censored data and multi‐state models

We aim to compare the life expectancy of a filling in a primary tooth between two types of treatments. We define the probabilities that a dental filling survives without complication until the permanent tooth erupts from beneath (exfoliation). We relate the time to exfoliation of the tooth to the age of the child and the time to failure of the filling to the duration since the treatment. We followed up fillings at repeated examinations where information is collected regarding the filling and the tooth. Several fillings can be placed in the same mouth, possibly by the same dentist. To deal with all these particularities, we propose to use a parametric four-state model with three random effects to take into account the hierarchical cluster structure. For inference, right and interval censoring as well as left truncation have to be dealt with. With the proposed approach, we can conclude that the estimated probability that a filling survives without complication until exfoliation is larger for one treatment than for the other, for all ages of the child at the time of treatment.

Continuous covariate frailty models for censored and truncated clustered data

Using some logarithmic and integral transformation we transform a continuous covariate frailty model into a polynomial regression model with a random effect. The responses of this mixed model can be ‘estimated’ via conditional hazard function estimation. The random error in this model does not have zero mean and its variance is not constant along the covariate and, consequently, these two quantities have to be estimated. Since the asymptotic expression for the bias is complicated, the two-large-bandwidth trick is proposed to estimate the bias. The proposed transformation is very useful for clustered incomplete data subject to left truncation and right censoring (and for complex clustered data in general). Indeed, in this case no standard software is available to fit the frailty model, whereas for the transformed model standard software for mixed models can be used for estimating the unknown parameters in the original frailty model. A small simulation study illustrates the good behavior of the proposed method. This method is applied to a bladder cancer data set.

Out of sight but not out of mind: Why failure to account for left truncation biases research on failure rates

We note at least three major issues in entrepreneurship theory that can be clarified by studying the survival chances of new ventures: the extent to which entrepreneurs are so constrained by initial founding conditions that they are unable to learn; the degree to which heterogeneity and innovative capabilities are lost due to the failure of new ventures; and the imprinting effects of new ventures' early days on their subsequent development. However, previous research on these issues has been inconclusive because of problems in research design and data analysis. In this paper, we shed light on new venture failure rates by assessing the validity and generalizability of previous findings. We argue that research using registration data to study new ventures is very likely to generate biased results and that research attempting to track new ventures from a very early stage can still suffer from selection bias due to left truncation. Using a sample of new ventures from the Panel Study of Entrepreneurial Dynamics II, we provide evidence for the extent of such biases. We offer a statistical solution to left truncation that can be easily applied in widely used statistical programs.

Effective Truncation of a Student’s <i>t</i>-Distribution by Truncation of the Chi Distribution in a Chi-Normal Mixture

A Student’s t-distribution is obtained from a weighted average over the standard deviation of a normal distribution, σ, when 1/σ is distributed as chi. Left truncation at q of the chi distribution in the mixing integral leads to an effectively truncated Student’s t-distribution with tails that decay as exp (-q2t2). The effect of truncation of the chi distribution in a chi-normal mixture is investigated and expressions for the pdf, the variance, and the kurtosis of the t-like distribution that arises from the mixture of a left-truncated chi and a normal distribution are given for selected degrees of freedom 5. This work has value in pricing financial assets, in understanding the Student’s t--distribution, in statistical inference, and in analysis of data.

A Generalized Time-Dependent Conditional Linear Model with Left-Truncated and Right-Censored Data

Consider the model φ(S(y | X)) = β(y) T X, where φ is a known link function, S(· | X) is the survival function of a response Y given a covariate X = (1, X, X 2,…, X p ), and β(y) is an unknown vector of time-dependent regression coefficients. The response Y is subject to left truncation and right censoring. We assume that given X, Y is independent of (C, T) where C and T are censoring and truncation variables with P(C ≥ T) = 1. In this article, with some modification of the assumptions in Lemmas 5 and 6 of Iglesias-Pérez and González-Manteiga (1999), we present an almost sure representation for the generalized product-limit estimator (GPL) of S(y | X). Based on the GPL and the approach of Teodorescu et al. (2010), a least squares estimator of β(y) is obtained and a bootstrap procedure is proposed to choose the optimum bandwidth.

Estimation of the Bivariate Cause-Specific Distribution Function with Left-Truncated Competing Risks Data

In this article, we consider estimating the bivariate cause-specific distribution function when both components are subject to left truncation and right censoring. Based on the estimators proposed by Shen and Yan (2008), we propose two types of estimators as the extension of Sankaran et al. (2006). The asymptotical properties of the proposed estimators are established. A simulation study is conducted to investigate the performance of the proposed estimators.

Proportional subdistribution hazards regression for left-truncated competing risks data

Fine and Gray [(1999), ‘A Proportional Hazards Model for the Subdistribution of a Competing Risk’, Journal of the American Statistical Association, 94, 496–509] considered a proportional subdistribution hazards model under the framework of competing risks. Using the partial likelihood principle and weighting techniques, they derived estimation and inference procedures for regression parameters under right censoring. In this article, we show that the partial likelihood approach to estimation is applicable when both right censoring and left truncation are present. We derive large sample properties of the proposed estimators and investigate their finite sample performances via simulation.