Singly and Doubly Censored Current Status Data: Estimation, Asymptotics and Regression

In biostatistical applications interest often focuses on the estimation of the distribution of time between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well-understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, with its distribution over an observed interval [A, B] known; the data is referred to as doubly censored current status data. This model has applications in AIDS partner studies. If the initial time is known to be uniformly distribute d, the model reduces to a submodel of the current status model with the same asymptotic information bounds as in the current status model, but the distribution of interest is essentially the derivative of the distribution of interest in the current status model. As a consequence the non-parametric maximum likelihood estimator is inconsistent. Moreover, this submodel contains only smooth heavy tailed distributions for which no moments exist. In this paper, we discuss the connection between the singly censored current status model and the doubly censored current status model (for the uniform initial time) in detail and explain the difficulties in estimation which arise in the doubly censored case. We propose a regularized MLE corresponding with the current status model. We prove rate results, efficiency of smooth functionals of the regularized MLE, and present a generally applicable efficient method for estimation of regression parameters, which does not rely on the existence of moments. We also discuss extending these ideas to a non-uniform distribution for the initial time.     

Semiparametric analysis of panel count data with correlated observation and follow-up times

This paper discusses regression analysis of panel count data that often arise in longitudinal studies concerning occurrence rates of certain recurrent events. Panel count data mean that each study subject is observed only at discrete time points rather than under continuous observation. Furthermore, both observation and follow-up times can vary from subject to subject and may be correlated with the recurrent events. For inference, we propose some shared frailty models and estimating equations are developed for estimation of regression parameters. The proposed estimates are consistent and have asymptotically a normal distribution. The finite sample properties of the proposed estimates are investigated through simulation and an illustrative example from a cancer study is provided.     

A semiparametric additive rates model for recurrent event data

Recurrent event data often arise in biomedical studies, with examples including hospitalizations, infections, and treatment failures. In observational studies, it is often of interest to estimate the effects of covariates on the marginal recurrent event rate. The majority of existing rate regression methods assume multiplicative covariate effects. We propose a semiparametric model for the marginal recurrent event rate, wherein the covariates are assumed to add to the unspecified baseline rate. Covariate effects are summarized by rate differences, meaning that the absolute effect on the rate function can be determined from the regression coefficient alone. We describe modifications of the proposed method to accommodate a terminating event (e.g., death). Proposed estimators of the regression parameters and baseline rate are shown to be consistent and asymptotically Gaussian. Simulation studies demonstrate that the asymptotic approximations are accurate in finite samples. The proposed methods are applied to a state-wide kidney transplant data set.     

A parametric model fitting time to first event for overdispersed data: application to time to relapse in multiple sclerosis

In this article, we propose a parametric model for the distribution of time to first event when events are overdispersed and can be properly fitted by a Negative Binomial distribution. This is a very common situation in medical statistics, when the occurrence of events is summarized as a count for each patient and the simple Poisson model is not adequate to account for overdispersion of data. In this situation, studying the time of occurrence of the first event can be of interest. From the Negative Binomial distribution of counts, we derive a new parametric model for time to first event and apply it to fit the distribution of time to first relapse in multiple sclerosis (MS). We develop the regression model with methods for covariate estimation. We show that, as the Negative Binomial model properly fits relapse counts data, this new model matches quite perfectly the distribution of time to first relapse, as tested in two large datasets of MS patients. Finally we compare its performance, when fitting time to first relapse in MS, with other models widely used in survival analysis (the semiparametric Cox model and the parametric exponential, Weibull, log-logistic and log-normal models).     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

Proportional cross-ratio model

Cross-ratio is an important local measure of the strength of dependence among correlated failure times. If a covariate is available, it may be of scientific interest to understand how the cross-ratio varies with the covariate as well as time components. Motivated by the Tremin study, where the dependence between age at a marker event reflecting early lengthening of menstrual cycles and age at menopause may be affected by age at menarche, we propose a proportional cross-ratio model through a baseline cross-ratio function and a multiplicative covariate effect. Assuming a parametric model for the baseline cross-ratio, we generalize the pseudo-partial likelihood approach of Hu et al. (Biometrika 98:341–354, 2011) to the joint estimation of the baseline cross-ratio and the covariate effect. We show that the proposed parameter estimator is consistent and asymptotically normal. The performance of the proposed technique in finite samples is examined using simulation studies. In addition, the proposed method is applied to the Tremin study for the dependence between age at a marker event and age at menopause adjusting for age at menarche. The method is also applied to the Australian twin data for the estimation of zygosity effect on cross-ratio for age at appendicitis between twin pairs.

     

Structural Equation Models for Dealing With Spatial Confounding

In regression analyses of spatially structured data, it is common practice to introduce spatially correlated random effects into the regression model to reduce or even avoid unobserved variable bias in the estimation of other covariate effects. If besides the response the covariates are also spatially correlated, the spatial effects may confound the effect of the covariates or vice versa. In this case, the model fails to identify the true covariate effect due to multicollinearity. For highly collinear continuous covariates, path analysis and structural equation modeling techniques prove to be helpful to disentangle direct covariate effects from indirect covariate effects arising from correlation with other variables. This work discusses the applicability of these techniques in regression setups, where spatial and covariate effects coincide at least partly and classical geoadditive models fail to separate these effects. Supplementary materials for this article are available online.     

Regression Parameter Estimation from Panel Counts

This paper considers a study where each subject may experience multiple occurrences of an event and the rate of the event occurrences is of primary interest. Specifically, we are concerned with the situations where, for each subject, there are only records of the accumulated counts for the event occurrences at a finite number of time points over the study period. Sets of observation times may vary from subject to subject and differ between groups. We model the mean of the event occurrence number over time semiparametrically, and estimate the regression parameter. The proposed estimation procedures are illustrated with data from a bladder cancer study ( Byar, 1980 ). Both asymptotics and simulation studies on the estimators are presented.     

Semiparametric inference for a general class of models for recurrent events

Procedures for estimating the parameters of the general class of semiparametric models for recurrent events proposed by Peña and Hollander [(2004). Models for recurrent events in reliability and survival analysis. In: Soyer R., Mazzuchi T., Singpurwalla N. (Eds.), Mathematical Reliability: An Expository Perspective. Kluwer Academic Publishers, Dordrecht, pp. 105–123 (Chapter 6)] are developed. This class of models incorporates an effective age function encoding the effect of changes after each event occurrence such as the impact of an intervention, it models the impact of accumulating event occurrences on the unit, it admits a link function in which the effect of possibly time-dependent covariates are incorporated, and it allows the incorporation of unobservable frailty components which induce dependencies among the inter-event times for each unit. The estimation procedures are semiparametric in that a baseline hazard function is nonparametrically specified. The sampling distribution properties of the estimators are examined through a simulation study, and the consequences of mis-specifying the model are analyzed. The results indicate that the flexibility of this general class of models provides a safeguard for analyzing recurrent event data, even data possibly arising from a frailty-less mechanism. The estimation procedures are applied to real data sets arising in the biomedical and public health settings, as well as from reliability and engineering situations. In particular, the procedures are applied to a data set pertaining to times to recurrence of bladder cancer and the results of the analysis are compared to those obtained using three methods of analyzing recurrent event data.     

Misclassification of current status data

We describe a simple method for nonparametric estimation of a distribution function based on current status data where observations of current status information are subject to misclassification. Nonparametric maximum likelihood techniques lead to use of a straightforward set of adjustments to the familiar pool-adjacent-violators estimator used when misclassification is assumed absent. The methods consider alternative misclassification models and are extended to regression models for the underlying survival time. The ideas are motivated by and applied to an example on human papilloma virus (HPV) infection status of a sample of women examined in San Francisco.     

A Semiparametric Binary Regression Model Involving Monotonicity Constraints

Abstract.  We study a binary regression model using the complementary log–log link, where the response variable Δ is the indicator of an event of interest (for example, the incidence of cancer, or the detection of a tumour) and the set of covariates can be partitioned as ( X ,  Z ) where Z (real valued) is the primary covariate and X (vector valued) denotes a set of control variables. The conditional probability of the event of interest is assumed to be monotonic in Z , for every fixed X . A finite-dimensional (regression) parameter β describes the effect of X . We show that the baseline conditional probability function (corresponding to X  =  0 ) can be estimated by isotonic regression procedures and develop an asymptotically pivotal likelihood-ratio-based method for constructing (asymptotic) confidence sets for the regression function. We also show how likelihood-ratio-based confidence intervals for the regression parameter can be constructed using the chi-square distribution. An interesting connection to the Cox proportional hazards model under current status censoring emerges. We present simulation results to illustrate the theory and apply our results to a data set involving lung tumour incidence in mice.     

Estimating Marginal Effects in Accelerated Failure Time Models for Serial Sojourn Times Among Repeated Events

Recurrent event data are commonly encountered in longitudinal studies when events occur repeatedly over time for each study subject. An accelerated failure time (AFT) model on the sojourn time between recurrent events is considered in this article. This model assumes that the covariate effect and the subject-specific frailty are additive on the logarithm of sojourn time, and the covariate effect maintains the same over distinct episodes, while the distributions of the frailty and the random error in the model are unspecified. With the ordinal nature of recurrent events, two scale transformations of the sojourn times are derived to construct semiparametric methods of log-rank type for estimating the marginal covariate effects in the model. The proposed estimation approaches/inference procedures also can be extended to the bivariate events, which alternate themselves over time. Examples and comparisons are presented to illustrate the performance of the proposed methods.     

Efficient estimation for the proportional hazards model with bivariate current status data

We consider efficient estimation of regression and association parameters jointly for bivariate current status data with the marginal proportional hazards model. Current status data occur in many fields including demographical studies and tumorigenicity experiments and several approaches have been proposed for regression analysis of univariate current status data. We discuss bivariate current status data and propose an efficient score estimation approach for the problem. In the approach, the copula model is used for joint survival function with the survival times assumed to follow the proportional hazards model marginally. Simulation studies are performed to evaluate the proposed estimates and suggest that the approach works well in practical situations. A real life data application is provided for illustration.     

The NPMLE for Doubly Censored Current Status Data

In biostatistical applications interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, but with its distribution over an observed interval ' A, B ' known to be uniformly distributed; the data is referred to as doubly censored current status data. These authors used this model to handle application in AIDS partner studies focusing on the NPMLE of the distribution G of T . The model is a submodel of the current status model, but the distribution G is essentially the derivative of the distribution of interest F in the current status model. In this paper we establish that the NPMLE of G is uniformly consistent and that the resulting estimators for the n ^1/2-estimable parameters are efficient. We propose an iterative weighted pool-adjacent-violator-algorithm to compute the estimator. It is also shown that, without smoothness assumptions, the NPMLE of F converges at rate n ^−2/5 in L ²-norm while the NPMLE of F in the non-parametric current status data model converges at rate n ^−1/3 in L ²-norm, which shows that there is a substantial gain in using the submodel information.     

Bayesian regression model for recurrent event data with event-varying covariate effects and event effect

In the course of hypertension, cardiovascular disease events (e.g. stroke, heart failure) occur frequently and recurrently. The scientific interest in such study may lie in the estimation of treatment effect while accounting for the correlation among event times. The correlation among recurrent event times comes from two sources: subject-specific heterogeneity (e.g. varied lifestyles, genetic variations, and other unmeasurable effects) and event dependence (i.e. event incidences may change the risk of future recurrent events). Moreover, event incidences may change the disease progression so that there may exist event-varying covariate effects (the covariate effects may change after each event) and event effect (the effect of prior events on the future events). In this article, we propose a Bayesian regression model that not only accommodates correlation among recurrent events from both sources, but also explicitly characterizes the event-varying covariate effects and event effect. This model is especially useful in quantifying how the incidences of events change the effects of covariates and risk of future events. We compare the proposed model with several commonly used recurrent event models and apply our model to the motivating lipid-lowering trial (LLT) component of the Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial (ALLHAT) (ALLHAT-LLT).     

Estimation of the additive hazards model with interval-censored data and missing covariates

The additive hazards model is one of the most commonly used regression models in the analysis of failure time data and many methods have been developed for its inference in various situations. However, no established estimation procedure exists when there are covariates with missing values and the observed responses are interval-censored; both types of complications arise in various settings including demographic, epidemiological, financial, medical and sociological studies. To address this deficiency, we propose several inverse probability weight-based and reweighting-based estimation procedures for the situation where covariate values are missing at random. The resulting estimators of regression model parameters are shown to be consistent and asymptotically normal. The numerical results that we report from a simulation study suggest that the proposed methods work well in practical situations. An application to a childhood cancer survival study is provided. The Canadian Journal of Statistics 48: 499–517; 2020 © 2020 Statistical Society of Canada     

Regression analysis of multivariate current status data under a varying coefficients additive hazards frailty model

This article discusses regression analysis of multivariate current status failure time data for which the observation time may be related to the underlying survival time. A local partial likelihood technique is used to estimate the varying coefficient covariate effect functions under the additive hazards frailty model. The asymptotic properties of the proposed estimators are established. An extensive simulation study is conducted for the evaluation of the proposed procedure, the results of which indicate that the proposed method works well in practice. Also, a real data study is provided to illustrate the performance of the proposed method.     

A class of semiparametric cure models with current status data

Current status data occur in many biomedical studies where we only know whether the event of interest occurs before or after a particular time point. In practice, some subjects may never experience the event of interest, i.e., a certain fraction of the population is cured or is not susceptible to the event of interest. We consider a class of semiparametric transformation cure models for current status data with a survival fraction. This class includes both the proportional hazards and the proportional odds cure models as two special cases. We develop efficient likelihood-based estimation and inference procedures. We show that the maximum likelihood estimators for the regression coefficients are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in finite samples. For illustration, we provide an application of the models to a study on the calcification of the hydrogel intraocular lenses.

     

Nonparametric estimation of a regression function from backward recurrence times in a cross-sectional sampling

This study considers the nonparametric estimation of a regression function when the response variable is the waiting time between two consecutive events of a stationary renewal process, and where this variable is not completely observed. In these circumstances, our data are the recurrence times from the occurrence of the last event up to a pre-established time, along with the corresponding values of a certain set of covariates. Estimation of the error density function and some of its characteristics are also considered. For the proposed estimators, we first analyze their asymptotic behavior and, thereafter, carry out a simulation study to highlight their behavior in finite samples. Finally, we apply this methodology to an illustrative example with biomedical data.     

Nonstationary nonlinear quantile regression

This study examines estimation and inference based on quantile regression for parametric nonlinear models with an integrated time series covariate. We first derive the limiting distribution of the nonlinear quantile regression estimator and then consider testing for parameter restrictions, when the regression function is specified as an asymptotically homogeneous function. We also study linear-in-parameter regression models when the regression function is given by integrable regression functions as well as asymptotically homogeneous regression functions. We, furthermore, propose a fully modified estimator to reduce the bias in the original estimator under a certain set of conditions. Finally, simulation studies show that the estimators behave well, especially when the regression error term has a fat-tailed distribution.