Left Truncation Research Articles

Asymptotic Properties of a Nonparametric Regression Function Estimator with Randomly Truncated Data

In this paper, we define a new kernel estimator of the regression function under a left truncation model. We establish the pointwise and uniform strong consistency over a compact set and give a rate of convergence of the estimate. The pointwise asymptotic normality of the estimate is also given. Some simulations are given to show the asymptotic behavior of the estimate in different cases. The distribution function and the covariable’s density are also estimated.

Lorenz Curve for Truncated and Censored Data

In this paper, we consider a nonparametric estimator of the Lorenz curve and Gini index when the data are subjected to random left truncation and right censorship. Strong Gaussian approximations for the associated Lorenz process are established under appropriate assumptions. A law of the iterated logarithm for the Lorenz curve is also derived. Lastly, we obtain a central limit theorem for the corresponding Gini index.

HLA typing associated with hepatitis B E antigen seroconversion in children with chronic hepatitis B virus infection: A long-term prospective sibling cohort study in Taiwan

To conduct a prospective cohort study to clarify the relationship between human leukocyte antigen (HLA) polymorphisms and the seroconversion of hepatitis B e antigen (HBeAg). In the prospective cohort study, 81 HBeAg-positive children with chronic hepatitis B virus (HBV) infection from 40 unrelated families were recruited and followed-up regularly for a mean period of 17.70 +/- 3.23 years. The association between HLA antigen and the age at HBeAg seroconversion was analyzed using Cox regression model with shared frailties under left truncation and right censorship. HLA-B61 and HLA-DQB1*0503 antigens predicted a higher HBeAg seroconversion rate (relative incidence = 6.17 and 3.22, P = .024 and .017, respectively). Within-family frailty in our sibling cohort study demonstrated a negligible or a low degree of within-family correlation with spontaneous HBeAg seroconversion in each HLA antigen. HLA class I antigen B61 and class II antigen DQB1*0503 are associated with earlier HBeAg seroconversion in Taiwanese children with chronic HBV infection.

Edgeworth Expansion and Bootstrap Approximation for Studentized Product-Limit Estimator with Truncated and Censored Data

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.

Virtual Bass Model and the left-hand data-truncation bias in diffusion of innovation studies

One of the most important applications of the Bass model is “guessing by analogy” — the parameter estimates of the Bass model for analogous products can be used to predict the diffusion pattern of a new product. However, estimates based on left-hand truncated data will be biased unless care is taken to adjust for the bias. We demonstrate the prevalence of left truncation in historical sales data and present a method for dealing with the bias by the Virtual Bass Model. We use the model to develop a database of parameter estimates that may be used in “guessing by analogy.”

Uniform Representation of Product‐Limit Integrals with Applications

Abstract. Let X be a d‐variate random vector that is completely observed, and let Y be a random variable that is subject to right censoring and left truncation. For arbitrary functions ϕ we consider expectations of the form E[ϕ(X, Y)], which appear in many statistical problems, and we estimate these expectations by using a product‐limit estimator for censored and truncated data, extended to the context where covariates are present. An almost sure representation for these estimators is obtained, with a remainder term that is of a certain negligible order, uniformly over a class of ϕ‐functions. This uniformity is important for the application to goodness‐of‐fit testing in regression and to inference for the regression depth, which we consider in more detail.

An inverse-probability-weighted approach to estimation of the bivariate survival function under left-truncation and right-censoring

In this note, we consider estimating the bivariate survival function when both survival times are subject to random left truncation and one of the survival times is subject to random right censoring. Motivated by Satten and Datta [2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. 55, 207–210], we propose an inverse-probability-weighted (IPW) estimator. It involves simultaneous estimation of the bivariate survival function of the truncation variables and that of the censoring variable and the truncation variable of the uncensored components. We prove that (i) when there is no censoring, the IPW estimator reduces to NPMLE of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131] and (ii) when there is random left truncation and right censoring on only one of the components and the other component is always observed, the IPW estimator reduces to the estimator of Gijbels and Gürler [1998. Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Statist. Sin. 1219–1232]. Based on Theorem 3.1 of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627], we prove that the IPW estimator is consistent under certain conditions. Finally, we examine the finite sample performance of the IPW estimator in some simulation studies. For the special case that censoring time is independent of truncation time, a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627]. For the special case (i), a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by Huang et al. (2001. Nonnparametric estimation of marginal distributions under bivariate truncation with application to testing for age-of-onset application. Statist. Sin. 11, 1047–1068).

Nonparametric Bivariate Estimation with Left-Truncated and Right-Censored Data

ABSTRACT In this article we consider estimating the bivariate survival function observations where one of the components is subject to left truncation and right censoring and the other is subject to right censoring only. Two types of nonparametric estimators are proposed. One is in the form of inverse-probability-weighted average (Satten and Datta, 2001) and the other is a generalization of Dabrowska's 1988 estimator. The two are then compared based on their empirical performances.

Semiparametric Analysis of Survival Data with Left Truncation and Dependent Right Censoring

Studies of chronic life-threatening diseases often involve both mortality and morbidity. In observational studies, the data may also be subject to administrative left truncation and right censoring. Because mortality and morbidity may be correlated and mortality may censor morbidity, the Lynden-Bell estimator for left-truncated and right-censored data may be biased for estimating the marginal survival function of the non-terminal event. We propose a semiparametric estimator for this survival function based on a joint model for the two time-to-event variables, which utilizes the gamma frailty specification in the region of the observable data. First, we develop a novel estimator for the gamma frailty parameter under left truncation. Using this estimator, we then derive a closed-form estimator for the marginal distribution of the non-terminal event. The large sample properties of the estimators are established via asymptotic theory. The methodology performs well with moderate sample sizes, both in simulations and in an analysis of data from a diabetes registry.

Relative hazard rate estimation for right censored and left truncated data

In medical research differences among treatment groups are a common focus of study. The concept of relative hazard rate is a tool for comparing two groups in terms of their difference in risk rates. A kernel estimator is proposed in the case where both samples are subject to left truncation and right censoring and an iid representation is obtained in this setup. The asymptotic distribution and the asymptotic mean squared error of the estimator are obtained. An application to the famous Channing House data set illustrates the theory.

Sequential Confidence Bands for Quantile Densities Under Truncated and Censored Data

In this paper an asymptotic distribution is obtained for the maximal deviation between the kernel quantile density estimator and the quantile density when the data are subject to random left truncation and right censorship. Based on this result we propose a fully sequential procedure for constructing a fixed-width confidence band for the quantile density on a finite interval and show that the procedure has the desired coverage probability asymptotically as the width of the band approaches zero.

Quantile process for left truncated and right censored data

In this paper, we consider the product-limit quantile estimator of an unknown quantile function when the data are subject to random left truncation and right censorship. This is a parallel problem to the estimation of the unknown distribution function by the product-limit estimator under the same model. Simultaneous strong Gaussian approximations of the product-limit process and product-limit quantile process are constructed with rate\(O(\frac{{(\log n)^{3/2} }}{{n^{1/8} }})\). A functional law of the iterated logarithm for the maximal deviation of the estimator from the estimand is derived from the construction.

Estimation of Survival Function Subject to Time-Dependent Covariates and Left Truncation

Satten et al. [Satten, G. A., Datta, S., Robins, J. M. (2001). Estimating the marginal survival function in the presence of time dependent covariates. Statis. Prob. Lett. 54: 397--403] proposed an estimator [denoted by Ŝ(t)] of survival function of failure times that is in the class of survival function estimators proposed by Robins [Robins, J. M. (1993). Information recovery and bias adjustment in proportional hazards regression analysis of randomized trials using surrogate markers. In: Proceedings of the American Statistical Association-Biopharmaceutical Section. Alexandria, VA: ASA, pp. 24--33]. The estimator is appropriate when data are subject to dependent censoring. In this article, it is demonstrated that the estimator Ŝ(t) can be extended to estimate the survival function when data are subject to dependent censoring and left truncation. In addition, we propose an alternative estimator of survival function [denoted by Ŝ w (t)] that is represented as an inverse-probability-weighted average Satten and Datta [Satten, G. A., Datta, S. (2001). The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. Ass. 55: 207--210]. Simulation results show that when truncation is not severe the mean squared error of Ŝ(t) is smaller than that of Ŝ w (t), except for the case when censoring is light. However, when truncation is severe, Ŝ w (t) has the advantage of less bias and the situation can be reversed.

Strong representation of a conditional quantile function estimator with truncated and censored data

We consider lifetime data with covariables which are subject to both left truncation and right censorship. In this context, it is interesting to study the conditional distribution function of the lifetime and the corresponding conditional quantile function. A generalized product-limit estimator (GPLE) of the conditional distribution function has been studied in Iglesias-Pérez and González-Manteiga (J. Nonparametric Statist. 10 (1999) 213). In the present paper, we define a conditional quantile function estimator via the mentioned GPLE and we derive an almost sure representation for this quantile estimator. This result extends strong quantile representations studied on conditional survival analysis for censored data (Dabrowska (Sankhya Ser. A 54 (1992) 252) and Van Keilegom and Veraverbeke (J. Statist. Plann. Inference 69 (1998) 115)). As consequence of this representation we establish the asymptotic normality of the conditional quantile estimator.

A prospective study of maternal smoking and spontaneous abortion.

Cigarette smoking has been considered a risk factor for spontaneous abortion, but conclusions from many studies may be impaired by methodological problems. Studies are required that use data on smoking habits collected before knowledge about outcome of pregnancy and in which adjustment is made for potential confounders. All Danish-speaking pregnant women booking for delivery at the Department of Obstetrics and Gynecology, Aarhus University Hospital, from September 1989 to August 1996 were invited to participate in the study (n=24,608 pregnancies, including 321 spontaneous abortions and 100 induced abortions). Information about smoking habits before pregnancy and during the first trimester was obtained from self-administered questionnaires before the knowledge about pregnancy outcome. Information about spontaneous and induced abortions was obtained from the Danish National Patient Registry. We used Cox regression analyses with delayed entry (left truncation) for analyses, and hazard ratios to express the association between smoking and spontaneous abortion. Multivariate analyses were performed using a model including alcohol and coffee intake during pregnancy, maternal age, marital status, occupation, education, prepregnancy body mass index, and parity. Overall, 104 (1.3%) of those contributing time at risk in the first trimester experienced a spontaneous abortion in the first trimester, and 217 (0.9%) of those contributing time at risk in the second trimester experienced a second-trimester abortion. We found no association between smoking and first- and second-trimester abortions. Adjustment for alcohol, coffee, maternal age, marital status, occupation, education, prepregnancy body mass index and parity did not change the result substantially. Our study did not support findings from previous studies of an association between smoking and spontaneous abortion.

Fixed Design Nonparametric Regression with Truncated and Censored Data

In this paper we consider a fixed design model in which the observations are subject to left truncation and right censoring. A generalized product-limit estimator for the conditional distribution at a given covariate value is proposed, and an almost sure asymptotic representation of this estimator is established. We also obtain the rate of uniform consistency, weak convergence and a modulus of continuity for this estimator. Applications include trimmed mean and quantile function estimators.

Estimating multiplicative and additive hazard functions by kernel methods

We propose new procedures for estimating the component functions in both additive and multiplicative nonparametric marker-dependent hazard models. We work with a full counting process framework that allows for left truncation and right censoring and time-varying covariates. Our procedures are based on kernel hazard estimation as developed by Nielsen and Linton and on the idea of marginal integration. We provide a central limit theorem for the marginal integration estimator. We then define estimators based on finite-step backfitting in both additive and multiplicative cases and prove that these estimators are asymptotically normal and have smaller variance than the marginal integration method.

Three-state frailty model for age at onset of dementia and death in Swedish twins.

We present a frailty model to estimate the relative importance of genetic and environmental factors on age at onset of dementia in a twin design. We use modern survival methodology to define a model that accounts simultaneously for longitudinal aspects, e.g., left truncation and right censoring in data, and the multivariate nature of twin data. Additionally, we present a novel three-state frailty model, with nondemented, demented, and dead states, describing variation in the onset of disease and mortality simultaneously in one model, while accounting for possible dependence for the two competing events. The frailty structure, i.e., the latent random effects structure, mimics the traditional twin model for continuous variables used in quantitative genetics, and as such describes within-pair dependence. This in turn leads to estimates for intrapair correlations, as well as for additive genetic, and shared and nonshared environmental components of variance. A hierarchical Bayesian model formulation and Gibbs sampling are used to estimate posterior distributions of the parameters. The models are applied to Swedish Twin Registry data on the onset of dementia in elderly twins. Based on the three-state frailty model, we estimate the intrapair correlations for dementia to be 0.87 [90% credible interval: 0.61,0.98] and 0.68[0.18,0.91] for MZ and DZ twins, respectively. Based on our model, we estimate that genetic effects account for about one third, and shared environmental effects for almost a half, of the variation in dementia hazards between individuals. More data, however, are needed to gain precision in these estimates.

Survival curve estimation with partial non-random exposure information.

The objective of this paper is to estimate survival curves for two different exposure groups when the exposure group is not known for all observations, and the data is subject to left truncation and right censoring. The situation we consider is when the probability that the exposure group is missing may depend on whether the observation is censored or uncensored, in which case the exposure is not missing at random. The problem was motivated by a study of Alzheimer's disease to estimate the distribution of ages at diagnosis for individuals with and without an apolipoprotein E4 allele (the exposure group). Genotyping for this risk factor was incomplete and performed more frequently on the cases of Alzheimer's disease (the uncensored observations) than the censored observations. The survival curves are estimated in discrete time using an EM algorithm. A bootstrapping procedure is proposed that guarantees each bootstrap sample has the same proportion of observations with missing exposure. A simulation is performed to evaluate the bias of the estimators and to investigate design and efficiency issues. The methods are applied to the Alzheimer's disease study.

A Simple Two-Sample Rank Test for Multivariate Survival Outcomes with Left Truncation and Right Censoring

Summary A distribution-free two-sample rank test is proposed for testing for differences between survival distributions in the analysis of biomedical studies in which two groups of subjects are followed over time for a particular outcome, which may recur. This method is motivated by an observational HIV (human immunodeficiency virus) study in which a group of HIV-seropositive women and a comparable group of HIV-seronegative women were examined every 6 months for the presence of cervical intraepithelial neoplasia (CIN), the cervical cancer precursor. Women entered the study serially and were subject to random loss to follow-up. Only women free of CIN at study entry were followed resulting in lefttruncated survival times. If a woman is found to be CIN infected at a later examination, she is treated and then followed until CIN recurs. The two groups of women were compared at both occurrences of CIN on the basis of rank statistics. For the first occurrence of CIN, survival times since the beginning of the study (based on calendar time) are compared. For a recurrence of CIN, survival times since the first development of CIN are compared. The proposed test statistic for an overall difference between the two groups follows a chi-square distribution with two degrees of freedom. Simulation results demonstrate the usefulness of the proposed test proposed test statistic, which reduces to the Gehan statistic if each person is followed only to the first failure and there is no serial enrollment.