New Algorithms for Evaluating the Log-Likelihood Function Derivatives in the AI-REML Method

Abstract

In this study, we propose several improvements of the Average Information Restricted Maximum Likelihood algorithms for estimating the variance components for genetic mapping of quantitative traits. The improved methods are applicable when two variance components are to be estimated. The improvements are related to the algebraic part of the methods and utilize the properties of the underlying matrix structures.

In contrast to previously developed algorithms, the explicit computation of a matrix inverse is replaced by the solution of a linear system of equations with multiple right-hand sides, based on a particular matrix decomposition. The computational costs of the proposed algorithms are analyzed and compared.

Keywords:

• Identity-by-descent (IBD) matrix
• Reduced computational complexity
• Restricted maximum likelihood algorithm (REML)
• Spectral decomposition
• Variance component estimation
• Woodbury's formula

Mathematics Subject Classification:

• 62P10
• 92D10
• 65Y20

Acknowledgments

K. Mishchenko expresses her gratitude to Aleksandr Zheltukhin and Sverker Holmgren for the valuable discussions and support. The authors are indebted to Lars Rönnegård for the helpful discussions to better understand the characteristics of the underlying real application and the behavior of the obtained solution. The critical remarks and advice given by Per Lödstedt regarding improvements of the algorithm developed in Mishchenko et al. (2007) were important and stimulated the presented work.

The authors also thank the anonymous referee for his/her careful reading and numerous suggestions which led to improvements both in the structure and style of the article.