Application of Maximum Likelihood Methods to Population Genetic Data for the Estimation of Individual Fertilities

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Abstract
The fitness of plants or animals within a population is largely determined by the number of offspring they produce. In natural populations lacking familial structure either one or both parents are often unknown. To circumvent this problem, parents and offspring can be genotyped for a set of genetic markers. Likelihood models are proposed to estimate the fertilty of either male or male and female parents in a population where fertility is defined broadly as a fraction of progeny in the population fathered or mothered by some individual. Models are developed for three cases: the mother is known and the male fertilities differ depending on the maternal parent; the mother is known and the male fertilities are the same for all maternal parents; neither parent is known and fertilities are estimated for parent pairs. It is established that a unique maximum likelihood solution exists under conditions that are commonly net. For situations in which these conditions are not met, a method is presented to determine estimates of a set of independent linear combinations of the parameters that can be uniquely estimated. Two algorithms are examined which can be used to find the parameter estimates. The variance of the estimator, likelihood ratio tests for constraints on the fertility parameters, and a goodness-of-fit test are developed. Finally, a worked example is presented.