Maximum Likelihood Estimation of Parameters in Linkage. Numerical Iterative Methods

Abstract

Taking into account the advantages of Maximum Likelihood Method (most precise estimation), the statistical properties of MLEs (unbiasedness, consistency, efficiency, invariance, asymptotic normality) this paper aim is to present MLE in the context of estimate the recombination fraction r in linkage analysis. Maximum Likelihood Method follows some steps: specifies the likelihood function; takes derivatives of likelihood with respect to the parameters; sets the derivatives equal to zero and finally generates a likelihood equation , that maximized provides the most precise estimation of the recombination fraction. Generally, it is solved by iterative procedures, if no, closed form solution exists for likelihood equation . In this work we discuss comparatively two iterative optimization methods useful in computing MLE of the recombination fraction: Newton-Raphson method and Fisher’s Method of Scoring. We implemented these two methods in Maple application and we illustrated them by an example: the estimation of the recombination fraction in the case of the Morgan (1909) experiment on fruit flies. The Maple code for these two methods connected with the Morgan example is given in the appendix. We can not guarantee which of the two presented methods give us an optimal maximum. Â