A Monte Carlo Newton-Raphson procedure for maximizing complex likelihoods on pedigree data

摘要

The Monte Carlo Newton-Raphson (MCNR) method is an iterative procedure that can be used to approximate the maximum of a likelihood function in situations where direct likelihood computation is infeasible because of the existence of unmeasured variables, missing data, or measurement error. We describe the method in the context of pedigree analysis, where genotype are unmeasured. Parameter values are set to an initial estimate of the MLE and are repeatedly updated using the following three steps: (1) A Monte Carlo Markov chain procedure such as the Gibbs sampler is used to generate multiple samples of genotypes of the current parameter values, (2) the genotype samples are used to estimate the score (gradient) and observed information matrix (Hessian), and (3) the estimated gradient and Hessian are used in a standard Newton-Raphson step to obtain updated parameter values. The resulting sequence of parameter values converges to a neighborhood of the MLE, then varies randomly around it. The parameter values that are generated after convergence are averaged to provide an approximation to the MLE, and an additional set of genotype samples can then be obtained and used to approximate standard errors of the estimates. We present simulation studies indicating that MCNR provides close approximations to the MLE in several standard genetic models. We also apply the method to an analysis of high-density lipids (HDL) in the Berkeley data set (Austin et al., 1988. Amer. J. Hum. Genet. 43, 838-846), showing evidence that the expression of HDL depends on a dominant major gene and an interaction between this gene and body mass index.