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Estimating Population Distributions When Some Data Are Below a Limit of Detection by Using a Reverse Kaplan-Meier Estimator

Gillespie, Brenda W.a,b; Chen, Qixuana,c; Reichert, Heidib; Franzblau, Alfredd; Hedgeman, Elizabethd; Lepkowski, Jamese; Adriaens, Peterf; Demond, Averyf; Luksemburg, Williamg; Garabrant, David H.d

doi: 10.1097/EDE.0b013e3181ce9f08
Methodologic Issues in Environmental Exposures: Mixtures and Limits of Detection: Original Article

Background: Data with some values below a limit of detection (LOD) can be analyzed using methods of survival analysis for left-censored data. The reverse Kaplan-Meier (KM) estimator provides an effective method for estimating the distribution function and thus population percentiles for such data. Although developed in the 1970s and strongly advocated since then, it remains rarely used, partly due to limited software availability.

Methods: In this paper, the reverse KM estimator is described and is illustrated using serum dioxin data from the University of Michigan Dioxin Exposure Study (UMDES) and the National Health and Nutrition Examination Survey (NHANES). Percentile estimates for left-censored data using the reverse KM estimator are compared with replacing values below the LOD with the LOD/2 or LOD/√2.

Results: When some LODs are in the upper range of the complete values, and/or the percent censored is high, the different methods can yield quite different percentile estimates. The reverse KM estimator, which is the nonparametric maximum likelihood estimator, is the preferred method. Software options are discussed: The reverse KM can be calculated using software for the KM estimator. The JMP and SAS (SAS Institute, Cary, NC) and Minitab (Minitab, Inc, State College, PA), software packages calculate the reverse KM directly using their Turnbull estimator routines.

Conclusion: The reverse KM estimator is recommended for estimation of the distribution function and population percentiles in preference to commonly used methods such as substituting LOD/2 or LOD/√2 for values below the LOD, assuming a known parametric distribution, or using imputation to replace the left-censored values.


From the aDepartment of Biostatistics, School of Public Health, University of Michigan, Ann Arbor, MI; bCenter for Statistical Consultation and Research, University of Michigan, Ann Arbor, MI; cDepartment of Biostatistics, Mailman School of Public Health, Columbia University, New York, NY; dDepartment of Environmental Health Sciences, University of Michigan, Ann Arbor, MI; eInstitute for Social Research, University of Michigan, Ann Arbor, MI; fDepartment of Civil and Environmental Engineering, University of Michigan College of Engineering, Ann Arbor, MI; and gVista Analytical Laboratory, El Dorado Hills, CA.

Submitted 1 October 2008; accepted 15 October 2009; posted 8 April 2010.

Supported by The Dow Chemical Company through an unrestricted grant to the University of Michigan.

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Correspondence: Brenda W. Gillespie, Center for Statistical Consultation and Research, 3550 Rackham Building, University of Michigan, Ann Arbor, MI 48109–1070. E-mail:

© 2010 Lippincott Williams & Wilkins, Inc.