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.

Type‐I‐error rate inflation in mixed models for repeated measures caused by ambiguous or incomplete model specifications

“…Instead Fisher scoring can be used alternatively which utilizes the expected information rather than the observed information (Newton–Raphson) for the Hessian matrix. This algorithm takes more computational time to converge, but the iteration process is more stable in case of a missing data problem 30,31 . In this study, no missing values are simulated and, thus, minimal discrepancies in the results between these two algorithms are expected.…”

“…This algorithm takes more computational time to converge, but the iteration process is more stable in case of a missing data problem. 30,31 In this study, no missing values are simulated and, thus, minimal discrepancies in the results between these two algorithms are expected.…”