#### Simulation

As a first set-up, we simulated 100 data sets each consisting of genotypes for 100 unlinked SNPs, typed in 1000 case-parent trios. In each of these data sets, the SNPs *S*_{3} and *S*_{7} were simulated such that the chance of being a case was 3 times larger for subjects exhibiting at least one copy of the variant allele at both *S*_{3} and *S*_{7}. Thus, a subject showing the interaction *S*_{3D}∧*S*_{7D} had a 3-fold increase in the chance of being a case, where the symbol ∧ denotes the logic AND-operator. The genotypes of the other 98 SNPs were drawn under the assumption of no association with the outcome.

We applied trioFS with *B*= 20 iterations to these 100 data sets and computed , and was identified as the (usually by far) most important interaction in all applications of trioFS when considering , and in all but one application when using to rank the interactions. In fact, *S*_{3D}∧*S*_{7D} is the only interaction that was detected in all applications. In general, *S*_{3D}∧*S*_{7D} was considered very important by all three metrics , and . However, with the exception of one application in which , the value of was typically substantially larger than the value of (Fig. 1). The reason for this is that although *S*_{3D}∧*S*_{7D} exhibits the largest importance, interactions composed of *S*_{3D}∧*S*_{7D} and one other SNP were also frequently identified in some of the iterations of trioFS, reducing the improvement due to *S*_{3D}∧*S*_{7D} as quantified in equation (2) in such an iteration to zero, and thus, decreasing the overall importance of *S*_{3D}∧*S*_{7D} (see Table 1 for an example output of trioFS). Adjusting for LD has no effect on in this application, since all SNPs were simulated independently from each other. Thus, no pair of SNPs found in interaction with *S*_{3D}∧*S*_{7D} exhibited an *r*^{2}-value larger than 0.7, which was used as the defining threshold for LD in equation (3) to quantify (see Table 1).

To investigate whether trioFS is also able to detect *S*_{3D}∧*S*_{7D} when this interaction has a smaller effect size, we simulated *S*_{3} and *S*_{7} such that the odds of being a case were 2.5, 2, or 1.5 times larger for subjects showing *S*_{3D}∧*S*_{7D}. For each of these three odds ratio, 100 data sets were generated, each consisting of 1000 case-parent trios typed at *S*_{3}, *S*_{7} and 98 additional independent SNPs intended to have no effect on the disease risk. Additionally, we simulated 100 data sets consisting of 500 trios for each of the four odds ratios 3, 2.5, 2, and 1.5. We then applied trioFS with *B*= 20 iterations to each of these data sets.

The simulation reveals that sample sizes of 1000 trios or fewer had to be considered insufficient for a study to detect interactions with odds ratios of 1.5 or smaller (Table 2).

TrioFS detected *S*_{3D}∧*S*_{7D} only in 8 of the 100 applications to the data sets with 500 trios and 33 times in the data sets consisting of 1000 trios, where in only the former applications *S*_{3D}∧*S*_{7D} ranked once under the five interactions with the largest values of either or . Interactions composed of *S*_{3D}∧*S*_{7D} and another SNP were detected in 40 or 83 of the applications, respectively, but they were detected just once amongst the five top-ranking interactions in both simulation scenarios.

In all but one application to the data sets from the simulation scenarios with odds ratios of 2.5 and 3, *S*_{3D}∧*S*_{7D} was detected and ranked first when considering . In a few of the analyses, *S*_{3D}∧*S*_{7D} ranked not first, but typically second or third when basing this ranking on , where the scenarios with the 500 trios performed worse than the scenarios with the 1000 trios. Usually, all top five interactions contained *S*_{3D}∧*S*_{7D} (Table 2).

Employing in the applications to the data sets from the simulation scenario with 1000 case-parent trios and an odds ratio of 2 led to the detection of *S*_{3D}∧*S*_{7D} as the top-ranking interaction in 97% of the cases, whereas *S*_{3D}∧*S*_{7D} itself or an extension of it was found as the most important interaction in 72 or 22 of the applications, respectively, if was used. In the six remaining applications, three-way interactions of the SNPs with no main effect showed up as most important, but in these cases at least three of the other four top ranking SNPs were either *S*_{3D}∧*S*_{7D} or an extension of it, that is, a three-way interaction containing *S*_{3D}∧*S*_{7D}. When considering the data sets consisting of 500 case-parent trios, *S*_{3D}∧*S*_{7D} itself was found in 87 of the applications, and in another 12 analyses it was identified in interaction with another SNP. When the ranking is based on , both *S*_{3D}∧*S*_{7D} itself and extensions of it ranked first in 20 of the applications, and represented the most important interaction in 44 and 13 of the analyses, when considering , respectively.

We also computed p-values based on 10,000 permutations of the case-pseudo-control status for all identified interactions and adjusted for multiple comparisons using the Bonferroni correction. The term *S*_{3D}∧*S*_{7D} was identified as significant in virtually all analyses based on 1000 trios, when the effect size was assumed to be 2.5 or larger (Table 2). Frequently, none of the 10,000 permuted importances were actually larger than the observed (un-permuted) importance of *S*_{3D}∧*S*_{7D}. The p-value was smaller than 0.05 in about 60% () or 67% () of the applications, when an odds ratio of 2 was assumed. When analyzing 500 trios, an odds ratio of 3 was necessary to systematically achieve significance. Virtually all interactions with a p-value smaller than 0.05 were either *S*_{3D}∧*S*_{7D} or an extension of it. An exception is *S*_{7D}, which showed up significant in some of the applications.

To evaluate whether trioFS is also able to detect three-way interactions, *S*_{3D}∧*S*_{5D}∧*S*_{7D} was simulated such that it exhibits an odds ratio of 3, 2.5, or 2, and 97 SNPs were randomly drawn under the assumption of no association with the outcome. In this way, six sets consisting of 100 data sets were generated, where the data sets in three of these sets contained 500 case-parent trios, and in the other sets 1000 trios. We then applied trioFS with *B*= 20 iterations to all of these data sets and computed and .

Not surprisingly, even larger effect sizes are required to detect the higher order interaction (Table 3). Even for odds ratios of 2, neither *S*_{3D}∧*S*_{5D}∧*S*_{7D} nor extensions of it were identified. However, in all but one simulation scenario with odds ratio of 2.5 and 3, *S*_{3D}∧*S*_{5D}∧*S*_{7D} was detected by trioFS. In almost any application to the data sets with 1000 trios, this interaction ranked first when employing , and it ranked first in most analyses when considering . Only in a few of the applications, the value of for *S*_{3D}∧*S*_{5D}∧*S*_{7D} was substantially larger than the value of , as just a few extensions of *S*_{3D}∧*S*_{5D}∧*S*_{7D} appeared in the applications of trioFS. Instead the two-way interactions contained in *S*_{3D}∧*S*_{5D}∧*S*_{7D}, that is, *S*_{3D}∧*S*_{5D}, *S*_{3D}∧*S*_{7D}, and *S*_{5D}∧*S*_{7D}, showed up in almost any application of trioFS. In the settings with 500 trios, frequently at least one of these two-way interactions had a higher importance than *S*_{3D}∧*S*_{5D}∧*S*_{7D}, whereas the studies with 1000 trios reliably identified the three-way interaction as the most important one. This can also be summarized using the permutation-based p-values for *S*_{3D}∧*S*_{5D}∧*S*_{7D}, which were smaller than 0.05 in virtually any application to the 1000 trios, and zero in most of the applications. On a positive note for the smaller studies, in many instances more than just one of the two-way interactions and *S*_{3D}∧*S*_{5D}∧*S*_{7D} appeared among the top five SNPs and with a p-value smaller than 0.05, suggesting that this three-way interaction might be important for the disease risk prediction.

In the final simulation set-up, we investigated the performance of trioFS and the differences between the importance measures when SNPs are in strong LD. We examined two specific settings for this simulation study, but also refer the reader to the autism case study discussed in the following section, which we believe is a particularly nice illustration of the differences between and when SNPs are in strong LD. For each of these settings, one considering an odds ratio of 2.5 for *S*_{3D}∧*S*_{7D}, the other an odds ratio of 3, we simulated 100 data sets consisting of 100 SNPs typed at 1000 trios. This time, we generated two LD-blocks of SNPs, one consisting of *S*_{2}, *S*_{3}, and *S*_{4}, and the other of *S*_{6}, *S*_{7} and *S*_{8}. The pairwise *r*^{2}-values within these blocks were larger than 0.99. The remaining 94 SNPs were randomly drawn.

Usually a minimum of three, and in most applications four of the top five interactions were composed of two SNPs, one from each of the two LD-blocks containing *S*_{3} and *S*_{7}, with permutation-based p-values typically equal to zero, but always less than 0.05 (see Table 4 for an example). The other top five interactions were always three-way interactions consisting of two SNPs from these LD-blocks and another SNP which only slightly contributed to the effect of the interaction. The term *S*_{3D}∧*S*_{7D} is sometimes found as the most important interaction, however, frequently another interaction consisting of either *S*_{2D}, *S*_{3D} or *S*_{4D}, and *S*_{6D}, *S*_{7D} or *S*_{8D} ranks first.

#### Case Study

In this section, we consider 461 autistic children and their parents from 289 families recruited by the Autism Genetic Resource Exchange (AGRE; http://www.agre.org), a collaborative gene bank created by Cure Autism Now (CAN) and the Human Biological Data Exchange (HBD) to advance genetic research in autism spectrum disorders by consolidating large numbers of families into one collection. Genetic biomaterials and clinical data were obtained for families with at least one offspring diagnosed with an Autism Spectrum Disorder based on evaluation by the Autism Diagnostic Interview-Revised (ADI-R) and the Autism Diagnostic Observational Schedule (Geschwind et al., 2001). Cases were included if they had an ADI-R diagnosis of Autism, and data for both parents were available.

Two of the available 331 SNPs were excluded from the analysis since they were almost monomorphic. Further, ten of the 461 trios were removed as more than 2% of the SNPs in each of these trios exhibited Mendelian errors. The haplotype-based procedure proposed by Li et al. (2010a) was used to impute the missing genotypes, and to transform the case-pseudo-control data into a format suitable for trio logic regression. We then applied trioFS with *B*= 20 iterations to these data.

The most important interaction detected by trioFS was a three-way interaction of rs11017112, rs7082126, and rs11017128, all showing the homozygous reference genotypes (Table 5). When considering the adjusted importance measure, a three-way interaction of the latter two SNPs and rs11017114 (also represented by the binary variable using dominant coding, that is, the variable indicating at least one variant allele) exhibits the second largest importance. All these SNPs are from the gene Glutaredoxin 3 (GLRX3) on chromosome 10.

Since epistasis is usually defined as interactions between SNPs in different genes and/or genomic regions, we also analyzed a subset of the 329 SNPs that was previously considered elsewhere (Bowers et al., in preparation). Briefly, this subset consists of 138 independent SNPs showing pair-wise *r*^{2}-values smaller than or equal to 0.2, and each glutathione-related gene is represented by the marker that has the largest estimated marginal effect size. In addition, all SNPs with a marginal p-value less than 0.1 were also included in the analysis. As before, we generated 10 case-pseudo-control data sets by applying the procedure of Li et al. (2010a) to the subset of 138 SNPs (the procedure is also applicable for “degenerate” haplotypes of size 1, that is, individual SNPs), and analyzed these data sets with trioFS. Even though we strongly biased the selection of SNPs, the application of trioFS did not reveal interesting interactions.