The analysis of mixed interval-censored and complete data

The article focuses mainly on a conditional imputation algorithm of quantile-filling to analyze a new kind of censored data, mixed interval-censored and complete data related to interval-censored sample. With the algorithm, the imputed failure times, which are the conditional quantiles, are obtained within the censoring intervals in which some exact failure times are. The algorithm is viable and feasible for the parameter estimation with general distributions, for instance, a case of Weibull distribution that has a moment estimation of closed form by log-transformation. Furthermore, interval-censored sample is a special case of the new censored sample, and the conditional imputation algorithm can also be used to deal with the failure data of interval censored. By comparing the interval-censored data and the new censored data, using the imputation algorithm, in the view of the bias of estimation, we find that the performance of new censored data is better than that of interval censored.     

Non-parametric maximum likelihood estimation of interval-censored failure time data subject to misclassification

The paper considers non-parametric maximum likelihood estimation of the failure time distribution for interval-censored data subject to misclassification. Such data can arise from two types of observation scheme; either where observations continue until the first positive test result or where tests continue regardless of the test results. In the former case, the misclassification probabilities must be known, whereas in the latter case, joint estimation of the event-time distribution and misclassification probabilities is possible. The regions for which the maximum likelihood estimate can only have support are derived. Algorithms for computing the maximum likelihood estimate are investigated and it is shown that algorithms appropriate for computing non-parametric mixing distributions perform better than an iterative convex minorant algorithm in terms of time to absolute convergence. A profile likelihood approach is proposed for joint estimation. The methods are illustrated on a data set relating to the onset of cardiac allograft vasculopathy in post-heart-transplantation patients.     

A Logspline Estimation for a Linear Regression Model with an Interval-Censored Continuous Covariate

The study of a linear regression model with an interval-censored covariate, which was motivated by an acquired immunodeficiency syndrome (AIDS) clinical trial, was first proposed by Gómez et al. They developed a likelihood approach, together with a two-step conditional algorithm, to estimate the regression coefficients in the model. However, their method is inapplicable when the interval-censored covariate is continuous. In this article, we propose a novel and fast method to treat the continuous interval-censored covariate. By using logspline density estimation, we impute the interval-censored covariate with a conditional expectation. Then, the ordinary least-squares method is applied to the linear regression model with the imputed covariate. To assess the performance of the proposed method, we compare our imputation with the midpoint imputation and the semiparametric hierarchical method via simulations. Furthermore, an application to the AIDS clinical trial is presented.     

Multiple imputation for competing risks regression with interval-censored data

ABSTRACT

We present here an extension of Pan's multiple imputation approach to Cox regression in the setting of interval-censored competing risks data. The idea is to convert interval-censored data into multiple sets of complete or right-censored data and to use partial likelihood methods to analyse them. The process is iterated, and at each step, the coefficient of interest, its variance–covariance matrix, and the baseline cumulative incidence function are updated from multiple posterior estimates derived from the Fine and Gray sub-distribution hazards regression given augmented data. Through simulation of patients at risks of failure from two causes, and following a prescheduled programme allowing for informative interval-censoring mechanisms, we show that the proposed method results in more accurate coefficient estimates as compared to the simple imputation approach. We have implemented the method in the MIICD R package, available on the CRAN website.     

Regression Analysis of Left-truncated and Case I Interval-censored Data with the Additive Hazards Model

In recent years the analysis of interval-censored failure time data has attracted a great deal of attention and such data arise in many fields including demographical studies, economic and financial studies, epidemiological studies, social sciences, and tumorigenicity experiments. This is especially the case in medical studies such as clinical trials. In this article, we discuss regression analysis of one type of such data, Case I interval-censored data, in the presence of left-truncation. For the problem, the additive hazards model is employed and the maximum likelihood method is applied for estimations of unknown parameters. In particular, we adopt the sieve estimation approach that approximates the baseline cumulative hazard function by linear functions. The resulting estimates of regression parameters are shown to be consistent and efficient and have an asymptotic normal distribution. An illustrative example is provided.     

Proportional hazards regression with interval-censored and left-truncated data

This paper considers the estimation of the regression coefficients in the Cox proportional hazards model with left-truncated and interval-censored data. Using the approaches of Pan [A multiple imputation approach to Cox regression with interval-censored data, Biometrics 56 (2000), pp. 199–203] and Heller [Proportional hazards regression with interval censored data using an inverse probability weight, Lifetime Data Anal. 17 (2011), pp. 373–385], we propose two estimates of the regression coefficients. The first estimate is based on a multiple imputation methodology. The second estimate uses an inverse probability weight to select event time pairs where the ordering is unambiguous. A simulation study is conducted to investigate the performance of the proposed estimators. The proposed methods are illustrated using the Centers for Disease Control and Prevention (CDC) acquired immunodeficiency syndrome (AIDS) Blood Transfusion Data.     

Estimation procedures for grouped data – a comparative study

Interval-censored data are very common in the reliability and lifetime data analysis. This paper investigates the performance of different estimation procedures for a special type of interval-censored data, i.e. grouped data, from three widely used lifetime distributions. The approaches considered here include the maximum likelihood estimation, the minimum distance estimation based on chi-square criterion, the moment estimation based on imputation (IM) method and an ad hoc estimation procedure. Although IM-based techniques are extensively used recently, we show that this method is not always effective. It is found that the ad hoc estimation procedure is equivalent to the minimum distance estimation with another distance metric and more effective in the simulation. The procedures of different approaches are presented and their performances are investigated by Monte Carlo simulation for various combinations of sample sizes and parameter settings. The numerical results provide guidelines to analyse grouped data for practitioners when they need to choose a good estimation approach.     

A two‐step proximal‐point algorithm for the calculus of divergence‐based estimators in finite mixture models

Estimators derived from the expectation‐maximization (EM) algorithm are not robust since they are based on the maximization of the likelihood function. We propose an iterative proximal‐point algorithm based on the EM algorithm to minimize a divergence criterion between a mixture model and the unknown distribution that generates the data. The algorithm estimates in each iteration the proportions and the parameters of the mixture components in two separate steps. Resulting estimators are generally robust against outliers and misspecification of the model. Convergence properties of our algorithm are studied. The convergence of the introduced algorithm is discussed on a two‐component Weibull mixture entailing a condition on the initialization of the EM algorithm in order for the latter to converge. Simulations on Gaussian and Weibull mixture models using different statistical divergences are provided to confirm the validity of our work and the robustness of the resulting estimators against outliers in comparison to the EM algorithm. An application to a dataset of velocities of galaxies is also presented. The Canadian Journal of Statistics 47: 392–408; 2019 © 2019 Statistical Society of Canada     

A study of moments and likelihood estimators of the odd Weibull distribution

The odd Weibull distribution is a three-parameter generalization of the Weibull and the inverse Weibull distributions having rich density and hazard shapes for modeling lifetime data. This paper explored the odd Weibull parameter regions having finite moments and examined the relation to some well-known distributions based on skewness and kurtosis functions. The existence of maximum likelihood estimators have shown with complete data for any sample size. The proof for the uniqueness of these estimators is given only when the absolute value of the second shape parameter is between zero and one. Furthermore, elements of the Fisher information matrix are obtained based on complete data using a single integral representation which have shown to exist for any parameter values. The performance of the odd Weibull distribution over various density and hazard shapes is compared with generalized gamma distribution using two different test statistics. Finally, analysis of two data sets has been performed for illustrative purposes.     

Parameter estimation of three-parameter Weibull distribution based on progressively Type-II censored samples

In this paper, the estimation of parameters for a three-parameter Weibull distribution based on progressively Type-II right censored sample is studied. Different estimation procedures for complete sample are generalized to the case with progressively censored data. These methods include the maximum likelihood estimators (MLEs), corrected MLEs, weighted MLEs, maximum product spacing estimators and least squares estimators. We also proposed the use of a censored estimation method with one-step bias-correction to obtain reliable initial estimates for iterative procedures. These methods are compared via a Monte Carlo simulation study in terms of their biases, root mean squared errors and their rates of obtaining reliable estimates. Recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of inference developed here.     

Consistency of the Generalized MLE with Interval-Censored and Masked Competing Risks Data

We consider nonparametric estimation based on interval-censored competing risks data with masked failure cause. The generalized maximum likelihood estimator of the joint survival function of the failure time and the failure cause is studied under mixed case interval censorship and random partition masking. Strong consistency in the L ₁(μ)-topology is established for some finite measure μ which is derived from the joint censoring and masking distribution. Under additional regularity assumptions we also establish the strong consistencies in the topologies of weak convergence, point-wise convergence, and uniform convergence.     

A Bezier curve method in interval-censored data

In this paper we propose a Bezier curve method to estimate the survival function and the median survival time in interval-censored data. We compare the proposed estimator with other existing methods such as the parametric method, the single point imputation method, and the nonparametric maximum likelihood estimator through extensive numerical studies, and it is shown that the proposed estimator performs better than others in the sense of mean squared error and mean integrated squared error. An illustrative example based on a real data set is given.     

Semiparametric Bayesian accelerated failure time model with interval-censored data

Many existing approaches to analysing interval-censored data lack flexibility or efficiency. In this paper, we propose an efficient, easy to implement approach on accelerated failure time model with a logarithm transformation of the failure time and flexible specifications on the error distribution. We use exact inference for the Dirichlet process without approximation in imputation. Our algorithm can be implemented with simple Gibbs sampling which produces exact posterior distributions on the features of interest. Simulation and real data analysis demonstrate the advantage of our method compared to some other methods.     

A generalization of Turnbull’s estimator for nonparametric estimation of the conditional survival function with interval-censored data

Simple nonparametric estimates of the conditional distribution of a response variable given a covariate are often useful for data exploration purposes or to help with the specification or validation of a parametric or semi-parametric regression model. In this paper we propose such an estimator in the case where the response variable is interval-censored and the covariate is continuous. Our approach consists in adding weights that depend on the covariate value in the self-consistency equation proposed by Turnbull (J R Stat Soc Ser B 38:290–295, 1976), which results in an estimator that is no more difficult to implement than Turnbull’s estimator itself. We show the convergence of our algorithm and that our estimator reduces to the generalized Kaplan–Meier estimator (Beran, Nonparametric regression with randomly censored survival data, 1981) when the data are either complete or right-censored. We demonstrate by simulation that the estimator, bootstrap variance estimation and bandwidth selection (by rule of thumb or cross-validation) all perform well in finite samples. We illustrate the method by applying it to a dataset from a study on the incidence of HIV in a group of female sex workers from Kinshasa.     

Nonparametric curve estimation with missing data: A general empirical process approach

A general nonparametric imputation procedure, based on kernel regression, is proposed to estimate points as well as set- and function-indexed parameters when the data are missing at random (MAR). The proposed method works by imputing a specific function of a missing value (and not the missing value itself), where the form of this specific function is dictated by the parameter of interest. Both single and multiple imputations are considered. The associated empirical processes provide the right tool to study the uniform convergence properties of the resulting estimators. Our estimators include, as special cases, the imputation estimator of the mean, the estimator of the distribution function proposed by Cheng and Chu [1996. Kernel estimation of distribution functions and quantiles with missing data. Statist. Sinica 6, 63–78], imputation estimators of a marginal density, and imputation estimators of regression functions.     

Nonconcave penalized estimation for partially linear models with longitudinal data

A nonconcave penalized estimation method is proposed for partially linear models with longitudinal data when the number of parameters diverges with the sample size. The proposed procedure can simultaneously estimate the parameters and select the important variables. Under some regularity conditions, the rate of convergence and asymptotic normality of the resulting estimators are established. In addition, an iterative algorithm is proposed to implement the proposed estimators. To improve efficiency for regression coefficients, the estimation of the covariance function is integrated in the iterative algorithm. Simulation studies are carried out to demonstrate that the proposed method performs well, and a real data example is analysed to illustrate the proposed procedure.     

Parametric fractional imputation for nonignorable missing data

Parameter estimation with missing data is a frequently encountered problem in statistics. Imputation is often used to facilitate the parameter estimation by simply applying the complete-sample estimators to the imputed dataset.In this article, we consider the problem of parameter estimation with nonignorable missing data using the approach of parametric fractional imputation proposed by Kim (2011). Using the fractional weights, the E-step of the EM algorithm can be approximated by the weighted mean of the imputed data likelihood where the fractional weights are computed from the current value of the parameter estimates. Calibration fractional imputation is also considered as a way for improving the Monte Carlo approximation in the fractional imputation. Variance estimation is also discussed. Results from two simulation studies are presented to compare the proposed method with the existing methods. A real data example from the Korea Labor and Income Panel Survey (KLIPS) is also presented.     

Estimation and goodness-of-fit for the Cox model with various types of censored data

The currently existing estimation methods and goodness-of-fit tests for the Cox model mainly deal with right censored data, but they do not have direct extension to other complicated types of censored data, such as doubly censored data, interval censored data, partly interval-censored data, bivariate right censored data, etc. In this article, we apply the empirical likelihood approach to the Cox model with complete sample, derive the semiparametric maximum likelihood estimators (SPMLE) for the Cox regression parameter and the baseline distribution function, and establish the asymptotic consistency of the SPMLE. Via the functional plug-in method, these results are extended in a unified approach to doubly censored data, partly interval-censored data, and bivariate data under univariate or bivariate right censoring. For these types of censored data mentioned, the estimation procedures developed here naturally lead to Kolmogorov-Smirnov goodness-of-fit tests for the Cox model. Some simulation results are presented.     

Ml estimation in the poisson binomial distribution with grouped data via the em algorithm

The maximum likelihood estimation of parameters of the Poisson binomial distribution, based on a sample with exact and grouped observations, is considered by applying the EM algorithm (Dempster et al, 1977). The results of Louis (1982) are used in obtaining the observed information matrix and accelerating the convergence of the EM algorithm substantially. The maximum likelihood estimation from samples consisting entirely of complete (Sprott, 1958) or grouped observations are treated as special cases of the estimation problem mentioned above. A brief account is given for the implementation of the EM algorithm when the sampling distribution is the Neyman Type A since the latter is a limiting form of the Poisson binomial. Numerical examples based on real data are included.     

Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution

In this paper, the Markov chain Monte Carlo (MCMC) method is used to estimate the parameters of a modified Weibull distribution based on a complete sample. While maximum-likelihood estimation (MLE) is the most used method for parameter estimation, MCMC has recently emerged as a good alternative. When applied to parameter estimation, MCMC methods have been shown to be easy to implement computationally, the estimates always exist and are statistically consistent, and their probability intervals are convenient to construct. Details of applying MCMC to parameter estimation for the modified Weibull model are elaborated and a numerical example is presented to illustrate the methods of inference discussed in this paper. To compare MCMC with MLE, a simulation study is provided, and the differences between the estimates obtained by the two algorithms are examined.