High-Order Association Mapping for Expression Quantitative Trait Loci

Abstract

Mapping expression quantitative trait loci (eQTLs) is an important avenue to identify putative genetic variants in regulatory regions. Famed eQTL mapping methods exploit the mean effects of locus-wise genetic variants on expression quantitative traits. Despite their successes, such methods are suboptimal because they neglect high-order heterogeneity inherent in genetic variants and covariates. High-order effects of observed loci are common due to their connections to various latent factors, i.e., latent interactions among genes and environmental factors. In this chapter, we introduce a new scheme to harmoniously integrate mean and high-order effects of genetic variants on expression quantitative trait. We rigorously evaluate its validity and utility of signal augmentation.

1. Introduction

Over the past decade, genetic studies on gene expression have enhanced our understanding of transcriptional regulation [1]. Expression quantitative trait loci (eQTL) refer to genomic loci that regulate the expression of mRNA. One key component of eQTL mapping is an association test between genetic loci and expression quantitative traits. The main advantage of eQTL mapping using association methods is the identification of candidate functional loci without requiring the knowledge about cis or trans regulatory regions [2].

Existing prominent eQTL mapping methods merely exploit the mean heterogeneity of a test locus on expression quantitative traits (referred to as MT in context). However, a true eQTL would lead to the alteration of the entire distribution instead of solely the mean of expression quantitative trait. High-order heterogeneity of an eQTL stems from the complicated causal network that involves biological disruption, linkage disequilibrium (LD), gene-by-gene, and/or gene-by-environment interactions. For example, expression quantitative traits may have different means and variances among different genotypic groups at a specific eQTL [3].

The 4-df versatile likelihood ratio test (LRT) [4] and the joint location-scale (JLS) test [5] would be able to reprioritize eQTLs. The LRT integrates mean and variance heterogeneities of a test locus, assuming normality on trait residual. This integrative method appeared to be more powerful than the MT test in the presence of mean and dispersion effects. The JLS applies Fisher’s linear combination of the ordinary MT (i.e., partial t test) on the mean effect of a test locus and Levene’s test [6] on variance heterogeneity of trait residuals among genotypic groups. This integrative method appeared more powerful than the MT test in the presence of latent gene-by-environment interactions. However, both methods suffer perceived limitations. Due to the need for an iterative search, the famed LRT is computationally slow. When the normality of trait residual is violated, it may not converge to a meaningless point or may not converge at all. The JLS extends normality of trait residuals to an exponential distribution family. Nevertheless, it requires a very large sample size or large-scale permutation to accurately compute the p value. Of note, both methods are suboptimal in that they require locus-wise genotypic partitions of sample and suffer unnecessarily large degrees of freedom. The two components of each of the famed methods are independent, no matter whether the test locus is a neutral or causal locus.

Here we introduce a new integrative high-order signal augmentation test (HSAT) to overcome these limitations. Unlike the famed methods, the proposed HSAT integrates harmonious primary test and auxiliary test at each test locus. Two tests are called to be harmonious if their test statistics have null independence and alternative dependence. To be specific, null independence requires that two statistics are independently distributed at a neutral locus. Alternative dependence means that the two statistics can be statistically dependent at a true eQTL (and its flanking loci due to linkage disequilibrium). The null independence is necessary and sufficient to properly control type I error rate. The merit of alternative dependence is a major novelty of the HSAT, and this merit is instructive for maximizing statistical power. The HSAT harmoniously integrates mean and high-order heterogeneities by a statistic of two degrees of freedom with no need to partition the sample into different genotypic groups. It computes p value at each locus numerically without any aid of iteration or resampling techniques. In the following sections, we describe this method, mathematically assess its validity, and demonstrate its power gains using published simulation models in supportive of the famed methods.

2. Methods

2.1. The HSAT

Let ϒ_i be the residual of expression quantitative trait for subject i after adjusting for various effects of covariates. To account for potential heteroscedasticity, the double generalized linear model (DGLM) [7] would be applied to adjust for both mean and variance effects of data structure and environmental covariates (see Note 1). In particular, for background population structure, the mean and dispersion effects on the trait should be adjusted for.

At a test locus, let G_i be the copy number of the minor allele for subject i (i.e., G_i = 0, 1, or 2). Let the symbols “mo”, “ho,” and “mh” stand for “modeling the mean effect only,” “modeling high-order effect only,” and “modeling mean and high-order effects jointly.” The primary test assumes the mean-effect model:

$${{\rm Y}}_i={\mu }_1+G_i{\beta }_1+e_i,$$

where μ₁ is the intercept, e_i is random error term, β₁ is the effect size of G_i on ϒ_i. The null hypothesis is H_0.mo: β₁ = 0. The primary test statistic is

$$T_1=\frac{\sqrt{n}{\widehat{\sigma }}_{{\rm Y},G}}{\sqrt{{{\widehat{\sigma }}^2}_{{\rm Y}}{{\widehat{\sigma }}^2}_G-{{\widehat{\sigma }}^2}_{{\rm Y},G}}},$$

where ${\widehat{\sigma }}_{{\rm Y},G}=\frac{1}{n}{\sum }_{i=1}^n({{\rm Y}}_i-\overline{{\rm Y}})(G_i-\overline{G})$, ${\widehat{\sigma }}_{{\rm Y}}^2=\frac{1}{n}{\sum }_{i=1}^n({{\rm Y}}_i-\overline{{\rm Y}}{)}^2$, ${\widehat{\sigma }}_G^2=\frac{1}{n}{\sum }_{i=1}^n(G_i-\overline{G}{)}^2$, $\overline{{\rm Y}}=\frac{1}{n}{\sum }_{i=1}^n{{\rm Y}}_i$, $\overline{G}=\frac{1}{n}{\sum }_{i=1}^n{G}_i$, and n is sample size. Mathematically, $T_1=\sqrt{n-2}{r}_{{\rm Y},G}∕\sqrt{1-{r_{{\rm Y},G}}^2}$, where r_{ϒ, G} is the sample Pearson coefficient of correlation between G_i’s and ϒ_i’s.

The auxiliary test assumes the secondary model:

$${{\rm Y}}_i^2={\mu }_2+G_i^2{\beta }_2+e_i^{\prime },$$

where $e_i^{\prime }$ is the random error term. The highlight of auxiliary test is to capture the second-order moment information. The null hypothesis of testing the high-order effect is H_0.ho : β₂ = 0. The auxiliary test statistic is

$$T_2=\frac{\sqrt{n}{\widehat{\sigma }}_{{{\rm Y}}^2,G^2}}{\sqrt{{{\widehat{\sigma }}^2}_{{{\rm Y}}^2}{{\widehat{\sigma }}^2}_{G^2}-{{\widehat{\sigma }}^2}_{{{\rm Y}}^2,G^2}}},$$

where ${\widehat{\sigma }}_{{{\rm Y}}^2,G^2}=\frac{1}{n}{\sum }_{i=1}^n\left({{\rm Y}}_i^2-\overline{{{\rm Y}}^2}\right)\left(G_i^2-\overline{G^2}\right)$, ${\widehat{\sigma }}_{{{\rm Y}}^2}^2=\frac{1}{n}{\sum }_{i=1}^n({{\rm Y}}_i^2-{{\rm Y}}^2{)}^2,\overline{\overline{{{\rm Y}}^2}=\frac{1}{n}{\sum }_{i=1}^n{{\rm Y}}_i^2}$, and $\overline{G^2}=\frac{1}{n}{\sum }_{i=1}^n{G}_i^2$. Mathematically, $T_2=\sqrt{n-2}{r}_{{{\rm Y}}^2,G^2}∕\sqrt{1-r_{{{\rm Y}}^2,G^2}}$, where r_ϒ²,G² is the sample Pearson coefficient of correlation between $G_i^2$’s and ${{\rm Y}}_i^2$’s.

The proposed HSAT integrates T₁ and T₂ to reject the joint null hypothesis H_0.mh : β₁ = β₂ = 0. To be specific, the HSAT statistic is defined as T² = T₁² + T₂². Of note, the definitions of sample correlation coefficients and T-statistics do not require normality. For large samples, sample correlation coefficients are consistent estimators for population correlation coefficients, and T statistics are very robust to non-normality. As mathematically proven, at a neutral locus, statistics T₁ and T₂ are asymptotically independent χ₁² variables and thus T² → χ₂² under mild moment conditions on the quantitative trait. As such, the p value of T² could be computed according to χ₂² distribution.

2.2. Theoretical Validation

Under the primary model ϒ_i = G_iβ₁ + e_i, the covariance between the trait and genotypic score is formulated by ${\sigma }_{{\rm Y},G}={\sigma }_G^2{\beta }_1$, where ${\sigma }_G^2$ is the variance of the genotypic score. From the primary model, we obtain ${{\rm Y}}_i^2=(G_i{\beta }_1+e_i{)}^2={\sigma }_e^2+G_i^2{\beta }_2+{\epsilon }_i$, where ${\sigma }_e^2$ is the variance of random error e_i, β₂ = β₁² and ${\epsilon }_i=2{\beta }_1{G}_i{e}_i+e_i^2-{\sigma }_e^2$. Of note, the newly defined ε_i has mean 0 and variance ${\sigma }_{\epsilon }^2=2(1+2{\beta }_1^2{\mu }_2){\sigma }_e^2$. The covariance between ${{\rm Y}}_i^2$ and $G_i^2$ is formulated by ${\sigma }_{{{\rm Y}}^2,G^2}={\beta }_2{\sigma }_{G^2}^2$ and the variance of ϒ² is ${\sigma }_{{{\rm Y}}^2}^2={\sigma }_{\epsilon }^2+{\beta }_2^2{\sigma }_{G^2}^2$. As such, if G is associated with ϒ(β₁ ≠ 0), then G² must be associated with ϒ². In other words, the mean heterogeneity of a locus alone already implies high-order association, i.e., the association between ϒ² and G². Famed integrative methods neglect this important advantage, whereas the proposed HSAT takes it. Let E(.) denote the expectation of an underlying variable. We formulate the harmonious property of the HSAT as below.

Proposition 1: Under primary model, if $E\left(e_i^4\right)<\infty$ and $E(e_i)=E\left(e_i^3\right)=0$, then T₁ − C₁ and T₂ − C₂ converge to a bivariate normal distribution:

$$\left(\begin{array}{c}\hfill T_1-C_1\hfill \\ \hfill T_2-C_2\hfill \end{array}\right)\stackrel{a.d.}{\to }{\mathcal{N}}_2\left(\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \end{array}\right),\left(\begin{array}{cc}\hfill 1\hfill & \hfill \rho \hfill \\ \hfill \rho \hfill & \hfill 1\hfill \end{array}\right)\right),$$

where $C_1={\beta }_1({\mu }_2-{\mu }_1\overline{G})∕\sqrt{{\widehat{\sigma }}_{{\rm Y}}^2{\widehat{\sigma }}_G^2-{\widehat{\sigma }}_{{\rm Y},G}^2}$, $C_2=({\beta }_1^2({\mu }_4-\overline{G^2}{\mu }_2)+\left({\mu }_2-\overline{G^2}\right){\sigma }_e^3)∕\sqrt{{\widehat{\sigma }}_{{{\rm Y}}^2}^2{\widehat{\sigma }}_{G^2}^2-{\widehat{\sigma }}_{{{\rm Y}}^2,G^2}^2}$, $\rho ={\beta }_1{\delta }_1^{-1}{\delta }_2^{-1}\left[{\sigma }_e^2(3{\mu }_4-3{\mu }_2^2+{\mu }_1^2{\mu }_2-{\mu }_1{\mu }_3)+{\beta }_1^2({\mu }_6-2{\mu }_2{\mu }_4+{\mu }_1{\mu }_5-{\mu }_1^2{\mu }_4+{\mu }_2^3-{\mu }_1{\mu }_2{\mu }_3+{\mu }_1^2{\mu }_2^2)\right]$, ${\delta }_1=\sqrt{{\sigma }_e^2{\sigma }_G^2}$, ${\delta }_2=\sqrt{2(1+2{\beta }_1^2{\mu }_2){\sigma }_e^2{\sigma }_{G^2}^2}$, and ${\sigma }_e^2$, ${\sigma }_G^2$, and ${\sigma }_{G^2}^2$, are the variances of e, G, and G², respectively.

Explicitly, Proposition 1 formulates the harmonious property of the statistics T₁ and T₂ under the primary model with very mild moment conditions (see Note 2). If the trait is independent of the test locus (β₁ = 0), then C₁ = C₂ = ρ = 0, and hence T₁ and T₂ asymptotically follow the standard bivariate normal distribution ${\mathcal{N}}_2(0,{\mathit{I}}_2)$. With the joint normality, zero correlation (ρ = 0) warrants the independence between statistics T₁ and T₂ under the null hypothesis. On the other hand, if the trait of interest is associated to the test locus (β₁ ≠ 0), then T₁ and T₂ turn to be asymptotically dependent (ρ ≠ 0). As such, the HSAT is more powerful than the LRT and JLS methods to identify a locus that only has mean effect on the trait. For such a locus, the LRT and JLS purely waste 2 degrees of freedom. In practice, high-order association can be driven by other factors, e.g., latent dispersion effects and latent gene-by-environment interactions.

2.3. Empirical Demonstration

2.3.1. Dispersion Design

For empirical comparisons, we first generated individual trait values and genotypes from ϒ = −0.03X₁ − 0.08X₂ + 0.23 [X₀ + δ(X₁ + 1.16X₂)]ε, where X_k is indicator of event “G = k” for k = 0, 1, 2, the genotypic score G follows the binomial distribution $\mathcal{B}$inom(2, MAF), MAF is the minor allele frequency, residual ε follows the standard normal distribution $\mathcal{N}(0,1)$. This data-generating model is exactly the dispersion model in [4] when δ = 25/23 and MAF = 0.4. By ref. 8, variants with MAF $<1∕\sqrt{2n}$ are taken as rare variants. In our simulations, we set MAF = 0.02 and sample size n = 1000. For power comparisons, we varied the dispersion coefficient δ from 1 to 2. To inspect type I error rate control, we generated individual trait values from null dispersion model ϒ = 0.23ε.

2.3.2. Latent Interaction Design

Following Soave et al. [5], we generated individual values of quantitative trait, genotype of a causal locus, and latent exposure from ϒ = Gβ_G + GEβ_GE + e, where genotypic score G follows binomial distribution $\mathcal{B}$inom(2, MAF), MAF is the minor allele frequency, the latent exposure follows Bernoulli $\mathcal{B}$(1, 0.3), and trait residual e follows the standard normal distribution $\mathcal{N}(0,1)$. In our simulations, we set MAF = 0.3 and sample size n = 1000. The model collapses to the null when setting β_G = β_GE = 0 to investigate type I error rate control. For power comparison, we set β_G = 0.01 and varied the interaction effect β_GE from 0 to 1 by a grid of 0.1. Under this design, individual values of the latent exposure and the interaction terms were hidden when computing the p values of all the tests.

2.3.3. Type I Error Rate Control

By large-scale simulations under the null models, our HSAT and the competitors appeared properly and controlled type I error rates at genome-wide significance level (5 × 10⁻⁸). Table 1 demonstrates their rigorous validity. Under each null model, the large number of p values of each test followed $\mathcal{U}$(0,1), granting the validity of the test. The rigorous validity of the proposed HSAT stems from the null independence between its two components, namely, they are independently distributed when the trait is independent of the test locus. Besides, our HSAT accurately computes the p values by using very robust numerical algorithms.

Table 1.

Type I error rates control

	Null dispersion model		Null interaction model
Method	Error ratea	P K–S b	Error ratea	P K–S b
MT	4 × 10−8	0.9946	6 × 10−8	0.9632
JLS	2 × 10−8	0.8756	4 × 10−8	0.8258
LRT	3 × 10−8	0.9567	1 × 10−8	0.1594
HSAT	5 × 10−8	0.9238	3 × 10−8	0.8136

^aUnder each null model, the type I error rate of each test was computed as the proportion 10⁸ null p values of <5 × 10⁻⁸

^bp-Value of the Kolmogorov–Smirnov test on the null hypothesis that the 10⁸ null p values follow the standard uniform distribution $\mathcal{U}$(0,1)

2.3.4. Power Comparisons

By large-scale simulations, the HSAT appeared significantly more powerful than all three competitors, where the LRT and JLS appeared equally powerful and both outperformed the classical MT. Under the dispersion model (Fig. 1), the power gain of the HSAT over the famed LRT and JLS reached about 40% when dispersion coefficient δ was around 1.6. Of note, the power gains appeared especially striking for identifying rare variants since the sample partitions in the LRT and JLS reduce power.

Fig. 1.

Power comparison under dispersion model with rare variant. Each point on each power curve was calculated from 10⁶ replicates of 1000 simulated subjects from the specific model setup at significance level 5 × 10⁻⁸

Under the latent interaction design, neither individual values of the latent exposure nor the interaction terms were directly modeled to compute the p values. The three integrative methods captured the interaction effects via high-order heterogeneity of the genotypic score in different ways, whereas the ordinary MT simply ignored the latent interaction effect. When interaction effect β_GE was around 0.5, the power gain of the HSAT over the MT, LRT, and JLS reached about 25%, 20%, and 15%, respectively (Fig. 2). The HSAT appeared more powerful than the competitors uniformly, whereas the LRT and JLS outperformed the ordinary MT when interaction effect was strong enough (e.g., β_GE > 0.4). The famed LRT and JLS appeared slightly less powerful than the ordinary MT when the interaction effect was mild (e.g., β_GE < 0.4).

Fig. 2.

Power comparison under interaction model with a latent exposure. Each point on each power curve was calculated from 10⁶ replicates of 1000 simulated subjects from the specific model setup at significance level 5 × 10⁻⁸

The HSAT overcame the deficiencies of the sample partition in the integrative LRT and JLS. Its uniform optimality and significant power gains stemmed from the dependence between its two components. They reinforced each other in the presence of high-order heterogeneities as driven by the dispersion and latent interaction effects of causal loci (see Note 3).

2.4. Implementation in R

In the following R function HSAT, the “geno” is the n × 1 vector of genotypic scores and “traitRes” is the n × 1 vector of trait residuals after adjusting for data structure and environmental covariates (see Note 1). This function returns the overall p value after harmoniously integrating the mean and high-order heterogeneities of a single test locus. This function is highly efficient in computation because it only calls the efficient correlation test function cor. test (.) and the pchi2(.) built in the R base package. It computes and combines two t-tests and evaluates the overall p value without any iterative or resampling techniques.

HSAT<-function(geno, traitRes, covariate=NULL)
{
x<-as.numeric(geno)
y<-as.numeric(traitRes)
T1<-cor.test(x, y)[1]
T2<-cor.test(x**2, y**2)[1]
Pval<-pchisq(q=T1**2+T2**2, df = 2, lower.tail=FALSE)
return (Pval)
}

3. Notes

1. Before applying the HSAT method in eQTL mapping, it is necessary to effectively calibrate mean and variance effects of data structure of environment covariates such as age and gender. The conventional linear adjustment only adjust for the mean effects of covariates and is often insufficient. The double generalized linear model (DGLM) [7] would work.

2. The validity of the proposed HSAT is warranted by large-sample theory (Proposition 1). The joint asymptotic normality of its two test statistics does not require normality of the error terms. It only requires $E\left(e_i^4\right)<\infty$ and $E(e_i)=E\left(e_i^3\right)=0$. These moment conditions are very mild and cover a broad range of symmetric and asymmetric error distributions.

3. The HSAT method outperforms existing integrative mean-variance methods such as JLS and LRT in terms of both power and computation efficiency. It combines primary test and high-order auxiliary test without partitioning sample into subgroups. Meanwhile, its mean and high-order components reinforce each other when the test variant is truly associated with the trait of interest. It numerically computes p values without using any resampling techniques. As such, it is an effective and efficient avenue for mapping eQTL associations.