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Linear Regression With an Independent Variable Subject to a Detection Limit

Nie, Leia; Chu, Haitaob; Liu, Chenglongc; Cole, Stephen R.d; Vexler, Alberte; Schisterman, Enrique F.f

doi: 10.1097/EDE.0b013e3181ce97d8
Methodologic Issues in Environmental Exposures: Mixtures and Limits of Detection: Original Article

Background: Linear regression with a left-censored independent variable X due to limit of detection (LOD) was recently considered by 2 groups of researchers: Richardson and Ciampi (Am J Epidemiol. 2003;157:355-363), and Schisterman et al (Am J Epidemiol. 2006;163:374-383).

Methods: Both groups obtained consistent estimators for the regression slopes by replacing left-censored X with a constant, that is, the expectation of X given X below LOD E(X|X<LOD) in the former group and the sample mean of X given X above LOD in the latter.

Results: Schisterman et al argued that their approach would be a better choice because the sample mean of X given X above LOD is available, whereas E(X|X<LOD) is unknown. Other substitution methods, such as replacing the left-censored values with LOD, or LOD/2,have been extensively used in the literature. Simulations were conducted to compare the performance under 2 scenarios in which the independent variable is normally and not normally distributed.

Conclusion: Recommendations are given based on theoretical and simulation results. These recommendations are illustrated with one case study.

From the aDivision of Biometrics IV, Office of Biometrics/OTS/CDER/FDA, Silver Spring, MD; bDepartment of Biostatistics and Lineberger Comprehensive Cancer Center, The University of North Carolina, Chapel Hill, NC; cDepartment of Medicine, Georgetown University, Washington, DC; dDepartment of Epidemiology, University of North Carolina at Chapel Hill School of Public Health, Chapel Hill, NC; eDepartment of Biostatistics, The State University of New York at Buffalo, Buffalo, NY; and fDivision of Epidemiology, Statistics, and Prevention Research, National Institute of Child Health and Human Development, National Institutes of Health, Rockville, MD.

Submitted 5 September 2008; accepted 3 June 2009; posted 23 March 2010.

Supported in part by the American Chemistry Council and the Intramural Research Program of the Eunice Kennedy Shriver National Institute of Child Health and Human Development.

Views expressed in this article are the author's professional opinions and do not necessarily represent the official positions of the US Food and Drug Administration.

Correspondence: Lei Nie, 10903 New Hampshire Ave, Silver Spring, MD 20993. E-mail:

© 2010 Lippincott Williams & Wilkins, Inc.