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Biostatistics. 2015 Oct;16(4):785-98. doi: 10.1093/biostatistics/kxv012. Epub 2015 Mar 21.

Semiparametric likelihood inference for left-truncated and right-censored data.

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Sidney Kimmel Comprehensive Cancer Center and Department of Biostatistics, Johns Hopkins University, Baltimore, MD 21205, USA
The University of Texas MD Anderson Cancer Center, Houston, TX 77030, USA.
National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, MD 20892, USA.


This paper proposes a new estimation procedure for the survival time distribution with left-truncated and right-censored data, where the distribution of the truncation time is known up to a finite-dimensional parameter vector. The paper expands on the Vardis multiplicative censoring model (Vardi, 1989. Multiplicative censoring, renewal processes, deconvolution and decreasing density: non-parametric estimation. Biometrika 76: , 751-761), establishes the connection between the likelihood under a generalized multiplicative censoring model and that for left-truncated and right-censored survival time data, and derives an Expectation-Maximization algorithm for model estimation. A formal test for checking the truncation time distribution is constructed based on the semiparametric likelihood ratio test statistic. In particular, testing the stationarity assumption that the underlying truncation time is uniformly distributed is performed by embedding the null uniform truncation time distribution in a smooth alternative (Neyman, 1937. Smooth test for goodness of fit. Skandinavisk Aktuarietidskrift 20: , 150-199). Asymptotic properties of the proposed estimator are established. Simulations are performed to evaluate the finite-sample performance of the proposed methods. The methods and theories are illustrated by analyzing the Canadian Study of Health and Aging and the Channing House data, where the stationarity assumption with respect to disease incidence holds for the former but not the latter.


Biased sampling; Cross-sectional studies; Prevalent sampling; Profile likelihood; Smooth tests of goodness of fit

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