Polygene models become progressively more complicated, particularly if one employs a profile likelihood interval mapping approach. Various authors have come up with approaches to find multiple QTL using variations on interval mapping (cf. jans:stam:1994, zeng:1994, chur:doer:1994 hale:knot:else:1994 and knap:1994 as well as references therein). Polygene models become progressively more complicated, particularly if one employs a profile likelihood interval mapping approach. land:bots:1989 suggested a forward selection approach, in which the locus with the highest LOD is fixed, the genome is scanned for a second locus, and so on. While appealing, this model selection method has the previously mentioned problem that it can sometimes mistakenly select a single ``ghost'' QTL rather than two linked QTLs (see Section 1.3 above).
The concept of simultaneously searching for more than one gene using interval mapping is fairly straight forward in a sense. The probability for a trait given the genotypes at markers 1 and 2, say,
prob( trait | geno1, geno2 ) ,
might take the form of the linear model for multiple genes at known loci as presented in Section 1.2. However, the probability when the genotypes are unknown, given the loci and map,
prob( trait | loci, map ) ,
involves the sum over all possible pairs of genotypes. Therefore, the full likelihood for a given set of loci,
like( loci ) = sum of log[ prob( trait | loci, map ) ] ,
is very complicated (see zeng:1994).
Some authors [\protect\citeauthoryearZengZeng1994][\protect\citeauthoryearJansen and StamJansen and Stam1994] have suggested instead to search for only one locus at a time across the genome while using markers in regions at some distance from the locus being considered as cofactors. That is, imagine that markers on other linkage groups, and those on the linkage group being examined which are at least 10-20cM away from the interval being scanned, are included in the model as if they represented genes. Simulations by these authors suggest that this approach can be very effective to uncover likely QTL while keeping the computation effort low. Both authors have computer packages - JoinMap and MapQTL (Stam) and QTLCart (Zeng) - which are available.