There is considerable interest in modeling gene × gene interaction for purposes of mapping and understanding complex traits (e.g., [1–3]). However, Vieland and Huang [4] showed that for two-locus (2L) models and data restricted to individuals who are affected (hereafter referred to as affecteds-only data) (e.g., ASPs), gene × gene interactions do not result in distinctive patterns in identity-by-descent (IBD) sharing, and therefore linkage analysis using ASPs cannot be used to distinguish interaction from independent gene effects (heterogeneity), with a few exceptions unlikely to be relevant to complex disorders. They suggested as a corollary that methods designed to exploit known interactions for purposes of mapping new genes would not be fruitful in ASPs. On the other hand, if the generating model involves more than two loci, then gene × gene interactions may predict specific structure in the (marginal) 2L IBD matrix even in affecteds-only data [2–5]. However, even in this case whether numerical effects would be substantial enough to provide meaningful benefits has not been systematically investigated.

In this paper, we ask whether it is possible to improve on our ability to map a new trait gene via linkage analysis by using the causal variants at a known risk locus, under conditions of gene × gene interaction, using a newly implemented extension of the posterior probability of linkage (PPL) to measure the strength of evidence for (or against) linkage.

In stark contrast with the dichotomous trait results, the quantitative trait linkage analyses at Locus E shows marked increases in the average evidence for linkage with the addition of DR information from 24% (SD ≈ 0.29) to 44% (SD ≈ 0.36) (see Table 1). The MODs are also larger when DR information is incorporated. In order to ensure the noticeably higher LC-PPL is not inflation simply due to the increased number of parameters in the model, we applied the same statistical analysis to the unlinked data obtained from the first markers on each chromosome 1–5 and 7; these markers were not annotated as being linked to any of the simulated phenotypes. The average QT-PPL was less than the 2% prior probability of linkage (data not shown), indicating evidence against linkage. This illustrates that the observed increase in the PPL at the original "linked" marker is not an artifact of including the additional penetrance parameters in the model. By contrast, the MODs increase when the additional parameters of the LCs are maximized over, at the unlinked markers as well as at the linked marker. This illustrates a key distinction in handling of nuisance parameters by integration (as with the PPL) versus maximization (as with the MOD).

Gene × gene interaction (or other covariate dependencies) can be directly represented in standard linkage likelihoods using LCs [22]. While theoretical considerations suggest that modeling gene × gene interactions in affecteds-only data is moot for dichotomous traits under two-locus models, in the context of quantitative trait analysis as well as under models with more than two loci, the situation could be fundamentally different. By extending the PPL to incorporate LCs for both dichotomous and quantitative trait analyses, we have shown that, under these generating conditions, utilizing genotypic information at the DR locus when evaluating the evidence for or against linkage to Locus A has no impact on our ability to detect Locus A using dichotomous trait analyses; whereas, incorporating information on DR is beneficial in detecting Locus E using quantitative trait analysis.

By extending the PPL to include a direct representation of gene × gene interaction (or other covariate dependencies), we have shown that under the generating conditions used in this simulation, incorporation of measured genotypes at a known locus does not improve our ability to detect linkage to another interacting locus for a dichotomous trait. These results are fully consistent with previous theoretical work on affecteds-only data under two-locus models, despite the fact that the generating model involves more than two loci and does include some unaffected individuals (among the parents). It appears that, nevertheless, there is virtually no information in nuclear families, that include only affected offspring, regarding the genetic architecture of the trait; and that as a result, utilizing information at one locus when evaluating a second is largely moot. This does not represent a specific limitation of the PPL, but rather, a limitation of the data structures provided in this simulation. See Kotti et al. [26] and Larkin et al. [27] for similar conclusions based on a variety of other statistical approaches.

We do have evidence that incorporation of gene × gene interactions in this manner may be more helpful in larger pedigrees, including even nuclear families with unaffected as well as affected offspring (data not shown). However, even in larger pedigrees, the utility of measured genotypes in forming LCs will be governed in part by the relative representation of different classes of individuals within the data set (the distribution of phenotypes, genotype at the marker being evaluated, and genotypes at the "risk" locus being conditioned on); and this in turn is a function of the underlying architecture of the trait together with the sampling frame and ascertainment criteria. Further research is needed to investigate other generating models that may potentially benefit from the use of LCs to model gene × gene interactions.

By contrast, the simulated data clearly support the use of LCs to model gene × gene interactions in quantitative trait linkage, even when the data are ascertained as ASPs for a related dichotomous trait. Apparently in this case sufficient variation in the phenotype still remains so that allowing genotypic means to depend on genotypes at a known risk locus can lead to stronger evidence for linkage at a second, interacting locus.