Maximum Likelihood Estimations and EM Algorithms With Length-Biased Data

• Published in: Journal of the American Statistical Association Vol. 106; no. 496; pp. 1434 - 1449
• Main Authors: Qin, Jing, Ning, Jing, Liu, Hao, Shen, Yu
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis 01.12.2011; American Statistical Association; Taylor & Francis Ltd
• Subjects: Algorithms; Applications; Asymmetry; Bias; Cancer; Censored data; Censorship; Cox regression model; Dementia; Distribution; Estimation; Estimation bias; Estimation methods; Estimators; Etiology; Exact sciences and technology; General topics; Increasing failure rate; Insurance, economics, finance; Mathematics; Maximum likelihood estimation; Maximum likelihood estimators; Maximum likelihood method; Medical screening; Modeling; Nonparametric likelihood; Nonparametric models; Normality; Parameter estimation; Parametric inference; Probability and statistics; Probability theory and stochastic processes; Profile likelihood; Regression analysis; Reliability; Right-censored data; Sampling techniques; Sciences and techniques of general use; Semiparametric modeling; Simulation; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Statistical methods; Statistics; Survival analysis; Theory and Methods; Nonparametric likelihood; Strong consistency; Estimator robustness; Censored sample; Non parametric estimation; Cox regression model; Statistical test; Estimator efficiency; Consistent estimator; Increasing function; Genetics; Sample survey; Increasing failure rate; Survival data; Censored data; Asymptotic behavior; Statistical estimation; Life test; Semiparametric method; Failure rate; Statistical method; Survival analysis; Profile likelihood; Estimating function; Cox model; Sampling theory; Estimating equation; Maximum likelihood; Econometrics; Reliability; EM algorithm; Right-censored data; Biased estimation
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1198/jasa.2011.tm10156

Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, and epidemiological, genetic, and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimation and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite-dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semiparametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.