Multiple QTL at Known Positions

Some traits may be attributed to several major gene effects. The simplest case would be a trait influenced by two genes,

   trait = mean + geno1 + geno2 + error .

If there are two genes affecting the expression of a trait, it is not sufficient to examine each gene effect on its own. That is, one cannot simply fit the models

   trait = mean + geno1 + error

   trait = mean + geno2 + error ,

concluding separately whether there is a gene at each of these loci. This is not appropriate since the effect of the ignored gene gets mixed into the error and the estimate of the effect of the included gene. If the genes are linked, then the gene left out can bias the estimated effects of the remaining one. Tests based on one gene when two are believed to be important are at best unreliable. At worst, they can be very misleading.

Instead, it is more proper to fit the model with both genes and ascertain whether their effects are significant in the presence of the other. That is, examine tests based on Type III sums of squares (litt:freu:spec:1991, sec 4.3) which adjust for the presence of the other gene, as outlined in the following SAS commands,

data genes;
   infile 'trait.dat';
   input geno1 $ geno2 $ days;
   trait = log10(days);
proc glm data=genes;  /* two genes in CRD */
   class geno1 geno2;
   model trait = geno1 geno2 / ss3;
   lsmeans geno1 geno2 / stderr pdiff;

Each genotype effect should be examined using by first adjusting for all other genes. This is known as Henderson's Type III approach, and is illustrated in the next section on point analysis.

Epistasis among genes can be examined by inserting an ``interaction'' term in the linear model, as follows,

   trait = mean + geno1 + geno2 + geno1*geno2 + error .

This is done in SAS commands as

proc glm;
   class geno1 geno2;
   model trait = geno1 geno2 geno1*geno2 / ss3;
   lsmeans geno1 geno2 geno1*geno2 / stderr pdiff;

Interaction may be difficult to uncover, especially with small sample sizes. Further, if there are several genes being considered, there are many more interactions to examine. Typically, epistasis is assumed to be negligible in the early stages of investigation. However, it can be important to examine evidence of epistasis in the later stages of analysis.

Two genes may not be enough. Sometimes it is useful to combine the effects of any other ``minor'' genes. Consider the following model, in which minor gene effectsare lumped together as the rest,

   trait = mean + geno1 + geno2 + rest + error .

The rest of the genes can affect results in profound ways if they are ignored [\protect\citeauthoryearKnapp, Bridges, and LiuKnapp et al.1992][\protect\citeauthoryearZengZeng1993][\protect\citeauthoryearJansenJansen1993][\protect\citeauthoryearHaley and KnottHaley and Knott1992]. Omitting unlinked genes which influence the trait from the linear model inflate the error variance, even though they do not bias the estimates of the remaining gene effects. Linked genes which affect a trait but are left out of the linear model can bias the effects of the remaining genes and can inflate the error estimate. In fact, this situation can lead to artificially large F values, giving one a false impression about the significance of putative genes. Thus it is very important to consider all candidate genes simultaneously to assess gene effects of each.

Sometimes it is desirable to combine material from several similar crosses in order to determine effects of genes at QTLs. One way to do this is to include a factor for cross,

   trait = mean + geno1 + geno2 + cross + error .

The idea is that each cross may differ at many genes other than the genes under immediate study. [However, the cross factor may be ignored in some studies if the parental lines are truly inbred, in which case the F1s are all genetically identical.] There are certain assumptions made even here. For instance, are the alleles at the gene loci the same for the different crosses? If not, or if this is unknown, it may be appropriate to examine interaction terms, such as geno1*cross.