In practice, we know the phenotypic trait but not the genotype at the QTLs. For convenience, the ideas are developed for one gene on one linkage group. Our aim is to determine the conditional probability distribution of the trait given a possible locus and the linkage map. The locus with the highest probability is the most likely estimate given the data, or the maximum likelihood estimate of the QTL.
The first section developed a linear model,
trait = mean + geno + error ,
which can be interpreted as describing the probability distribution of the trait given the genotype,
prob( trait | geno ) .
Here, the genotype may be unknown. However, suppose a locus is suggested for the gene, as some position on a linkage map. It is possible to assign probabilities to the possible genotypes at the locus based on the genotypes of the markers which comprise the map. These mixture probabilities,
prob( geno | locus, map ) ,
have a very simple form if recombination events are assumed independent, as in the Haldane map function. In that case, only the flanking markers, on either side of the putative locus, come in to play. The calculations are just those for 3-point linkage.
For each experimental unit, one can determine the probability for the phenotypic trait given the putatitive locus and the linkage map as the sum over all possible genotypes,
prob( trait | locus, map ) = sum of
prob( trait | geno ) * prob( geno | locus, map ) .
The joint probability for a sample of independent offspring is the product of individual probabilities. Usually one takes logarithms to turn this into a sum, which is called the likelihood (or sometimes the log-likelihood),
like( locus ) = sum of log[ prob( trait | locus, map ) ] .
That is, the likelihood is a sum of logs of products of terms. Thus, the maximum likelihood estimate of locus is difficult to evaluate in practice.