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Notes on the maximum likelihood estimation of haplotype frequencies
1Biological Information Research Center, National Institute of Advanced Industrial Science and Technology, Tokyo, Japan. smano@is.icc.u-tokai.ac.jp
Annals of Human Genetics
|June 8, 2004
Summary
Maximum likelihood estimation (MLE) for haplotype frequencies can err due to local maxima. This study identifies conditions for unimodal likelihood surfaces and provides a general algorithm for accurate haplotype frequency estimation.
Area of Science:
- Population Genetics
- Statistical Genetics
- Bioinformatics
Background:
- Maximum Likelihood Estimation (MLE) is a standard method for estimating population haplotype frequencies from genotype data with unknown linkage phases.
- The Expectation-Maximization (EM) algorithm is commonly used for MLE but risks converging to local maxima or saddle points, leading to inaccurate haplotype frequency estimates.
Purpose of the Study:
- To theoretically identify the necessary and sufficient conditions for the appearance of local maxima or saddle points on the likelihood surface.
- To establish a sufficient condition for a unimodal likelihood surface, ensuring accurate MLE of haplotype frequencies.
- To develop a general algorithm for obtaining the global maximum of the likelihood surface.
Main Methods:
- Theoretical analysis of the likelihood surface to derive conditions for local extrema.
- Development of an analytic solution for the biallelic two-locus haplotype frequency estimation problem.
- Construction of a general algorithm to find the global maximum of the likelihood function.
Main Results:
- Necessary and sufficient conditions for the occurrence of local maxima and saddle points on the likelihood surface were determined.
- A rule of thumb is presented: if the difference between coupling and repulsive haplotype frequencies in phase-known individuals is 3/2 times larger than the frequency of phase-ambiguous individuals, the likelihood surface is unimodal.
- An analytic solution for the biallelic two-locus problem was derived, and a general algorithm for global maximum estimation was constructed.
Conclusions:
- Understanding the conditions leading to non-unimodal likelihood surfaces is crucial for reliable haplotype frequency estimation using MLE.
- The derived sufficient condition provides a practical guideline to assess the unimodality of the likelihood surface.
- The developed algorithm offers a robust method for obtaining accurate global maximum likelihood estimates of haplotype frequencies.
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