Analysis of Gene-Gene Interactions Using Gene-Trait Similarity Regression

Abstract

Objective

Gene-Gene interactions (GxG) are important to study because of their extensiveness in biological systems and their potential in explaining missing heritability of complex traits. In this work, we propose a new similarity-based test to assess GxG at gene level, which permits the study of epistasis at biologically functional units with amplified interaction signals.

Methods

Under the framework of gene-trait similarity regression (SimReg), we propose a gene-based test for detecting gene-gene interactions. SimReg uses a regression model to correlate trait similarity with genotypic similarity across a gene. Unlike existing gene-level methods based on leading principal components (PCs), SimReg summarizes all information on genotypic variation within a gene and can be used to assess the joint/interactive effects of two genes as well as the effect of one gene conditional on another.

Results

Using simulations and a real data application on warfarin study, we show that the SimReg GxG tests have satisfactory power and robustness under different genetic architecture when compared to existing gene-based interaction tests such as PC analysis or partial least squares (PLS). A genomewide association study with ~20,000 genes may be completed on a parallel computing system in 2 weeks.

Introduction

Gene-Gene interactions (GxG) are important to study since they are believed to be widespread in biological systems [1, 2] and are likely involved in gene regulation, signal transduction, biochemical networks, as well as other physiological and developmental pathways [3-5]. GxG further can provide insight into the missing heritability of complex traits [6-8] and explain replication failures of initial GWAS findings [9-10]. Regarding this latter point, GxG help explain between-study differences in marginal genetic effects, which can be due to between-study differences in frequency of modifier genetic variant(s). The study of GxG has provided insight into biological mechanisms for many complex diseases, including Alzheimer’s disease, diabetes, cardiovascular disease, autism, multiple sclerosis, and cancer [11-17].

Analysts often implement GxG tests using a regression model accounting for the main effects of two single nucleotide polymorphisms (SNPs) and also the two-way interaction between the two SNPs. The interaction effect can be assessed on the additive scale or the multiplicative scale---the former examines the effect on the phenotype on the linear scale, and the latter examines the effect on the log scale of the phenotype. The additive scale is often more relevant to public health importance [18], while the multiplicative scale more naturally corresponds to the biological mechanisms [19]. Regardless of scale, one can test whether the interaction parameter is different from zero or, alternatively, consider a test similar to that of Chapman and Clayton [20] to assess the effect of a SNP in the presence of interaction with a second SNP. While analysts primarily applied such SNP-SNP interaction tests in small-scale candidate-gene studies, there has also been keen interest in performing exhaustive interaction testing of SNPs in a GWAS [21]. One can perform such genomewide analyses using standard tools like regression [6]. However, one can also apply more innovative techniques, such as two-stage screening procedures [22], Bayes networks [23], Bayesian model averaging [24], logic regression [25] and data-mining procedures [26-29].

All of these existing interaction methods consider the analysis on the level of a SNP. However, there is increasing interest in performing such analyses on the broader level of a gene [30-32]. Several factors motivate the paradigm shift from SNP to gene. First, genes are the basic units in the biological mechanism and SNPs within a gene tend to work concordantly. Thus, gene-level results may be more biologically insightful and easier to interpret. Second, a gene-level analysis incorporates linkage disequilibrium (LD) information from all SNPs simultaneously within the gene. Consequently, such joint analysis of SNPs should have improved ability to tag untyped causal variants compared to the analysis of individual SNPs, leading to improved power. Finally, if a gene harbors multiple causal variants, then joint analysis of SNPs in aggregate should be more powerful than separate analysis of each individual SNP (owing in part to the gene-based test often having less degrees of freedom than its individual-SNP counterpart).

A few gene-based methods for interaction testing exist. Chatterjee et al. [33] proposed Tukey’s 1-df method to investigate an interaction between two candidate genes. The approach calculates the sum of the main effects of SNPs contained within each gene and then uses the product of the two sums as the GxG interaction term. Thus, the approach models one interaction parameter at the gene level rather than several interaction parameters on the SNP level. Owing to the reduced degrees of freedom of the Tukey test, 5 the authors showed their approach lead to a great improvement in power to detect GxG compared to individual-SNP analysis. Motivated by this idea, Wang et al. [32] considered two different interaction tests that summarized SNP information within a gene. For the first test, the authors used principal component analysis (PCA) to summarize LD information of SNPs within a gene. They then created an interaction test using the top (first) principal component from each gene. For their second test, the authors applied partial least squares (PLS) to extract components that summarize both the LD information among SNPs in a gene as well as the correlation between such SNPs and the outcome of interest. The authors then constructed their interaction test using the top PLS component from each gene. Using simulated data, the PCA and PLS methods often had better performance than the Tukey 1-df method, particularly when causal SNPs had no or negligible marginal effect.

In this article, we propose a new gene-based test for detecting gene-gene interactions in complex traits using similarity regression (SimReg) [34]. SimReg is an analytic procedure that uses a regression model to correlate trait similarity with genotypic similarity across a gene. SimReg is inspired by Haseman-Elston regression from linkage analysis [35, 36] and haplotype similarity tests for regional association [37, 38]. In SimReg, the trait similarity is quantified by the trait covariance adjusting for the covariates. The multi-marker information of a gene is first summarized by genetic similarity, which is measured using a pre-specified metric such as the proportion of alleles shared identity by state (IBS) across the gene. The gene-gene interaction is then modeled by taking the product of the genetic similarities of the two genes, and its significance is assessed by testing the significance of the corresponding regression coefficient. Compared to the Tukey/PCA/PLS based methods, we believe SimReg has several advantages. First, SimReg utilizes all the information within a gene while the PCA/PLS procedures may lose some information due to consideration of only the top component of each gene within the interaction analysis. Second, as the top PCA/PLS component from each gene is a weighted sum of SNP information, the use of the product of the two linear combinations as the interaction term in the model only captures limited forms of non-additive effect. In contrast, SimReg provides a tool to model a variety of effects through different measures of summarized gene-similarity information. At the same time, SimReg does not involve a large number of parameters so that it has a good power performance. Finally, as we show later, we can obtain an analytic p-value for the SimReg procedure without the need for intensive resampling procedures like those required for the Tukey approach.

We arrange the remainder of the paper as follows. We first present the SimReg approach and describe how to use the framework to construct three score tests of interest (interaction test, joint test and conditional main effect test). We derive the distributions of these tests under the null hypothesis by connecting the approach to a variance component model. We use simulated data to example the performance of the proposed method against the PCA and PLS methods, as well as a standard SNP-based test of interaction. We also apply SimReg to the Warfarin data from the study of Wysowski et al. [39]. Finally, we discuss the further research directions of the proposed approach.

Method

The Gene-Trait Similarity Model

We assume a sample of N subjects. For subject i (i = 1, …, N), we denote Y_i as the trait value, X_i a K × 1 vector of covariates (including the intercept term), and G_m,i a coding of the subject’s genotype at SNP m.

Define $S_{\mathit{ij}}^A$ to be the genetic similarity of gene A between subjects i and j (i ≠ j). There are many ways to describe the genetic similarity between individuals. Here, we model similarity as the weighted sum of the proportion of alleles shared identical by state (IBS) across the M_A SNPs in gene A. That is, for subjects i and j, the similarity level is $S_{\mathit{ij}}^A={\sum }_{m=1}^{M_A}{w}_m{s}_{m,\mathit{ij}}$ where s_m,ij = x∕2 if G_m,i and G_m,j share x alleles IBS [40, 41], and the weight, w_m, can be used to up-weight or down-weight a variant based on allele frequencies, the degree of evolutionary conservation, or the functionality of the variations [40, 42, 43]. In this work, to up-weight similarities that are contributed by rare alleles, we set $w_m=q_m^{-1}$, where q_m is the minor allele frequency of marker m [44, 45]. We can define the similarity level in gene B $S_{\mathit{ij}}^B$ in the same manner. The trait similarity between individual i and j, denoted by Z_ij, is computed by Z_ij = (Y_i − μ_i)(Y_j−μ_j), where μ_i = E(Y_i∣X_i) = X_iγ, which is the conditional mean of trait with no genotype effect and γ is the effect of the covariates. The gene-trait similarity regression for gene-gene interactions

$$E(Z_{\mathit{ij}}|X,G)={\tau }_A{S}_{\mathit{ij}}^A+{\tau }_B{S}_{\mathit{ij}}^B+{\tau }_{\mathit{AB}}{S}_{\mathit{ij}}^A{S}_{\mathit{ij}}^B.$$ (1)

Note that the regression model has zero intercept because we incorporated the covariate effects when quantifying trait similarity [34].

The Interaction Test

The interaction test examines the hypothesis H_0,Int: τ_AB = 0. We derive the score test of this hypothesis from model (1) by taking advantage of the connection between the similarity model and a variance component model [34]. Below, we first show the regression coefficients in (1) can be viewed as the variance components under a mixed model. We then construct the test of τ _AB using the same test from the corresponding variance component model. Specifically, consider the following working mixed model:

$$Y_i=X_i\gamma +g_{A,i}+g_{B,i}+g_{\mathit{AB},i}+e_i,$$ (2)

where e_i ~ N(0, σ), and g_A,i, g_B,i and g_AB,i are subject- specified genetic effects for gene A, gene B and gene-gene interactions, respectively. Let g_A = [g_A,1,⋯, g_A,n], g_B = [g_B,1,⋯, g_B,n], and g_AB = [g_AB,1,⋯, g_AB,n], and and assume that

$$g_A\sim \mathit{MN}(0,v_A{S}_A),g_B\sim \mathit{MN}(0,v_B{S}_B),g_{\mathit{AB}}\sim \mathit{MN}(0,v_{\mathit{AB}}{S}_{\mathit{AB}}),$$

where $S_A=\left\{S_{\mathit{ij}}^A\right\}$, $S_B=\left\{S_{\mathit{ij}}^B\right\},$ and $S_{\mathit{AB}}=\left\{S_{\mathit{ij}}^A\times S_{\mathit{ij}}^B\right\}$. Under model (2), we can obtain the marginal trait covariance by

$$\begin{array}{ll}\mathit{cov}(Y_i,Y_j|X,G)& ={\mathit{cov}}_{g.}\{E(Yi|\mathbf{X},G,g_A,g_B,g_{\mathit{AB}}),E(Y_i|\mathbf{X},G,g_A,g_B,g_{\mathit{AB}})\}\\ & ={\mathit{cov}}_{g.}\{X_i\gamma +g_{A,i}+g_{B,i}+g_{\mathit{AB},i},X_j\gamma +g_{A,j}+g_{B,j}+g_{\mathit{AB},j}\}\\ & =v_A{S}_{\mathit{ij}}^A+v_B{S}_{\mathit{ij}}^B+v_{\mathit{AB}}{S}_{\mathit{ij}}^A{S}_{\mathit{ij}}^B.\end{array}$$ (3)

Comparing (3) with (1), we have τ_A = v_A, τ_B = v_B, and τ_AB = v_AB.

We derive the score function of the REML log-likelihood function of (2) in the Appendix A, and obtain the score statistic under H₀,_Int as

$$T_{\mathit{Int}}=\frac{1}{2}\mathbf{Y}\prime P_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}\mathbf{Y}{|}_{{\tau }_A={\widehat{\tau }}_A,{\tau }_B={\widehat{\tau }}_B,\sigma =\widehat{\sigma }},$$

where = (Y₁,…, Y_n), $P_{\mathit{Int}}=V_{\mathit{Int}}^{-1}-V_{\mathit{Int}}^{-1}\mathbf{X}{({{\mathbf{X}}^{\prime }}^{-1}{V}_{\mathit{Int}}^{-1}\mathbf{X})}^{-1}{\mathbf{X}}^{\prime }{V}_{\mathit{Int}}^{-1}$, V_Int = τ_AS_A + τ _BS_B + σI and (τ̂_A, τ̂_B, σ̂) are the maximum REML estimates obtained under H₀,_Int: τ_AB = 0. We describe an EM algorithm to obtain (τ̂_A, τ̂_B, σ̂) in Appendix B.

Under the alternative hypothesis τ_AB ≠ 0, T_Int is a strictly increasing function of τ_AB. Therefore larger values of T_Int provide stronger evidence against H₀,_Int. This suggests that the testing procedure should be one sided. As shown in the Appendix A, the distribution of T_Int follows a weighted χ² distribution. That is, define $C_{\mathit{Int}}=\frac{1}{2}{V}_{\mathit{Int}}^{1/2}{P}_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}{V}_{\mathit{Int}}^{1/2}$, and then $T_{\mathit{Int}}~{\sum }_{j=1}^c{\lambda }_{j,\mathit{Int}}{\chi }_1^2$, where λ _j,Int is the ordered none zero eigenvalues of matrix C_Int. We can calculate the p-values analytically using moment-matching approximations [46].

The Joint Test

Instead of performing a test specifically for interactions, one may be interested in assessing whether the two genes have any marginal or interactive effects on the trait [47]. This can be done using the joint test to examine the null hypothesis H₀,_Joint: τ_A = τ_B = τ_AB = 0 under the full model $E(Z_{\mathit{ij}}|X,G)={\tau }_A{S}_{\mathit{ij}}^A+{\tau }_B{S}_{\mathit{ij}}^B+{\tau }_{\mathit{AB}}{S}_{\mathit{ij}}^A{S}_{\mathit{ij}}^B$. As shown in Appendix A, the test statistic is given as

$$T_{\mathit{Joint}}=\frac{1}{2}{\mathbf{Y}}^{\prime }{P}_{\mathit{Joint}}\left(S_A+S_B+S_{\mathit{AB}}\right)P_{\mathit{Joint}}\mathbf{Y}{|}_{\sigma =\stackrel{⌣}{\sigma }},$$

where P_Joint = σ⁻¹{I − X(X′X)⁻¹X′} and σ̆ = Y′{I − X(X′X)⁻¹X′}Y/(n − K). The distribution of T_Joint also follows a weighted χ² distribution, i.e., $T_{\mathit{Joint}}~{\sum }_{j=1}^c{\lambda }_{j,\mathit{joint}}{\chi }_1^2$ with λ_j,Joint the ordered nonzero eigenvalues of $C_{\mathit{Joint}}=\frac{1}{2}{V}_{\mathit{Joint}}^{1⁄2}{P}_{\mathit{Joint}}\left(S_A+S_B+S_{\mathit{AB}}\right)P_{\mathit{Joint}}{V}_{\mathit{Joint}}^{1⁄2}$. As with the interaction test, we calculate the p-value of the joint test analytically using moment-matching procedures.

The Conditional Main Effect Test

We also construct the score test for the main effect of a gene conditioning on the effect of the other gene, assuming no gene-gene interactions. This is because in scenarios where the interaction effects do exist, the main effects are typically not well defined, and its significance depends on the scale of the interacting variables. When there is no gene-gene interaction, the full model is

$$E(Z_{\mathit{ij}}|X,G)={\tau }_A{S}_{\mathit{ij}}^A+{\tau }_B{S}_{\mathit{ij}}^B$$

Here to test the main effect of Gene A accounting for the effect of Gene B, we evaluate the null hypothesis H₀,_A: τ_A = 0. We similarly can test the main effect of Gene B accounting for the effect of Gene A by evaluating H₀,_B: τ_B = 0. Similar to the score tests in the interaction test and joint test, we derive the score test statistic for conditional main effect of Gene A conditional on Gene B as:

$$T_A=\frac{1}{2}{\mathbf{Y}}^{\prime }{P}_A{S}_A{P}_A\mathbf{Y}{|}_{{\tau }_B={\stackrel{\sim }{\tau }}_B,\sigma =\stackrel{\sim }{\sigma }},$$

where $P_A=V_A^{-1}-V_A^{-1}\mathbf{X}{\left({{\mathbf{X}}^{\prime }}^{-1}{V}_A^{-1}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\prime }{V}_A^{-1}$, V_A = τ_BS_B + σI, and (τ̃_B, σ̃) are the maximum REML estimates obtained under H₀,_A: τ_A = 0. We describe the EM algorithms to obtain (τ̃_B, σ̃) in Appendix B. The test statistic T_B can be defined similarity for examining H₀,_B: τ_B = 0 under the constrain of τ_AB = 0. As shown in Appendix A, T_A has the same distribution as ${\sum }_{j=1}^c{\lambda }_{j,A}{\chi }_1^2$ with λ_j,A the ordered nonzero eigenvalues of $C_A=\frac{1}{2}{V}_A^{1/2}{P}_A{S}_A{P}_A{V}_A^{1/2}$. As with the interaction and joint tests, we analytically derive the p-value of the test statistic using moment-matching procedures.

Simulation study

Design

We study the performance of the proposed methods using simulated data, and benchmark them against 3 approaches: (1) LR: linear regression; (2) PCA: the Principal Component method of Wang, et al, [32] and (3) PLS: the Partial Least-Square of Wang et al, [32]. The LR incorporates all SNPs from each of the 2 genes as well all pairwise interactions of SNPs across genes. When performing conditional main effect testing, we exclude the interaction terms from the LR analysis. We also only consider the proposed test and LR for the conditional main effect test, since PCA and PLS are identical to LR when there is no interaction term.

To simulate genotype data with realistic LD patterns, we use genotype data of Gene RBJ (8 SNPs) and Gene GPRC5B (15 SNPs) of the Phase III CEU samples downloaded from the International HapMap Project (http://hapmap.ncbi.nlm.nih.gov/). We show the LD structure of the two genes in Supplementary Figure 1 and the MAF and the average R² of each SNP in Supplementary Table 1. The average R² is calculated by averaging the R² between the target SNP and the remaining SNPs in the same gene. We consider two causal SNPs from each gene and simulate the trait values based on the model listed below.

$$\begin{array}{ll}Y=& {\beta }_A\times \left({\mathit{SNP}}_1^A+{\mathit{SNP}}_2^A+{\mathit{SNP}}_1^A\times {\mathit{SNP}}_2^A\right)\\ & +{\beta }_B\times \left({\mathit{SNP}}_1^B+{\mathit{SNP}}_2^B+{\mathit{SNP}}_1^B\times {\mathit{SNP}}_2^B\right)\\ & +{\beta }_{\mathit{AB}}\times \left({\mathit{SNP}}_1^A+{\mathit{SNP}}_1^B+{\mathit{SNP}}_2^A\times {\mathit{SNP}}_2^B\right)+e,\end{array}$$ (4)

where ${\mathit{SNP}}_1^A$ and ${\mathit{SNP}}_2^A$ are the number of the minor alleles carried by a subject at the first and second causal loci in Gene A; ${\mathit{SNP}}_1^B$ and ${\mathit{SNP}}_2^B$ are defined similarity; e follows a normally-distributed variable with mean 0 and variance 1. The trait value Y is defined based on 3 parts: the gene effect from Gene A, the gene effect from Gene B and the interaction effect between Gene A and Gene B. We assume no LD between the genes RBJ and GPRC5B and consequently sample the genotypes of the former gene independent of the genotypes for the latter gene.

To evaluate the type-I error rate of the interaction test, we considered three situations of no gene-gene interaction effects: (1) both genes have no genetic effect (β_A = β_B = β_AB = 0), (2) only Gene A has a main effect to the trait (β_A ≠ 0, β_B = β_AB = 0), and (3) both Gene A and Gene B have main effect to the trait (β_A ≠ 0, β_B ≠ 0, β_AB = 0). To evaluate type-I error rate for the joint test, we set β_A = β_B = β_AB = 0. To evaluate type-I error for the conditional main-effect test, we examined size strictly for Gene B (H₀: τ_B = 0). We considered two scenarios of no effect of Gene B: (1) both Gene A and Gene B have no main effect (β_A = β_B = β_AB = 0) and (2) Gene A has a main effect but not Gene B (β_A ≠ 0, β_B = 0, β_AB = 0). For each scenario, we simulated datasets comprised of 300 subjects and evaluated type-I error rates using 1000 replicates of the data.

For power analysis, we consider three different scenarios based on the causal SNPs. Specifically, we pick up “representative” SNPs from SNPs with similar LD and MAF patterns and form 3 different causal SNP combinations (Table 1). When generating trait values using the model above, we consider different β′s value so that power of different methods at a significance threshold of 5% are between 20%~80%. The power was calculated based on 200 replications.

Table 1.

The causal SNPs for Gene A and Gene B for the three scenarios considered in power analysis

Scenario	${\mathit{SNP}}_1^A$		${\mathit{SNP}}_2^A$		${\mathit{SNP}}_1^B$		${\mathit{SNP}}_2^B$
	avgR2	MAF	avgR2	MAF	avgR2	MAF	avgR2	MAF
1	0.58	0.12	0.13	0.48	0.26	0.46	0.23	0.48
2	0.57	0.12	0.16	0.06	0.14	0.16	0.23	0.48
3	0.58	0.12	0.16	0.06	0.07	0.04	0.14	0.16

Results

Table 2 shows empirical type I error rates for all the methods at a significance level of α = 0.05. The type I error rates of all approaches are around the nominal level in all different settings. For the interaction test, the type I error rates of the proposed method are slightly conservative but we observed that, as the variance due to the main effects of Gene A and/or Gene B increase, the type-I error rate generally approaches the nominal level. As discussed in the appendix, the conservative nature of the interaction test arises because of the bias in the EM-algorithm estimates of the variance components when their true values are 0. The type I error for interaction test using the competing methods (PCA, PLS, LR) are generally appropriate. For joint and conditional-main testing, we observed the type-I error rates of all methods considered in our analyses to be quite similar to the nominal levels.

Table 2.

The type I error rate at significance level 0.05 for 3 tests: interaction test, joint test and conditional main effect test

(Effect Size) Test	(βA, βA, βAB)
	Interaction Test			Joint Test	Conditional Test (B ∣ A)
Method	(0,0,0)	(0.1,0,0)	(0.1,0.1,0)	(0,0,0)	(0,0,0)	(0.3,0,0)
SimReg	0.038	0.04	0.046	0.049	0.048	0.052
LR	0.060	0.062	0.064	0.060	0.060	0.056
PCA	0.056	0.06	0.058	0.055
PLS	0.046	0.048	0.049	0.054

We next performed power analysis under the models considered in Table 1. Because different Scenario has different causal SNP combination, we adjust different β′s so that the power of different methods are around 0.2~0.8. The results are summarized in Table 3.

Table 3.

The power analysis for different approaches at significance level 0.05 under the scenarios listed in Table 1. In each scenario, the effect size (β_A, β_B and β_AB) are set so that the power range between 20% and 80%. The shaded values indicate the method with the largest power under a certain scenario, and the bolded values indicate methods with overlapping confidence intervals with the best methods.

Interaction test
Method	(βA, βB, βAB)
	S1 (0.1,0.1,0.3)	S2 (0.1,0.1,0.5)	S3 (0.1,0.1,1.5)
SimReg	0.690 (0.626, 0.754)*	0.460 (0.391, 0.529)	(0.761, 0.869)
LR	0.375 (0.308, 0.442)	0.285 (0.222, 0.348)	0.760 (0.701, 0.819)
PCA	0.715 (0.652, 0.778)	0.350 (0.284, 0.416)	0.120 (0.075, 0.165)
PLS	(0.685, 0.805)	(0.421, 0.559)	0.470 (0.401, 0.539)
Joint test
Method	(βA, βB, βAB)
	S1(0.05,0.05,0.05)	S2(0.1,0.1,0.1)	S3(0.3,0.3,0.3)
SimReg	(0.605, 0.735)*	(0.829, 0.921)	(0.512, 0.648)
LR	0.325 (0.260, 0.390)	0.690 (0.626, 0.754)	0.325 (0.260, 0.390)
PCA	0.535 (0.466, 0.604)	0.765 (0.706, 0.824)	0.500 (0.431, 0.569)
PLS	0.545 (0.476, 0.614)	0.785 (0.728, 0.842)	0.505 (0.436, 0.574)
Conditional main effect test
Method	(βA, βB, βAB)
	S1(0.07,0.07,0)	S2 (0.1,0.1,0)	S3(0.3,0.3,0)
SimReg	(0.584, 0.716)*	(0.621, 0.749)	0.495 (0.426, 0.564)
LR	0.415 (0.347, 0.483)	0.475 (0.406, 0.544)	0.485 (0.416, 0.554)

^*95% confidence interval of the power, obtained by $\widehat{p}\pm 1.96\times \sqrt{\widehat{p}\times \left(1-\widehat{p}\right)/200}$ with p̂ the empirical power estimated based on 200 replications.

Overall, we observed the proposed method generally has optimal power relative to the other methods considered under various genetic architectures of the causal SNPs For the interaction test, Scenario 1 has the causal SNPs with relatively high LD and large MAF. PCA and PLS perform the best, closely followed by the proposed method. LR has the least power due to the large degree of freedom used. In Scenario 2 where the causal SNP had a relatively smaller LD and MAF, PLS still has the best power, closely followed by the proposed method. The power of PCA drops a lot in this scenario. LR again has the least power. In Scenario 3, the proposed method has the best power, followed by LR, PLS and PCA. PCA and PLS have a significant low power performance in Scenario 3. As the LD and MAF become smaller, the power of PCA drops dramatically, because the first PC can only capture limit amount of information on the causal SNPs. The first PC aims to maximize the SNP variation captured and tends to be dominated by SNPs with high LD or common MAF, which are non-causal SNPs in Scenario 3. PLS shares the same issue as PCA and hence it also suffers from power loss when LD and MAF of the causal SNPs become smaller. However PLS also accounts for the trait information, which makes PLS perform better than PCA. The power advantage of PLS over PCA is more substantial in S2 and S3, where an increased proportion of causal SNPs have low MAFs. Consequently, the first principal component of PCA (which explains the largest variance among the genotypes) is less likely to harbor such causal SNPs and therefore have reduced power for interaction testing compared to PLS [32].

For joint test, under three different scenarios, the proposed method performs the best. PCA and PLS perform similarly with a slightly better power of PLS. The observation is consistent with Wang, et al. [32] and is because the first PC from PLS considers LD and correlation between trait and gene. LR has the lowest power, which it may be due to the large degree of freedoms used in the joint test.

For conditional main effect test, the proposed method always has a better power than linear model, but the difference between proposed method and linear model becomes smaller when the LD and MAF of the causal SNPs become smaller. The finding is consistent with the finding in the interaction test.

Real Data Analysis

Warfarin is a widely used oral anticoagulant. In 2004, more than 30 million prescriptions contained this drug in United State [39]. The optimal dose of warfarin is different from patient to patient, and an inappropriate dosage can lead to severe consequence such as bleeding, swelling of face, throat. Extensive research has been conducted to develop methods for predicting the appropriate dose.

We have conducted a genetic analysis using the data from the Warfarin study [48]. In this data set, we studied the relationship between stable warfarin dose and 2 genes: VKORC1 (containing 7 SNPs) and CYP2C9 (a tri-allelic locus). We further adjusted for 4 covariates associated with warfarin therapy: age, sex, height, and weight. After quality control, the dataset consisted of 301 individuals. We applied the proposed method and the benchmark methods to evaluate the association between warfarin dose and the two genes, adjusting for covariates. We summarize the results in Table 4. All methods identified significant association between warfarin dose and the two genes. The smallest p-value of the joint test was obtained from SimReg (i.e., p-value 5.6 × 10⁻²⁰), closely followed by LR (i.e., p-value 1.92 × 10⁻¹⁶). The p-values by PCA and PLS are similar to each other and are many orders of magnitude larger (i.e., 10⁻⁵) than those derived from SimReg and LR. We next examined the interaction effect between VKORC1 and CYP2C9 and observed no significant interaction using any of the methods considered. Finally, we performed the conditional main-effect for each gene using SimReg and LR. The result suggests that both genes had significant effects on warfarin dose, with VKORC1 demonstrating a stronger effect (i.e., p-values on the order of 10⁻¹⁶~10⁻²⁰) than CYP2C9 (i.e., p-values on the order of 10⁻⁷~10⁻⁸).

Table 4.

The p-values of the four approaches in the warfarin analysis

Methods	Joint test	Interaction test	Conditional main effect test
			VKORC1	CYP2C9
SimReg	5.6 × 10−20	0.450	1.67 × l0−20	1.02 × 10−7
LR	1.92 × 10−16	0.753	1.94 × 10−16	2.65 × 10−8
PCA	8.07 × 10−5	0.530	-	-
PLS	7.43 × 10−5	0.470	-	-

The main effect results were consistent with current literature [49-51]. Previous studies indicate contradictory results regarding the interaction between the VKORC1 and CYP2C9 on anticoagulant effect and hence the dose requirement [52-56]. Our results agreed with recent reports [55, 56], which suggest no evidence of a gene-gene interaction between CYP2C9 and VKORC1.

Discussion

In this article, we described a novel similarity-based test to assess gene-gene interactions in large-scale association studies. Unlike the majority of existing procedures in this area, our approach considers the analysis at the level of the gene rather than the SNP, which we show can lead to considerable improvements in power as a result. We also compared the performance of our similarity approach to existing gene-based interaction tests (PCA/PLS) and showed our joint/interaction/conditional mean tests nearly always had optimal power across simulation models. Relative to PCA/PLS approaches, we believe our approach can improve power since it summarizes all information on genotypic variation within a gene. PCA and PLS, on the other hand, only use a subset of this variation within their analytic frameworks. The R code that implements the SimReg testing procedure is available from the authors’ website (http://www4.stat.ncsu.edu/~jytzeng/Software/SimReg-GxG-QT/). Using a personal computer equipped with Intel Core i7-3770 CPU 40GHz and 20GB RAM, 200 runs of the interaction test (which is the slowest test because it requires estimating two nuisance variance components) with the simulated datasets required 767 seconds. While the approach is not scalable to genomewide interaction testing using a single processor, we believe the procedure can be applied to genomewide data in reasonable time using a parallel-computing cluster that possesses a large number (1000) of CPUs.

In the SimReg model, we define the similarity function for two subjects as the weighted average of SNP alleles shared IBS by the pair across a gene. The weights are SNP specific and are user defined. It is common to weight SNPs based on minor-allele frequency such that rarer variants are assigned more weight than common variants; common weights in this setting include the reciprocal of a SNPs minor-allele frequency or its square root [40, 41]. Additionally, we can weight SNPs using functional information; perhaps providing more weight to SNPs that demonstrate evidence of being a cis-acting expression quantitative-trait locus [57]. While we considered only common SNP variants in this work, we note that our similarity-regression approach also can integrate information from rare variants ascertained using next-generation sequencing technology. While weighting rare variants by minor-allele frequency has obvious value, additional weighting of such variants using some measure that predicts the probability the variant is deleterious and lowers fitness likely has value as well. A variety of computational algorithms exists for such prediction based on evolutionary, biochemical, and/or structural information (see [58] for an overview).

We applied our similarity approach in the context of a candidate-gene study but we can further expand the approach to consider interactions on a genomewide scale. A gene-based interaction test has appealing features over SNP-based interaction tests when applied to a genomewide association study (GWAS). Given the number of genes is many fold smaller than the typical number of GWAS SNPs, the number of tests that need to be evaluated using similarity regression would be substantially smaller than a SNP-based interaction test like LR. This could substantially reduce the computational burden. Additionally, the issue of adjusting for multiple testing when conducting genomewide interaction testing of SNPs is challenging. Within GWAS, most studies adjust for multiple testing using permutations since a Bonferroni correction leads to conservative inference due to LD among SNPs. However, while permutations are valid for multiple-testing adjustment in tests of main effects, they are not valid when applied to tests of interactions because such random shuffling of phenotypes does not preserve the observed main effects in the sample [59]. In contrast, multiple-testing adjustment of interaction tests based on similarity regression is more straightforward. While application of gene-based similarity regression to GWAS data will lead to correlated tests (due to repeated testing of the same gene) such that a Bonferroni correction is inappropriate, we can adjust for multiple testing using a computationally-efficient perturbation procedure similar to that proposed by Wu et al. [60] that preserves the main effects of the genes under consideration. We will explore this work in a future manuscript.

Appendix A. Derivation of the score tests and their distributions

Consider the matrix presentation of model (2):

$$\mathbf{Y}=\mathbf{X}\gamma +g_A+g_B+g_{\mathit{AB}}+e.$$

The corresponding REML log-likelihood function, denoted as L(τ_A, τ_A, τ_AB, σ), is:

$$L\left(\theta \right)=-\frac{1}{2}[\log |V|+\log |{{\mathbf{X}}^{\prime }}^{-1}V\mathbf{X}|+{\mathbf{Y}}^{\prime }P\mathbf{Y}],$$

where V = Var(Y) = τ_AS_A + τ_BS_B + τ_ABS_AB + σI is the marginal variance of Y and P = V⁻¹ − V⁻¹X(X′⁻¹V⁻¹X)⁻¹X′V⁻¹ is the projection matrix for the model. The score functions of τ_A, τ_B, and τ_AB based on L(θ) are

$$\begin{array}{l}{U}_{{\tau }_A}({\tau }_A,{\tau }_B,{\tau }_{\mathit{AB}},\sigma )=\frac{\partial L({\tau }_A,{\tau }_B,{\tau }_{\mathit{AB}},\sigma )}{\partial {\tau }_A}=\frac{1}{2}\left[{\mathbf{Y}}^{\prime }{\mathit{PS}}_{A}P\mathbf{Y}-\mathit{tr}\left({\mathit{PS}}_A\right)\right],\\ U_{{\tau }_B}({\tau }_A,{\tau }_B,{\tau }_{\mathit{AB}},\sigma )=\frac{\partial L({\tau }_A,{\tau }_B,{\tau }_{\mathit{AB}},\sigma )}{\partial {\tau }_B}=\frac{1}{2}\left[{\mathbf{Y}}^{\prime }{\mathit{PS}}_{B}P\mathbf{Y}-\mathit{tr}({\mathit{PS}}_B)\right],\mathrm{and}\\ U_{{\tau }_{\mathit{AB}}}({\tau }_A,{\tau }_B,{\tau }_{\mathit{AB}},\sigma )=\frac{\partial L\left({\tau }_A,{\tau }_B,{\tau }_{\mathit{AB}},\sigma \right)}{\partial {\tau }_{\mathit{AB}}}=\frac{1}{2}\left[{\mathbf{Y}}^{\prime }{\mathit{PS}}_{\mathit{AB}}P\mathbf{Y}-\mathit{tr}\left({\mathit{PS}}_{\mathit{AB}}\right)\right].\end{array}$$

We construct the statistics based on the first terms of the score functions. Here after we define matrices P_h and V_h as P and V evaluated under H₀,_h. Then for the interaction test H₀,_Int: τ_AB = 0, we set the test statistic as

$$T_{\mathit{Int}}=\frac{1}{2}{\mathbf{Y}}^{\prime }{P}_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}\mathbf{Y}{|}_{{\tau }_A={\widehat{\tau }}_A,{\tau }_B={\widehat{\tau }}_B,\sigma =\widehat{\sigma }},$$

where the $P_{\mathit{Int}}=V_{\mathit{Int}}^{-1}-V_{\mathit{Int}}^{-1}\mathbf{X}{\left({{\mathbf{X}}^{\prime }}^{-1}{V}_{\mathit{Int}}^{-1}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\prime }{V}_{\mathit{Int}}^{-1}$, V_Int = τ_AS_A + τ_BS_B + σI, and (τ̂_A, τ̂_B, σ̂) are the maximum REML estimates obtained under H₀,_Int: τ_AB = 0.

For the conditional main effect test H₀,_A: τ_A = 0 under the constrain of no interaction (i.e., τ_AB = 0), we set the test statistic as

$$T_A=\frac{1}{2}{\mathbf{Y}}^{\prime }{P}_A{S}_A{P}_A\mathbf{Y}{|}_{{\tau }_B={\widehat{\tau }}_B,\sigma =\stackrel{\sim }{\sigma }},$$

where $P_A=V_A^{-1}–V_A^{-1}\mathbf{X}{\left({{\mathbf{X}}^{\prime }}^{-1}{V}_A^{-1}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\prime }{V}_A^{-1}$, V_A = τ_BS_B + σI, and (τ̃_B,σ̃) are the maximum REML estimates obtained under H₀,_A: τ_A = 0 The test statistic T_B can be defined similarily for examining H₀,_B: τ_B = 0 under the constrain of τ_AB = 0 We describe the EM algorithms that we use to obtain (τ̂_A, τ̂_B, σ̂) and (τ̃_B, σ̃) and in Appendix B.

For the joint test H₀,_Joint: τ_A = τ_B = τ_AB = 0, because τ’s are non-negative variance components, τ_A = τ_B = τ_AB = 0 if and only if τ_A + τ_B + τ_AB = 0. This motivates us to construct the test statistic based on the sum of the three score functions. That is,

$$T_{\mathit{Joint}}=\frac{1}{2}{\mathbf{Y}}^{\prime }{P}_{\mathit{Joint}}\left(S_A+S_B+S_{AB}\right)P_{\mathit{Joint}}\mathbf{Y}{|}_{\sigma =\stackrel{⌣}{\sigma }},$$

where the $P_{\mathit{Joint}}=V_{\mathit{Joint}}^{-1}-V_{\mathit{Joint}}^{-1}\mathbf{X}{\left({{\mathbf{X}}^{\prime }}^{-1}{V}_{\mathit{Joint}}^{-1}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\prime }{V}_{\mathit{Joint}}^{-1}$, V_Joint = σI and σ̆ = Y′{I − X(X′X)⁻¹X′}Y/(n − K).

The distributions of the test statistics can be shown to follow a weighted chi-squared distribution via the fact that these statistics are quadratic form of Y. To illustrate, consider $T_{\mathit{Int}}=\frac{1}{2}{\mathbf{Y}}^{\prime }{P}_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}\mathbf{Y}$. Because P_Int is a projection matrix, P_IntXγ = 0. Therefore,

$$\begin{array}{ll}{T}_{\mathit{Int}}& =\frac{1}{2}{\mathbf{Y}}^{\prime }{P}_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}\mathbf{Y}=\frac{1}{2}{\left(\mathbf{Y}-\mathbf{X}\gamma \right)}^{\prime }{P}_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}\left(\mathbf{Y}-\mathbf{X}\gamma \right)\\ & =\frac{1}{2}{\left(\mathbf{Y}-\mathbf{X}\gamma \right)}^{\prime }{V}_{\mathit{Int}}^{-1/2}{V}_{\mathit{Int}}^{1/2}{P}_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}{V}_{\mathit{Int}}^{1/2}{V}_{\mathit{Int}}^{-1/2}\left(\mathbf{Y}-\mathbf{X}\gamma \right)\sim \sum _{i=1}^c{\lambda }_i{\chi }_1^2\end{array}$$

Define $\mathbf{z}=V_{\mathit{Int}}^{-1/2}\left(\mathbf{Y}-\mathbf{X}\gamma \right)$ and $C_{\mathit{Int}}=\frac{1}{2}{V}_{\mathit{Int}}^{1/2}{P}_{\mathit{Int}}{S}_{\mathit{AB}}{P}_{\mathit{Int}}{V}_{\mathit{Int}}^{1/2}$, we have $T_{\mathit{Int}}={\mathbf{z}}^{\prime }{C}_{\mathit{Int}}\mathbf{z}\sim {\sum }_{j=1}^c{\lambda }_{j,\mathit{Int}}{\chi }_1^2$, where λ_j,Int is the ordered none zero eigenvalues of matrix C_Int. By the same mannar, one can obtain that $T_A\sim {\sum }_{j=1}^c{\lambda }_{j,A}{\chi }_1^2$ with λ_j,A the ordered nonzero eigenvalues of $C_A=\frac{1}{2}{V}_A^{1/2}{P}_A{S}_A{P}_A{V}_A^{1/2}$, and $T_{\mathit{joint}}\sim {\sum }_{j=1}^c{\lambda }_{j,\mathit{joint}}{\chi }_1^2$ with λ_j,Joint the ordered nonzero eigenvalues of $C_{\mathit{Joint}}=\frac{1}{2}{V}_{\mathit{Joint}}^{1/2}{P}_{\mathit{Joint}}\left(S_A+S_B+S_{\mathit{AB}}\right)P_{\mathit{Joint}}{V}_{\mathit{Joint}}^{1/2}$.

Appendix B. The EM algorithm to obtain the maximum REML estimates

We first describe the EM algorithm for (τ̂_A, τ̂_B, σ̂), i.e., the maximum REML estimates under H₀,_Int: τ_AB = 0. Under H₀,_Int, the LMM is

$$\mathbf{Y}=\mathbf{X}\gamma +g_A+g_B+e,$$

where e~N(0, σI), g_A~N(0, τ_AS_A) and g_B~N(0, τ_BS_B) as τ_A = v_A and τ_B = v_B. Define U = A^TY with the restriction that A^T A = I_n−k and AA^T = I − X(X′X)⁻¹X. Then U∣g_A, g_B ~N(A′g_A + A′_B,σI_n−K), which is independent of the fixed effect γ̂ = (X^TX)⁻¹X^TY. Therefore the maximum REML estimates can be obtained by maximizing the marginal distribution of U, i.e., f(U) = ∫ f(U∣g_A,g_B)f(g_A)f(g_B)dg_Adg_B. This motivated an expectation-maximization algorithm based on U (i.e., the observed data) and (g_A, g_B) (i.e., the missing data). The complete-data log likelihood is based on f (U, g_A, g_B) is

$$\begin{array}{l}\log f\left(U,g_A,g_B;{\tau }_A,{\tau }_B,\sigma \right)\\ =\log f\left(U|g_A,g_B;{\tau }_A,{\tau }_B,\sigma \right)+\log f\left(g_A;{\tau }_A,{\tau }_B,\sigma \right)+\log f\left(g_B;{\tau }_A,{\tau }_B,\sigma \right)\\ =-\frac{n–K}{2}\log \sigma -\frac{1}{2\sigma }\left(U-A^{\prime }{g}_A-A^{\prime }{g}_B\right)\prime \left(U-A^{\prime }{g}_A-A^{\prime }{g}_B\right)\\ -\frac{q_A}{2}\log {\tau }_A-\frac{1}{2}\log \left({|S_A|}_{+}\right)-\frac{1}{2{\tau }_A}{g}_A^T{S}_A^{-}{g}_A\\ -\frac{q_B}{2}\log {\tau }_B-\frac{1}{2}\log \left({|S_B|}_{+}\right)-\frac{1}{2{\tau }_B}{g}_B^T{S}_B^{-}{g}_B\end{array}$$

where q_A and q_B are the rank for matrix S_A and S_B respectively, |S_A|₊ is the pseudo-determinant, and $S_A^{-}$ is the generalized inverse (as S_A and S_B may be singular).

In the expectation step, we compute

$$\begin{array}{l}Q\left({\tau }_A,{\tau }_B,\sigma ;{\widehat{\tau }}_A^{(t)},{\widehat{\tau }}_B^{(t)},{\widehat{\sigma }}^{(t)}\right)\equiv E\left\{\log f\left(U,g_A,g_B;{\tau }_A,{\tau }_B,\sigma \right)|U;{\widehat{\tau }}_A^{(t)},{\widehat{\tau }}_B^{(t)},{\widehat{\sigma }}^{(t)}\right\}\\ =-\frac{n-K}{2}\log \sigma -\frac{1}{2\sigma }E\left\{{(U-A^{\prime }{g}_A-A^{\prime }{g}_B)}^{\prime }(U-A^{\prime }{g}_A-A^{\prime }{g}_B)|U;{\widehat{\tau }}_A^{(t)},{\widehat{\tau }}_B^{(t)},{\widehat{\sigma }}^{(t)}\right\}\\ -\frac{q_A}{2}\log {\tau }_A-\frac{1}{2}\log ({|S_A|}_{+})-\frac{1}{2{\tau }_A}E\left(g_A^T{S}_A^{-}{g}_A|U;{\widehat{\tau }}_A^{(t)},{\widehat{\tau }}_B^{(t)},{\widehat{\sigma }}^{(t)}\right)\\ -\frac{q_B}{2}\log {\tau }_B-\frac{1}{2}\log ({|S_B|}_{+})-\frac{1}{2{\tau }_B}E\left(g_B^T{S}_B^{-}{g}_B|U;{\widehat{\tau }}_A^{(t)},{\widehat{\tau }}_B^{(t)},{\widehat{\sigma }}^{(t)}\right).\end{array}$$

In the maximization step, we solve for ∂Q/∂τ_A = 0, ∂Q/∂τ_B = 0 and ∂Q/∂σ = 0, and obtain

$${\widehat{\tau }}_A=\frac{1}{q_A}E\left(g_A^T{S}_A^{-}{g}_A|U;{\stackrel{⌢}{\tau }}_A^{\left(t\right)},{\stackrel{⌢}{\tau }}_B^{\left(t\right)},{\stackrel{⌢}{\sigma }}^{\left(t\right)}\right)=\frac{1}{q_A}\left\{{\stackrel{\sim }{\mathbf{g}}}_A^{{\left(t\right)}^{\prime }}{S}_A^{-}{\stackrel{\sim }{\mathbf{g}}}_A^{\left(t\right)}+tr\left(S_A^{-}{\stackrel{\sim }{\mathbf{v}}}_A^{\left(t\right)}\right)\right\},$$

where ${\stackrel{\sim }{\mathbf{g}}}_A^{\left(t\right)}=E\left(g_A|g_B,U;{\stackrel{⌢}{\tau }}_A^{\left(t\right)},{\stackrel{⌢}{\tau }}_B^{\left(t\right)},{\stackrel{⌢}{\sigma }}^{\left(t\right)}\right)={\tau }_A{S}_A{P}_{Int}{|}_{{\stackrel{⌢}{\tau }}_A^{\left(t\right)},{\stackrel{⌢}{\tau }}_B^{\left(t\right)},{\stackrel{⌢}{\sigma }}^{\left(t\right)}}$, and ${\stackrel{\sim }{\mathbf{v}}}_A^{\left(t\right)}=Var\left(g_A|g_B,U;{\stackrel{⌢}{\tau }}_A^{\left(t\right)},{\stackrel{⌢}{\tau }}_B^{\left(t\right)},{\stackrel{⌢}{\sigma }}^{\left(t\right)}\right)={\tau }_A{S}_A-{\tau }_A^2{S}_A{P}_{\mathit{Int}}{S}_A{|}_{{\stackrel{⌢}{\tau }}_A^{\left(t\right)},{\stackrel{⌢}{\tau }}_B^{\left(t\right)},{\stackrel{⌢}{\sigma }}^{\left(t\right)}}$. Similarly, we have

$${\stackrel{⌢}{\tau }}_B=\frac{1}{q_B}E\left(g_B^T{S}_B^{-}{g}_B|U;{\stackrel{⌢}{\tau }}_A^{\left(t\right)},{\stackrel{⌢}{\tau }}_B^{\left(t\right)},{\stackrel{⌢}{\sigma }}^{\left(t\right)}\right)=\frac{1}{q_B}\left\{{\stackrel{\sim }{\mathbf{g}}}_B^{{\left(T\right)}^{\prime }}{S}_B^{-}{\stackrel{\sim }{\mathbf{g}}}_B^{\left(t\right)}+tr\left(S_B^{-}{\stackrel{\sim }{\mathbf{v}}}_B^{\left(t\right)}\right)\right\}$$

Finally,

$$\begin{array}{l}{\stackrel{⌢}{\sigma }}^{\left(t\right)}=\frac{1}{n-K}E\left\{{\left(U-A^{\prime }{g}_A-A^{\prime }{g}_B\right)}^{\prime }\left(U-A^{\prime }{g}_A-A^{\prime }{g}_B\right)|U;{\stackrel{⌢}{\tau }}_A^{\left(t\right)},{\stackrel{⌢}{\tau }}_B^{\left(t\right)},{\stackrel{⌢}{\sigma }}^{\left(t\right)}\right\}\\ =Y^{\ast T}A{A}^{\prime }{Y}^{\ast }+tr[A{A}^{\prime }({\stackrel{⌢}{\tau }}_A^{\left(t\right)}{S}_A-{({\stackrel{⌢}{\tau }}_A^{\left(t\right)})}^2{S}_A{\mathit{PS}}_A+{\stackrel{⌢}{\tau }}_B^{\left(t\right)}{S}_B-{({\stackrel{⌢}{\tau }}_B^{\left(t\right)})}^2{S}_B{\mathit{PS}}_B-2{\stackrel{⌢}{\tau }}_A^{\left(t\right)}{\stackrel{⌢}{\tau }}_B^{\left(t\right)}{S}_A{\mathit{PS}}_B)]\end{array}$$

The EM algorithm for obtaining (τ̃_B, σ̃) under H₀,_A: τ_A = 0 is similar to the above algorithm except that τ_A is set to be 0.

When applying the above EM algorithm, we add an additional testing step as detailed below. Because the EM algorithm provides non-negative estimates, when τ_A and τ_B are 0 or close to 0, the EM estimates obtained from the above algorithm can be biased as $E\left(\stackrel{⌢}{{\tau }_A}\right)>{\tau }_A$ and $E\left(\stackrel{⌢}{{\tau }_B}\right)>{\tau }_B$. To solve this problem, we first apply conditional main effect tests for H₀: τ_A = 0 and H₀: τ_B = 0 to examine if these nuisance variance components are significantly different from 0. If we fail to reject the null hypotheses, then we set the corresponding τ̂’s as 0. If τ for a gene is significantly different from 0, then we obtain its estimate using the EM algorithm described above. By applying this additional step, E(τ̂) would be closer to 0 when the τ is 0 or close to 0. When τ is relatively large, the conditional main effect test would reject H₀: τ = 0 and the final estimate is the same as the estimate from the original EM algorithm.