Nonparametric Estimation for Censored Mixture Data With Application to the Cooperative Huntington’s Observational Research Trial

Abstract

This work presents methods for estimating genotype-specific outcome distributions from genetic epidemiology studies where the event times are subject to right censoring, the genotypes are not directly observed, and the data arise from a mixture of scientifically meaningful subpopulations. Examples of such studies include kin-cohort studies and quantitative trait locus (QTL) studies. Current methods for analyzing censored mixture data include two types of nonparametric maximum likelihood estimators (NPMLEs; Type I and Type II) that do not make parametric assumptions on the genotype-specific density functions. Although both NPMLEs are commonly used, we show that one is inefficient and the other inconsistent. To overcome these deficiencies, we propose three classes of consistent nonparametric estimators that do not assume parametric density models and are easy to implement. They are based on inverse probability weighting (IPW), augmented IPW (AIPW), and nonparametric imputation (IMP). AIPW achieves the efficiency bound without additional modeling assumptions. Extensive simulation experiments demonstrate satisfactory performance of these estimators even when the data are heavily censored. We apply these estimators to the Cooperative Huntington’s Observational Research Trial (COHORT), and provide age-specific estimates of the effect of mutation in the Huntington gene on mortality using a sample of family members. The close approximation of the estimated noncarrier survival rates to that of the U.S. population indicates small ascertainment bias in the COHORT family sample. Our analyses underscore an elevated risk of death in Huntington gene mutation carriers compared with that in noncarriers for a wide age range, and suggest that the mutation equally affects survival rates in both genders. The estimated survival rates are useful in genetic counseling for providing guidelines on interpreting the risk of death associated with a positive genetic test, and in helping future subjects at risk to make informed decisions on whether to undergo genetic mutation testing. Technical details and additional numerical results are provided in the online supplementary materials.

Keywords:

• Censored data
• Finite mixture model
• Huntington’s disease
• Kin-cohort design
• Quantitative trait locus

Acknowledgments

This research is supported in part by the National GEM (Graduate Degrees for Minorities in Engineering and Science) Consortium, the Philanthropic Education Organization (PEO) Scholarship Award, and grants from the U.S. National Science Foundation (0906341 and 1206693), the National Institutes of Health (NS073671-01 and AG031113-01A2), and the National Cancer Institute (R25T-CA090301). Samples and data from the Cooperative Huntington’s Observational Research Trial (COHORT) study, which receives support from HP Therapeutics, Inc., were used in this study. The authors thank the Huntington Study Group COHORT investigators and coordinators who collected data and/or samples used in this study, as well as participants and their families who made this work possible.

Notes

NOTE: Sample size n=500, 20% and 50% censoring rate, and 1000 simulations.

NOTES: Upper half of the table: the true censoring distribution is independent of ; G(t) is estimated using a common Kaplan–Meier estimator of the censoring distribution. Lower half of the table: the true censoring distribution is subgroup-specific; G(t) is estimated using a common Kaplan–Meier estimator (denoted by †) or a subgroup-specific Kaplan–Meier estimator (denoted by *).

NOTES: The censoring distribution is subgroup-specific, and G(t) is estimated using a common Kaplan–Meier estimator (denoted by †) or a subgroup-specific Kaplan–Meier estimator (denoted by *). Sample size n = 500, 20% and 50% censoring rate, and 1000 simulations.

NOTE: Survival rates are compared with Kaplan–Meier estimated survival rates for the general male and female U.S. populations (USpop) in 2003.

^aComputed under a subsample by removing subjects in groups with small sample sizes.

*Integrated Absolute Bias.