Introduction

With continued decreasing cost of high throughput genotyping technology, genome-wide association studies (GWAS) are becoming increasingly popular for gene mapping of complex human diseases. Most of the published GWAS papers report results from single-marker-based analysis in which each SNP is analyzed individually. Although this simple approach has led to the discovery of disease susceptibility genes for many diseases, the identified SNPs often only explain a small fraction of the phenotypic variation, suggesting a large number of disease variants are yet to be discovered. There is growing evidence that gene–gene interactions are important contributors to genetic variation in complex human diseases.1, 2, 3, 4, 5, 6 However, detecting gene–gene interactions remains a challenge due to the lack of powerful statistical methods. The most commonly used statistical approach for studying gene–gene interactions is to use a regression framework in which a pair of markers and their interaction terms are included as predictors. When a large number of markers are available, one might consider doing a stepwise regression7 or a two-stage analysis.8 Although such methods have been proven useful in simulation studies, they may lose power when multiple interacting variants exist in each gene.

One potential solution to the aforementioned problem is to perform interaction analysis at the gene level. There is increasing recognition for the importance of gene-based analysis.9 Several methods have been developed to test whether a gene is associated with the trait of interest.10, 11, 12, 13 The central idea of these methods is to summarize marker genotypes into a few components so that the overall degrees of freedom are reduced while most information in the data is retained. Extensive simulations demonstrate that gene-based association analysis can increase the power of detecting genetic association compared with single-marker-based analysis. It is therefore reasonable to expect that gene-based interaction analysis may outperform SNP-based interaction analysis and lead to identification of novel disease susceptibility genes.

Recently, Li et al13 proposed a novel gene-based association test – ATOM, by combining optimally weighted markers within a gene. For each marker in the gene, either genotyped or untyped, an optimally weighted score is derived based on observed genotypes and linkage disequilibrium (LD) information in a reference data set such as the HapMap.14, 15 To reduce the dimensionality of the data, ATOM tests for association using selected principal components (PCs) of these derived scores. Simulations and analysis of real data showed improved power of ATOM over methods that do not incorporate external LD information, especially when the disease loci are not directly genotyped.

The success of ATOM motivated us to extend it to the analysis of gene–gene interactions. Here we describe a PC framework for gene-based interaction analysis. We analytically derive the optimal weight for both quantitative and binary traits based on pairwise LD information. We then use PCs to summarize the information in each gene, and test for interactions between the PCs. We further extend this gene-based interaction analysis procedure to allow the use of imputation dosage scores obtained from popular imputation software packages MACH16 or IMPUTE,17 which incorporates multilocus LD information. We evaluate the performance of the proposed tests by extensive simulations and the analysis of a candidate gene study on high-density lipoprotein cholesterol (HDL-C).

Methods

We consider the problem of gene-based interaction analysis between two genes with multiple markers in each gene. We first present the analytical solutions for quantitative and binary traits assuming the interacting trait loci are known. We then extend the method to the more realistic situation in which the interacting trait loci are unknown.

Quantitative trait

Suppose the quantitative trait of interest, Y, is influenced by the interaction between two diallelic quantitative trait loci (QTLs) located in two different genes. Let Tj and tj (with frequencies and , respectively) denote the two alleles at QTL j (=1, 2). Assume the mean of the trait value Y given genotypes and can be written as

where is the number of allele Tj at QTL j. To detect interaction between the two QTLs, we test . However, as and may not be directly observed, the test of interaction is often accomplished through examination of interactions between genotyped markers. Assume a diallelic marker j in gene j (with alleles Aj and aj and allele frequencies and , respectively) is in LD with QTL j. We will show that the mean of Y given genotypes g1 and g2 at markers 1 and 2 can be written as

Equation (2) allows indirect assessment of interaction between the QTLs by testing H0: β1,2=0.

The regression coefficients and reflect the magnitude of interaction effects between the QTLs and between the markers, respectively, and their relationship depends on the degree of LD between the QTLs and the markers. To explicitly derive their relationship, we note that can be written as

where , j = 1, 2. Li et al13 have shown that when the QTLs and the markers are in Hardy–Weinberg equilibrium in the population,

where is the LD coefficient between QTL j and marker j. Therefore,

If we replace the items in equation (3) accordingly by those in equations (4) and (5), it becomes apparent that equation (2) holds, with

Equation (6) indicates that the two interaction coefficients, and β1,2, differ only by a factor , which is a function of the marker allele frequencies and the LD coefficients between the QTLs and the markers. The above derivation can be readily extended to binary traits such as disease affection status (see Supplementary Materials).

From the above derivation, we can see that if we define a weighted genotype score for marker j, , then the corresponding mean model for Y given the weighted genotype scores becomes

where the interaction coefficient becomes the same as that in equation (1). This indicates that for any pair of markers with one from each of the two genes, using weighted genotype scores will result in models that share the same interaction coefficient , a fact that we will use below to combine multiple markers within a gene.

Suppose diallelic markers in gene j are genotyped, with alleles and for marker lj (1≤lj≤mj) and allele frequencies and , respectively. The above derivations suggest that for individual i (1≤i≤n) and marker lj (1≤lj≤mj) in gene j, we may consider the weighted genotype score , where is the LD coefficient between QTL j and marker lj in gene j, and denotes the number of allele carried by individual i. Then the mean of the trait value Yi given weighted genotype scores at marker l1 in gene 1 and marker l2 in gene 2 will be . When all possible marker combinations in the two genes are considered, then all m1 × m2 interaction terms share a common interaction coefficient . This suggests that for individual i, we can aggregate the information from mj markers in gene j by defining a weighted genotype score

and then assess the interaction between the two QTLs by examining the cross product of their scores, . In situations in which the trait locus is in complete or strong LD with a genotyped marker or is itself genotyped, the score in equation (8) may not work well as the other markers will simply add noise and dilute the association signal. Given this consideration, an alternative weighted genotype score is

where lj,max is the genotyped marker that has the strongest LD with the trait locus as measured by r2.

Estimation of weights

In the previous sections, we assumed the trait loci are known. In real data analysis, the locations of the trait loci are unknown. It is reasonable to assume that each of the known polymorphisms in the gene, either genotyped or untyped in the study sample, is equally likely to be the trait locus. For each such locus, we can estimate the weights for all the genotyped markers and calculate a score for the locus. Following Li et al,13 we propose to estimate the weight using LD information obtained from a reference database such as that generated by the HapMap, other publicly available dense SNP data sets or resequencing data from a subset of the study sample. Suppose Mj markers are available for gene j in the reference data set, and they are a superset of the markers genotyped in the study sample. If marker kj (1≤kj≤Mj) in the reference data set is the trait locus, then the weight for marker lj in the study sample is

where is the LD coefficient between markers kj and lj, and and are allele frequencies at marker lj in gene j. These quantities can be estimated from the reference data set. For individual i and each marker kj in the reference data set, we can calculate a weighted genotype score

or an alternative weighted genotype score

where lj,max is the genotyped marker that has the strongest LD with marker kj.

We note that the weighted genotype scores in equations (11) and (12) share similarity with imputation dosage scores, and they can be considered as a simple version of the multilocus LD-based imputation dosage scores obtained from software packages such as MACH and IMPUTE. Although using pairwise-LD information only, the weighted genotype scores in equations (11) and (12) provide an intuitive justification of why incorporating external LD information may provide power gain for association testing. In the following sections, we will consider both the pairwise LD-based weighted genotype scores and multilocus LD-based imputation dosage scores in the testing procedure.

Gene-based interaction analysis

Once we have calculated the scores, either weighted genotype scores in equation (11) or (12) or imputation dosage scores in MACH or IMPUTE, for each marker in the reference data set, we can then test for gene–gene interaction based on the scores for gene j, where and n is the total number of individuals in the study. As the trait loci are unknown, a simple test of interaction could be to include all pairwise interactions of the imputation dosage scores in a regression framework and then test for their overall significance. However, this approach may suffer from low power due to the large number of degrees of freedom involved. To efficiently aggregate all information while reducing the degrees of freedom, we propose to test for gene–gene interaction using PCs obtained from the scores. Without loss of generality, for gene j, we order its PCs such that has the largest variance and has the second largest variance and so on.

Once the PCs are computed, we can then test for gene–gene interaction by conducting a regression analysis with a set of selected PCs and their interactions as covariates. As the PCs are ordered by the magnitude of explained variance, for each gene, we select the first several PCs that explain a prespecified fraction of the total variance. Suppose Lj PCs are selected for gene j. For a binary trait, we can fit the data with the following logistic regression model (see Supplementary Material)

For a quantitative trait, we can fit the data with a linear regression model. Under the null hypothesis of no interaction between the two genes, for l1=1,…, L1 and l2=1,…, L2. We can test this null hypothesis by a likelihood ratio test, and the corresponding test statistic is approximately distributed as a χ2 distribution with L1 × L2 degrees of freedom. We call this test as a global test. Alternatively, we can conduct pairwise interaction analysis between all selected PCs and choose the statistic for the most significant pair as the test statistic and evaluate its significance by Bonferroni correction. We call this test as a pairwise test. In our analyses, we used 90% threshold for the fraction of variance as it generally provides better power than other variance thresholds in scenarios we considered. We note that for binary traits, the null hypothesis tested by logistic regression is only an approximation to the null hypothesis that (f2,2–f2,0)–(f0,2–f0,0)=0, and thus the weighted genotype scores we derived earlier may not be ‘optimal’. However, as shown in the Supplementary Material, this approximation is probably valid as long as the interaction effects are not too strong and the disease is not common.

Results

In this section, we evaluate the performance of the gene-based interaction tests for binary traits and compare with SNP-based interaction test. We considered four gene-based interaction tests: (1) ATOM-AVG, which uses weighted genotype scores from equation (11); (2) ATOM-MAX, which uses weighted genotype scores from equation (12); (3) MACH, which uses imputation dosage scores from MACH; and (4) PCA, which uses genotyped markers only. For each test T, we considered two versions: (1) the global version, which tests for the joint interaction effect of all selected PCs; and (2) the pairwise version, which tests for the pairwise interaction among all selected PCs. Significance for the pairwise version is adjusted by Bonferroni correction. For the SNP-based interaction analysis, we only considered the pairwise version as the power of the global version is extremely low due to the large number of degrees of freedom.

Comparison of type I error and power based on simulated data

We simulated data based on the LD structures of two genes CHI3L2 (Figure 1) and PTPN22 (Figure 2), both are located on chromosome 1 but are in linkage equilibrium with each other. For each gene, we considered common SNPs with minor allele frequency ≥0.05 and selected tagSNPs using the program Tagger18 with pairwise tagging at r2 ≥0.8. We identified 25 common SNPs for CHI3L2 and selected seven tagSNPs; for PTPN22, 29 common SNPs and 9 tagSNPs. We assumed that only the tagSNPs were genotyped and available for analysis, a common scenario in both candidate gene and GWAS studies. To simulate case–control data with LD, we first estimated the haplotype frequencies of the tagSNPs for each gene, and then simulated the genotype data according to the estimated haplotype frequencies. We considered two situations: (1) each gene has only one disease locus; and (2) each gene has two disease loci.

Figure 1
figure 1

LD structure of CHI3L2 on chromosome 1 in the HapMap CEU samples. Displayed is estimated r2 for 25 SNPs with MAF ≥0.05. SNPs within the black boxes are tagSNPs selected using the Tagger program at r2 threshold of 0.8.

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Figure 2
figure 2

LD structure of PTPN22 on chromosome 1 in the HapMap CEU samples. Displayed is estimated r2 for 29 SNPs with MAF ≥0.05. SNPs within the black boxes are tagSNPs selected using the Tagger program at r2 threshold of 0.8.

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For the first situation, we designated one locus in each gene as the disease locus, and the case–control status for individual i was simulated according to the following model

where α is determined in a way such that the overall disease prevalence is 5%. Power was estimated based on 1000 replicate data sets each consisting of 2000 cases and 2000 controls and significance was assessed at the 1% level. The type I error rate was evaluated based on 10 000 data sets by setting the interaction effect in the above logit model to 0. Since gene-based interaction tests based on ATOM and MACH require external LD information, we simulated 60 individuals (mimicking the HapMap CEU samples) as a reference data set and then calculated the weighted genotype scores or the imputation dosage scores using the LD information estimated from these 60 individuals.

As the performance of different tests may vary depending on whether the disease loci are genotyped or not, we considered three scenarios: (1) both disease loci are genotyped; (2) only one of the disease loci is genotyped; and (3) both disease loci are untyped. A thorough evaluation of all tests would require consideration of 25 × 29=725 combinations. To avoid extensive simulations for all marker combinations, we classified the markers in each gene into three categories according to LD levels. Specifically, a marker is classified into the ‘strong LD’ category if five or more markers in the gene have r2 >0.8 with it; a marker is in the ‘moderate LD’ category if three to five markers in the gene have r2 >0.8 with it; the rest are in the ‘weak LD’ category. On the basis of this classification, markers in CHI3L fall into either strong or weak LD categories. By classifying markers in this manner, we were able to investigate the performance of various tests under a wide range of settings, yet avoided simulations of all marker combinations.

Table 1 displays the estimated type I error rates under two-locus interaction model. The type I error rates of all tests are under control. Not surprisingly, for each test, the pairwise version of the test is more conservative than the global version due to the correction of a large number of pairwise comparisons. Table 2 shows the estimated power. As expected, when there is a single disease locus in each gene, consistently outperforms the other tests. Among the other tests we considered, , and , which incorporate external LD information, offer better power, followed by . We note that the powers of ATOM- and MACH-based tests are similar, despite that MACH is much more computationally intensive. For example, it requires ∼210 s to finish one simulation for MACH-based tests with 2000 cases and 2000 controls; however, the required computing time for ATOM-based tests is only ∼5 s, 40 times faster.

Table 1 Type I error rates (%) under a two-locus interaction model in which one locus in CHI3L2 interacts with one locus in PTPN22
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Table 2 Comparison of power (%) under a two-locus interaction model in which one locus in CHI3L2 interacts with one locus in PTPN22
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For complex diseases, it might be an oversimplification to consider only one disease locus per gene. To evaluate the performance of different tests under a more complicated setting, we considered a model in which two loci in CHI3L2 interact with two loci in PTPN22. Specifically, we simulated case–control status according to the model:

Again, the overall disease prevalence was set at 5% by adjusting the value of α. For type I error estimation, we set the coefficient for the interaction effect at 0.

As shown in Table 3, the type I error rates of all interaction tests are under control. Table 4 shows the power comparison results. These results indicate that all gene-based interaction tests outperform the SNP-based test. For example, the power advantage of over as measured by mean power difference ranged from 12.7 to 27.3%. This is much higher than the mean power difference (4.1–9.7%) between the two tests under the simpler disease models in Table 2. This indicates that SNP-based interaction analysis is not sufficient when multiple loci in a gene interact with multiple loci in another gene. Among all gene-based tests we considered, is generally the most powerful test, followed by , and . It is worth noting that the power of is only slightly lower than despite that MACH is much more computationally intensive. The pairwise versions of these three tests are typically less powerful than the global versions of the tests. Moreover, our results clearly indicate the advantage of incorporating external LD information in the analysis. The power gain of over as measured by the mean power difference ranged from 3.3 to 7.9%, and the power gain of over ranged from 4.6 to 10.7%.

Table 3 Type I error rates (%) under a four-locus interaction model in which two loci in CHI3L2 interact with two loci in PTPN22
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Table 4 Comparison of power (%) under a four-locus interaction model in which two loci in CHI3L2 interact with two loci in PTPN22
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Application to the HDL data set

We applied the three gene-based interaction tests to an ongoing candidate gene study on subjects with extreme levels of HDL-C. In this study, 625 subjects of European ancestry with HDL >90th percentile were considered as cases and 606 subjects with HDL <30th percentile were considered as controls. All study subjects were genotyped using the IBC 50K SNP array.19 Our previous SNP pairwise interaction analysis on this data set reveals that a number of SNPs in CETP significantly interact with several SNPs in BCAT1. It is well known that CETP promotes the transfer of cholesteryl esters from HDL to low-density lipoprotein, and individuals that are genetically deficient for CETP often have extremely high HDL levels.20, 21 In a recent GWAS on biochemical traits, BCAT1 is shown to be significantly associated with serum albumin concentration.18 As albumin is correlated with HDL,22 it is possible that CETP and BCAT1 interact in modulating the level of HDL-C.

Figures 3 and 4 display the LD structures of CETP and BCAT1 estimated using the HDL data set. We downloaded genotype data at these two genes for the CEU samples from the HapMap website. For CETP, there are 31 common SNPs in the HapMap, whereas the HDL data set has 57, with 27 common SNPs in both data sets. As the HapMap data set does not provide much additional LD information, for ATOM-based tests, we calculated the weighted genotype scores using the LD information provided by the 57 SNPs in the HDL controls. For BCAT1, 164 common SNPs are in the HapMap and 79 are in the HDL data set, with 56 in both. For the 164 SNPs in the HapMap, we calculated their weighted genotype scores using the LD information provided by the HapMap; for the 23 SNPs in the HDL data set but not in the HapMap, we used their observed genotypes in the HDL data set.

Figure 3
figure 3

LD structure of CETP on chromosome 16 in the HDL data set. Displayed is estimated r2 for 57 SNPs with MAF ≥0.05 based on the controls.

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Figure 4
figure 4

LD structure of BCAT1 on chromosome 12 in the HDL data set. Displayed is estimated r2 for 79 SNPs with MAF ≥0.05 based on the controls.

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The BCAT1 SNPs are in several LD blocks with weak LD between some of the blocks, requiring 23 PCs to explain 90% of the variance. Testing interaction using all SNPs in BCAT1 may have low power due to the large number of degrees of freedom. To reduce the dimensionality, we divided the SNPs in BCAT1 into four blocks (Figure 4) and tested interaction between CETP and each of the four blocks. We found significant interaction between CETP and the third block of BCAT1. The P-value of is 0.0034. In comparison, the P-values of , and are 0.22, 0.25 and 0.072, respectively. The P-values of the pairwise versions of the four tests are 0.035, 0.062, 0.38 and 0.029, respectively. The P-value of is 0.078. Compared with other gene-based interaction tests, clearly revealed stronger evidence of association.

Discussion

We have proposed a PC framework for gene-based interaction analysis. Our tests are based on the aggregation of information from weighted genotype scores using pairwise LD information or imputation dosage scores using multilocus LD information in a gene. To reduce dimensionality, the scores within a gene are further summarized into PCs and then used in a regression framework for interaction analysis. By extensive simulations under various settings and the analysis of a real data set, we demonstrated that gene-based interaction tests are a powerful alternative to the conventional SNP-based interaction test and to approaches that do not incorporate external LD information.

The gene-based interaction tests consider each gene as a testing unit and tests for interaction at the gene level. Compared with methods that operate at the marker level, a key advantage of gene-based interaction tests lies in their ability to capture all potential risk conferring variants in a gene. This makes gene-based interaction tests particularly attractive when multiple disease loci in a gene interact with multiple disease loci in another gene. We note that when a single locus in a gene interacts with a single locus in another gene, or when some of the interaction effects are weak when more than two loci interact, the SNP-based interaction test may perform well, as such a simple test can capture the interaction effect more effectively than the gene-based interaction tests.

Another advantage of gene-based interaction analysis over the conventional SNP-based interaction analysis is that it requires much less number of tests. For example, for the IBC data with 50 000 SNPs genotyped in 2000 candidate genes, the conventional SNP pairwise interaction analysis will involve ∼1.25 billion tests. In contrast, using gene-based interaction analysis, the number of tests is reduced to 2 million. For large-scale candidate gene and GWAS data sets, gene-based interaction tests can be used as a screening tool. After a pair of significant interacting genes is identified, one can then conduct further investigation to evaluate which SNPs within the genes significantly interact.

Our method concerns with gene-based tests of interaction effect. We note that there exist gene-based methods that jointly test for the main effect and the interaction effect.23, 24 Although the goals of these tests are slightly different from ours, they all aim to incorporate information contributed by multiple markers in a gene. How to extend the proposed PC framework to jointly test for the main and interaction effects would merit further research.