A Powerful Retrospective Multiple Variant Association Test for Quantitative Traits by Borrowing Strength from Complex Genotypic Correlations

Abstract

High-throughput sequencing has often been used in pedigree-based studies to identify genetic risk factors associated with complex traits. The genotype data in such studies exhibit complex correlations attributed to both familial relation and linkage disequilibrium. Accounting for these genotypic correlations can improve power for assessing the contribution of multiple genomic loci. However, due to model restrictions, existing multiple variant association testing methods cannot make efficient use of the correlation information appropriately. Recognizing this limitation, we develop PC-ABT, a novel principal-component-based adaptive-weight burden test for gene-based association mapping of quantitative traits. This method uses a retrospective score test to incorporate genotypic correlations, and employs “data-driven” weights to obtain maximized test statistic. In addition, by adjusting the number of principal components that essentially reveals the effective number of tests in the target gene region, PC-ABT is able to reduce the degree of freedom of the null distribution to improve power. Simulation studies show that PC-ABT is generally more powerful than other multiple variant tests that allow related individuals. We illustrate the application of PC-ABT by a gene-based association analysis of systolic blood pressure using data from the NHLBI “Grand Opportunity” Exome Sequencing Project.

Appendices

Appendix 1: Description of MASTOR and Theoretical Justification of the Null Distribution of S _ABT

MASTOR (Jakobsdottir and McPeek 2013) is a retrospective, quasi-likelihood score test for testing single-variant association with a quantitative trait in samples with related individuals. Considering a biallelic genetic variant X of interest (an example in the general setting described in Sect. 4.2.1 is to let X = G _j, 1 ≤ j ≤ m), the MASTOR statistic (for complete data) takes the form

$$\displaystyle \begin{aligned} S_{MAS}=\frac{(\boldsymbol{V}^T\boldsymbol{X})^2}{(\boldsymbol{V}^T\boldsymbol{\varPhi}\boldsymbol{V})\widehat{\sigma}_X^2}. \end{aligned}$$

In this expression, $\boldsymbol {V}=\widehat {\boldsymbol {\varSigma }}_0^{-1}(\boldsymbol {Y}-\boldsymbol {Z}\widehat {\boldsymbol {\beta }}_0)$ is the transformed phenotypic residual obtained from the null model Y  = Zβ ₀ + 𝜖, 𝜖 ∼ N(0, Σ ₀), where β ₀ represents the coefficient of regressing quantitative trait Y on non-genetic covariates Z, and Σ ₀ is the trait covariance matrix under the null, usually with a variance component form $\sigma _e^2\boldsymbol {I}+\sigma _a^2\boldsymbol {\varPhi }$. The variance of variant X is denoted by $\sigma _X^2$. When Hardy-Weinberg equilibrium is assumed for this variant, $\sigma _X^2$ can be estimated by $\widehat {\sigma }_X^2=\widehat {p}(1-\widehat {p})/2$, where $\widehat {p}=(\boldsymbol {1}^T\boldsymbol {\varPhi }^{-1}\boldsymbol {1})^{-1}\boldsymbol {1}^T\boldsymbol {\varPhi }^{-1}\boldsymbol {X}$ is the best linear unbiased estimator (McPeek et al. 2004) of the allele frequency p of X, and 1 denotes a vector with every element equal to 1.

Now in Sect. 4.2.2, we have obtained the ABT statistic

$$\displaystyle \begin{aligned} S_{ABT}=\frac{\boldsymbol{V}^T\boldsymbol{G}(\widehat{\boldsymbol{D}}\boldsymbol{R}\widehat{\boldsymbol{D}})^{-1}\boldsymbol{G}^T\boldsymbol{V}}{\boldsymbol{V}^T\boldsymbol{\varPhi}\boldsymbol{V}}. \end{aligned}$$

Let $\widetilde {\boldsymbol {G}}=\boldsymbol {G}(\widehat {\boldsymbol {D}}\boldsymbol {R}\widehat {\boldsymbol {D}})^{-1/2}$ be a decorrelated version of the genotype matrix in which the across-column covariance has been transformed to identity, and let $\widetilde {\boldsymbol {G}}_j$ be the jth column of $\widetilde {\boldsymbol {G}}$. By linear algebra,

$$\displaystyle \begin{aligned} S_{ABT}=\sum_{j=1}^m\frac{\left(\boldsymbol{V}^T\widetilde{\boldsymbol{G}}_j\right)^2}{\boldsymbol{V}^T\boldsymbol{\varPhi}\boldsymbol{V}}. \end{aligned}$$

This is essentially the summation of m independent MASTOR statistics (in observing the uncorrelatedness and joint normality of $\boldsymbol {V}^T\widetilde {\boldsymbol {G}}_j$), each formulated from a transformed variant $\widetilde {\boldsymbol {G}}_j$ (note the variance estimate is 1 after transformation). Hence S _ABT follows $\chi _m^2$ distribution under the null hypothesis.

Appendix 2: Additional Simulation Results Show That the Data-Driven Weights W ^∗ Is Adaptive to the Direction of True Genetic Effects

In order to understand how the data-driven weights W ^∗ (defined in Eq. (4.5) of the main text) help gain power in association testing, we compare the signs of W ^∗ to those of the genetic effects γ using the simulated data sets in the power analysis. Figure 4.5, Panels a–d, present boxplots of the weights W ^∗ based on 5000 simulated data replicates in Scenario S2 with genetic effect Setting III, for LD Configurations C1–C4, respectively. We note that, in this setting, the first 30% components of γ are set to be positive, the next 30% are negative, and the remaining 40% are zeros. The boxplots clearly demonstrates that on average, the weights W ^∗ is able to track the direction of true genetic effects, thus result in stronger association on the weighted sum genetic score.

Fig. 4.5

Boxplot of W ^∗ based on 5000 simulated data replicates in Scenario S2 with genetic effect Setting III. The adaptive weights of risk, protective, and neutral variants are marked with red, green, and white color, respectively. Panel a: Configuration C1; Panel b: Configuration C2; Panel c: Configuration C3; Panel d: Configuration C4

Appendix 3: Additional Simulation Results Show the Relation Between the ABT Statistic and the famSKAT Statistic

We show in Fig. 4.6, Panels a–d, the scatter plots of the numerator of the ABT statistic vs. the famSKAT statistic based on 5000 simulated data replicates in Scenario S3 with genetic effect Setting II, for LD Configurations C1–C4, respectively. We observe that, when the LD correlation is negligible (Panel a), the numerator of the ABT statistic behaves similarly as the famSKAT statistic because in Eq. (4.6) of the main text, $(\widehat {\boldsymbol {D}}\boldsymbol {R}\widehat {\boldsymbol {D}})^{-1}$ is equivalent to the Madsen-Browning weights used in calculating the famSKAT statistic. As the LD correlation increases (Panels b, c, and d), the two statistics become less and less consistent because in calculating the famSKAT statistic, the Madsen-Browning weights only depend on individual variants, whereas in calculating the ABT statistic, the weight of an individual variant statistic is also affected by other variants on linked sites, as seen from the weight matrix $(\widehat {\boldsymbol {D}}\boldsymbol {R}\widehat {\boldsymbol {D}})^{-1}$ in Eq. (4.6) of the main text.

Fig. 4.6

Comparison between S _famSKAT and the numerator of S _ABT based on 5000 simulated data replicates in Scenario S3 with genetic effect Setting II. Panel a: Configuration C1; Panel b: Configuration C2; Panel c: Configuration C3; Panel d: Configuration C4

Appendix 4: Additional Simulation Results to Validate the Asymptotic Null Distribution of S _PC-ABT via Permutation Based Approach

We perform 1000 permutations to the simulated data under Scenario S1 (unrelated individuals and common variants) and configuration C3 (strong LD with η = 0.7). Figure 4.7 shows the asymptotic null distributions of S _PC-ABT for the number of principal components q = 1, 25, and 50, together with the corresponding empirical CDFs obtained via permutation. Note that two different asymptotic distributions are shown in this figure, one is $\chi _q^2$, the other is a mixture of $\chi _1^2$ distribution, obtained by applying adaptive weights W ^# in the famSKAT method. In Fig. 4.8, panels a, b, and c, we compare in log scale the empirical p-values via permutation based approach against the p-values from the asymptotic distribution (mixture of $\chi _1^2$) for the number of principal components q = 1, 25, and 50, respectively. Panel d of Fig. 4.8 further reports the correlation between −log₁₀(empirical p-values via permutation) and −log₁₀(p-values based on the asymptotic distribution) for the number of principal components q = 1, 2, ⋯ , 50.

Fig. 4.7

Asymptotic null distribution and permutation based empirical distribution of S _PC-ABT (for the number of principal components q = 1, 25, and 50) under Scenario S1 and LD configuration C3

Fig. 4.8

Comparing empirical p-values via permutation and p-values based on the asymptotic distribution. Panel a: scatter plot in log scale for the number of principal components q = 1; Panel b: for q = 25; Panel c: for q = 50; Panel d: correlation between −log₁₀(empirical p-values via permutation) and −log₁₀(p-values based on the asymptotic distribution) for q = 1, 2, ⋯ , 50

Appendix 5: Additional Simulation Results for Type I Error Evaluation

We provide additional simulation results for type I error evaluation. Table 4.4 lists the empirical type I error rates of five testing methods: FBT, famSKAT, ABT, MONSTER, and PC-ABT for the combinations of four scenarios (S1, S2, S3, and S4) and four LD configurations (C1, C2, C3, and C4), based on 20,000 simulated data replicates. Figures 4.9, 4.10, 4.11, and 4.12 show the Q-Q plots of the PC-ABT p-values under the null hypothesis for Scenarios S1, S2, S3, and S4, respectively. The number of principal components is chosen to guarantee that the total percent variance explained (PVE) >90%.

Fig. 4.9

Q-Q plots of the PC-ABT p-values under the null hypothesis for Scenario S1 based on 20,000 simulated data replicates. In each simulation, the number of principal components is chosen such that PVE >90%

Fig. 4.10

Q-Q plots of the PC-ABT p-values under the null hypothesis for Scenario S2 based on 20,000 simulated data replicates. In each simulation, the number of principal components is chosen such that PVE >90%

Fig. 4.11

Q-Q plots of the PC-ABT p-values under the null hypothesis for Scenario S3 based on 20,000 simulated data replicates. In each simulation, the number of principal components is chosen such that PVE >90%

Fig. 4.12

Q-Q plots of the PC-ABT p-values under the null hypothesis for Scenario S4 based on 20,000 simulated data replicates. In each simulation, the number of principal components is chosen such that PVE >90%

Table 4.4 Empirical type I error of five testing methods