Linkage analysis methods to identify gene × gene interactions in complex diseases have been developed [1–4], however, in the absence of already characterized candidate genes, their ability to identify epistasis is unknown. Moreover, analyses from the Genetic Analysis Workshop (GAW) 14 simulated data further support the difficulty in locating gene × gene interaction [5]. The GAW15 simulated rheumatoid arthritis (RA) data set affords another opportunity to compare the statistical power of three non-parametric linkage approaches using affected sibling pairs (ASPs) to identify gene × gene interactions between two unlinked loci: 1) locus DR, which was simulated to represent the risk of DRB1 locus of HLA on RA, and 2) locus A, which was simulated as an effect modifier on DR.

First, we examined a mean test variant of the conditional methodology presented by Cox et al. [1]. The motivation behind this methodology is that correlations between identity-by-descent (IBD) allele sharing at unlinked loci can be used to identify the relationship between loci. We adapted this methodology to the mean test for linkage at another locus by excluding ASPs with no evidence for linkage to the first locus.

We also examined the power and type I error of two other covariate based approaches to detect epistasis with varying covariate coding schemes of the genotyped locus (DR). The first is a regression-based mean test that can be used to test for the presence of gene × environment interactions in ASPs [2]. By treating the DR locus as an "environmental variable," it is possible to use this method to identify gene × gene interaction. Second, we explored the conditional logistic model developed by Olson and colleagues [4, 6], which is an alternative parameterization of the LOD score model presented by Risch [7]. Significant increases in linkage between a baseline model without covariates and a model with the DR locus as a covariate suggest epistasis between the DR locus and the locus where linkage is assessed.

The conditional method using the subsetting approach did not provide additional power to detect linkage above and beyond using the entire data set to detect a main linkage effect of Locus A. Figure 1 shows that selecting a subset of ASPs with evidence for linkage to the DR locus did not vary by the arbitrary cut-points chosen. At 26 cM (the approximate location of the A locus) only 27 of the replicates detected linkage at an α = 0.05 level using all ASPs in each replicate. Restricting the sample sizes in each replicate by selecting ASPs with the proportion of alleles shared IBD at the DR locus greater than or equal to 0.5 resulted in slightly decreased power to detect linkage at the A locus than the complete data set (24%). DR allele sharing cutoffs of 0.7 or 0.9 did not increase power to detect linkage (24% and 20%, respectively). The type I error of these tests averaged, 10%, 9%, 9%, and 13% for cutpoints of 0.5, 0.7, 0.9, and the complete data set, respectively. To examine heterogeneity, we restricted the sample to those sib pairs with allele sharing less than 0.3 and 0.1, and the power again barely exceeded the type I error rate (data not shown).

Similarly, the mean interaction test had limited power to detect evidence for linkage to the A locus or evidence for a significant interaction using a test of the β coefficient. Figure 2 shows the percent of replicates that were detected across chromosome 16 for the ASP covariate sum that models the DR4 allele additively. The power of the test for linkage was 18% at the A locus and the power of the interaction term by itself was low (4%). Type I error was 5% and 9% for the interaction term alone and the joint test of linkage, respectively. The other models which used an alternative coding scheme for the covariate coding the DR locus (as described in the Methods section) produced similar results.

The conditional logistic methodology did not detect the interaction at the A locus, with the power ranging from 1 to 9% in the 40-cM region around the A locus. This power never exceeded the type I error. Even coding the covariates in a manner that mimicked the actual simulated risk parameters through a linear coding scheme did not improve the power to detect linkage.

None of the methods we examined had enough power at a type I error rate of α = 0.05 to detect linkage to the A locus in the presence of linkage to DR. There was no difference between the conditional method, which used a restricted sample, and the mean test and conditional logistic models, demonstrating that this lack of power was not due to insufficient sample size. Our results were quite similar to those of Brock et al. [9], who analyzed the GAW14 data. Of the 36 multipoint models that used the conditional logistic model, the highest power achieved was 34%. These results support the theoretical work by Vieland and Huang, which suggests that the establishment of epistasis in ASP data is impossible due to insufficient penetrance structure [10]. Using Gauderman and Siegmund's mean test [2], we found that the power of the test of the linkage-only model (π > 0.5) was greatest, followed by the joint test of π and β, and that the interaction only model had no less than 10% power. The power to detect interaction only is generally much lower than the power to detect linkage when allowing for interaction [3], so our results are not surprising. Furthermore, the work of Elston et al. [11] suggests that discordant pairs are necessary to detect gene × gene interaction, which was further demonstrated using simulated data [12]. Ultimately, our analyses are limited by the fact that these statistical definitions may not reflect biological reality in real data.

Our analyses of the GAW15 data also illustrate the challenges of testing for gene × gene interaction for complex diseases. First, we found that our analysis was hampered by the complexity of the simulated model. While the interaction was simulated to be large for certain genotypes, the prevalence of those genotypes was low. For example, the largest simulated multiplicative interaction between DR and the "A" allele of locus A was the DR4/DR4 genotype; however, the prevalence of DR4/DR4 was approximately 0.1. Gauderman and Siegmund demonstrate that, assuming a prevalence of exposure is at least 0.5, the increased risk must be greater than 3 to have sufficient power to detect gene × "environment" interaction [2]. In our analyses, we used the DR locus as the "environmental exposure"; thus, the low prevalence of the high risk genotype combinations likely affected our inability to detect the gene × gene interaction. The power of the conditional logistic model (in LODPAL) is greatest when using a dichotomous risk categorization as the covariate [3]; thus, we suspect that the reason we did not observe an increase in power when we used the DR locus as the covariate was because of the low frequency of the high risk DR genotypes. Second, a challenge in this data was that the DR locus was tri-allelic. Holmans [3] also discusses situations in which there are varying levels of risk associated with the candidate locus, or when certain genotypes may confer their increased risk in conjunction with different genotypes at the test locus. In these cases, the coding of the covariate is not trivial, and incorrect recoding may greatly reduce power. These complexities are certainly true of the simulated relationship between DR and A, and likely explain our loss of power. Finally, these results also raise another issue regarding the importance of the definition of the main effect and how main effects should be incorporated into such an analysis [13]. The first methodology that conditions on allele sharing at the first locus detects interaction that is a departure from the multiplicative penetrance model. An implicit assumption in detecting this kind of interaction is the presence of joint Hardy-Weinberg proportions or gametic phase equilibrium. While the A and the DR loci are on different chromosomes, it is possible that additional allelic association not due to linkage disequilibrium exists. Schaid et al. also indicate that for the mean interaction test to be valid the two loci must be uncorrelated in the general population [14], which would be true if joint Hardy-Weinberg equilibrium holds. This limitation holds for the conditional method, as well.

In summary, we observed that the most commonly used current methods for detecting gene × gene interaction in ASP data had low power to detect the interaction between DR and A in the simulated data set. This lack of power is likely due to the lack of information in ASP data compared to having discordant pair data and/or the low prevalence of the high-risk DR genotypes and the complex nature of the simulated risk multipliers. Although complicated, this simulated data probably more accurately depicts real complex disease data; thus, we believe further research on linkage methods that can more powerfully detect epistasis while minimizing type I error is warranted.