Maximum Full Likelihood Approach to Randomly Truncated Data

Truncated data are commonly observed in economics, epidemiology, and other fields. The analysis of truncated data is challenging because the observed data are usually a biased sample of the target population due to truncation. Existing methods of handling truncated data largely depend on conditional likelihood which is the joint distribution of the data given that they are observed, and may be unreliable or have potential efficiency loss. In this paper, the authors develop a maximum full likelihood inference method for truncated data under a parametric model for the conditional distribution of the target variable given covatiates. The distribution of the truncation variable is left unspecified. The authors establish the asymptotic normalities of the maximum likelihood estimators (MLE) for various parameters, and the likelihood ratio statistics have central chisquare limiting distributions. As a by-product, the proposed method provides a natural MLE for the total number of the observed and unobserved data, which may shed light on the extent of truncation bias. A score test is provided to check the correctness of the assumed parametric model. Our simulation results indicate that the proposed estimation method generally produces more reliable point and interval estimates. For illustration, the authors apply the proposed approaches to analyze a breast cancer data in the Rotterdam Tumor Bank.