Testing and Estimation in Marker-Set Association Study Using Semiparametric Quantile Regression Kernel Machine

Summary

We consider quantile regression for partially linear models where an outcome of interest is related to covariates and a marker set (e.g., gene or pathway). The covariate effects are modeled parametrically and the marker set effect of multiple loci is modeled using kernel machine. We propose an efficient algorithm to solve the corresponding optimization problem for estimating the effects of covariates and also introduce a powerful test for detecting the overall effect of the marker set. Our test is motivated by traditional score test, and borrows the idea of permutation test. Our estimation and testing procedures are evaluated numerically and applied to assess genetic association of change in fasting homocysteine level using the Vitamin Intervention for Stroke Prevention Trial data.

1. Introduction

In this paper, we consider the problem of testing for association between a phenotype of interest and a marker-set (e.g. a gene or pathway), where the response values may contain outliers or have a skewed or heavy-tailed distribution, and the number of markers is typically large. We consider a semiparametric model where clinical and demographic covariates are modeled parametrically and the set of genetic variables is modeled jointly in a nonparametric way using statistical kernel machine. Instead of modeling the random error as a mean zero random variable, as is usually done in least squares regression setting, we consider the case where the (pre-specified) quantile of the errors is assumed to be zero. We are interested in two aspects of the model, namely the estimation of the effects of both the genetic and clinical covariates, and testing for association between the genetic covariates and the response.

As a popular approach to semiparametrically model multi-dimensional covariates in non-parametric and semiparametric models, kernel machine (KM) methods (Vapnik, 1998; Scholkopf and Smola, 2001) have emerged as a powerful tool in association studies. KM methods can accommodate high number of covariates as well as complex relationships easily while allowing to perform hypothesis testing. For the semiparametric models with continuous response, Liu et al. (2007) and Kwee et al. (2008) developed least squares kernel machine (LSKM) regression models and showed its attractive connection with linear mixed models. They also proposed a score based testing procedure for the pathway effect. There has been a growing number of applications and extensions of such models in recent years (Wu et al., 2010, 2011; Maity and Lin, 2011; Monsees et al., 2011; Meyer et al., 2012; Maity et al., 2012).

While LSKM is a powerful regression tool, it has some serious limitations. First, the parameter estimation in LSKM is performed by minimizing a penalized least squares criterion. This criterion can be very sensitive if the data are generated from skewed/heavy tailed distributions. Second, LSKM only models the conditional mean of the response given the covariates. Thus one can only detect association in terms of mean. However, any association between the response and covariates in different quantile levels may not be detected. This issue is particularly important when the covariates only have effect on particular subsets of samples having higher or lower response levels rather than the overall mean. To this end, we propose a semiparametric quantile regression model using the kernel machine framework.

Quantile regression has been widely used in many areas, such as survival analysis (Yang, 1999), microarray study (Wang and He, 2007) and economics (Hendricks and Koenker, 1992). For an extensive review of the quantile regression, we refer to Koenker (2005). Compared to the ordinary least squares estimates, quantile regression, for example median regression, provides estimates that are robust to outliers. Quantile regression reveals how the covariates influence the location, scale and shape of the response distribution, and one can investigate the effect of the covariates onto different quantiles of response, which is more informative than the least squares. There is a rich literature on nonparametric and semiparametric quantile regression. Yu and Jones (1998) used kernel smoothing based local polynomial regression (Fan and Gijbels, 1996) to develop an estimation procedure for nonparametric quantile regression. Such techniques are later generalized for partially linear quantile regression models (Lee, 2003; Sun, 2005; Liang and Li, 2009; Song et al., 2012). Most of these procedures only consider the case where the nonparametrically modeled variable is univariate. While Lee (2003) and Sun (2005) discuss theoretical results for multivariate covariates, it is well known that the performance deteriorates sharply as dimension increases, see e.g., Bickel and Li (2007). Other nonparametric estimation procedures for quantile regression include smoothing splines models (Koenker et al., 1994; Nychka et al., 1995; Yuan, 2006; Wu and Yu, 2014), elastic and plastic splines (Koenker and Mizera, 2002) for bivariate covariates, and estimation based on total variation regularization (Koenker and Mizera, 2004). These methods work well when the nonparametrically modeled covariate is scalar or low-dimensional. However, for moderate or high dimensional covariates, such methods are often computationally not feasible or very intensive. In addition, none of the above mentioned techniques provide any procedure to perform hypothesis testing for the effect of the nonparametrically modeled covariate. In the context of quantile regression using Reproducing Kernel Hilbert Space approach, Takeuchi et al. (2006) and Li, Liu, and Zhu (2007) considered a purely nonparametric quantile regression model and developed a solution path algorithm with respect to the tuning parameter to estimate the model components. Liu and Wu (2011) considered the simultaneous non-crossing quantile regression for the same model. However they neither develop any testing procedure to test for association nor propose any estimator of standard errors of the resulting estimates. In this article, we will develop estimation procedure for the model components along with their standard errors, and propose a testing procedure to assess association between the genetic covariate and the response.

Our research is motivated by the Vitamin Intervention for Stroke Prevention (VISP) trial (Toole et al., 2004), a multicenter, double-blind, randomized, controlled clinical trial that aimed to study the effect of vitamins on preventing recurrent stroke. Details about the study are provided in Section 5. We are interested in evaluating the effect of 9 candidate genes involved in the Hcy metabolic pathway on the Hcy level (Hsu et al., 2011) as it has been suggested that Hcy level can be used to predict risk of recurrent stroke (Toole et al., 2004; Pettigrew et al., 2008), and genetic variations might be attributed to mild to moderate hyperhomocystinemia (Sharma et al., 2006; Fredriksen et al., 2007). When we used LSKM, no genes in the Hcy pathway are found significant after multiple testing adjustment. This motivated us to propose a quantile regression kernel machine (QRKM) to test for gene effects on one of the quantiles. When using our QRKM to explore further, we found that Hcy level is significantly associated with gene CBS at quantile levels of 0.5 and 0.8, and with gene TCN1 at quantile level of 0.8, after adjusting for multiple tests performed at different genes and quantiles.

We make three major contributions in this article. First, we develop a simple and fast algorithm to solve the semiparametric model for a fixed tuning parameter. Second, we introduce a bootstrap based tuning method, which provides stable selection results and can provide the standard errors of the estimates of the model components with no extra computation cost. Finally, we develop a procedure for testing the joint effect of genetic variables under the semiparametric quantile regression framework. Since the loss function of the quantile regression model is nonsmooth, we can not use the score test in kernel machine literature. Instead, we propose a test statistic based on the subgradient of the check function, and develop a permutation method to compute p-values. To the best of our knowledge, this is first such methodology in the quantile regression kernel machine literature.

2. Penalized Quantile Regression Estimation using Kernel Machines

Suppose we observe n independent triples (y_i, X_i, Z_i) where X_i is a vector of p covariates, y_i is a continuous response and Z_i is a vector of q covariates. In our motivating data, y_i denotes change in Hcy level, X_i denotes genotype of a set of SNPs and Z_i is a vector of age and sex of the individual. We consider a partially linear model to relate the response to the clinical covariates and the genetic covariates:

$$y_i=f({\mathbf{X}}_i)+{\mathbf{Z}}_i^{\top }\mathit{\beta }+{\beta }_0+{\epsilon }_i,$$ (2.1)

where β = (β₁,…, β_q)^⊤ denotes the covariate effect, β₀ denotes the intercept effect, f(·) is an unknown centered function quantifying the genetic effect, and ε_i is the random error.

We consider a quantile regression model, where for a fixed value τ we assume the τth quantile of ε_i conditional on X_i and Z_i is assumed to be zero. As we have an intercept term in the model, we assume that E(f(·)) = 0 for identifiability. The parameters β, β₀ and f(·) can be estimated from the following minimization problem

$$\underset{{\beta }_0,\mathit{\beta },f}{min}{\sum }_{i=1}^n{\rho }_{\tau }(y_i-f({\mathbf{X}}_i)-{\mathbf{Z}}_i^{\top }\mathit{\beta }-{\beta }_0),$$ (2.2)

where ρ_τ(r) = τr1(r ≥ 0) + (τ – 1)r1(r < 0) is the check function and 1(·) denotes the indicator function. Typically, one assumes a parametric form for f(·) and model the SNP effects. For example, $f({\mathbf{X}}_i)={\mathbf{X}}_i^{\top }\mathit{\eta }$ for some unknown parameter vector η corresponds to a linear model with main SNP effects only. Such parametric assumptions may be too strong and may not work well if the true underlying effect is nonlinear. To allow for more flexibility, we assume f(·) has a nonparametric form. However, the solution of (2.2) is not unique if we do not put any constraints on the unknown function f(·).

To solve the problem of over-fitting, we consider the penalized version of (2.2)

$$\underset{{\beta }_0,\mathit{\beta },f\in H_K}{min}{\sum }_{i=1}^n{\rho }_{\tau }(y_i-f({\mathbf{X}}_i)-{\mathbf{Z}}_i^{\top }\mathit{\beta }-{\beta }_0)+P_{\lambda }(f),$$

where P_λ (f) is a penalty function which reflects the smoothness of f(·). One commonly used penalty function is λ ∫ f″(x)² dx, which is analogous to classical least squares smoothing spline model pioneered by Wahba (1990) and Gu (2002). Here λ is a penalty parameter controlling smoothness of f(·). In this article, we assume the function f(·) resides in a functional space H_K generated by a positive definite kernel function K (·, ·). From Mercer's Theorem (Cristianini and Shawe-Taylor, 2000), there is a one-to-one correspondence between a positive definite kernel function and H_K under some regularity conditions. We can expand f(·) using the basis functions in H_K, where the basis functions can be represented using the kernel function. By Representer Theorem (Kimeldorf and Wahba, 1971), the solution for the nonparametric function f(·) can be written using the dual representation

$$f(\mathbf{X})={\lambda }^{-1}{\sum }_{i=1}^n{\theta }_{i}K(\mathbf{X},{\mathbf{X}}_i),$$ (2.3)

where θ = (θ₁, …, θ_n)^⊤ are unknown parameters.

For the τth quantile, our estimation of f(·), β and β₀ is based on minimizing

$$\underset{{\beta }_0,\mathit{\beta },f\in H_K}{min}{\sum }_{i=1}^n{\rho }_{\tau }(y_i-f({\mathbf{X}}_i)-{\mathbf{Z}}_i^{\top }\mathit{\beta }-{\beta }_0)+\lambda {\Vert f\Vert }_{H_K}^2/2,$$ (2.4)

where the penalty function $\frac{\lambda }{2}{\Vert f\Vert }_{H_K}^2$ is determined by the kernel function to control the roughness of the function. Combining (2.3) and (2.4), the optimization problem becomes

$$\underset{\mathit{\beta },{\beta }_0,\mathit{\theta }}{min}{\sum }_{i=1}^n{\rho }_{\tau }(y_i-{\beta }_0-{\lambda }^{-1}{\sum }_{i\prime =1}^n{\theta }_{i\prime }K({\mathbf{X}}_i,{\mathbf{X}}_{i\prime })-Z_i^{\top }\mathit{\beta })+{(2\lambda )}^{-1}{\sum }_{i,i\prime =1}^n{\theta }_i{\theta }_{i\prime }K({\mathbf{X}}_i,{\mathbf{X}}_{i\prime }).$$ (2.5)

With a slight abuse of notation, define a matrix K such that the (i, i′)-th element is K (X_i, X_i′). We show that the solution of (2.5) can be obtained from a quadratic programming problem.

Theorem 1: For a fixed λ, the minimizer of (2.5) can be found by solving

$$\underset{\mathit{\theta }}{max}\{-{(2\lambda )}^{-1}{\mathit{\theta }}^{\top }\mathbf{K}\mathit{\theta }+{\mathit{\theta }}^{\top }\mathbf{y}\},$$ (2.6)

subject to −(1 – τ) ≤ θ_i ≤ τ, for 1 ≤ i ≤ n, ${\sum }_{i=1}^n{\theta }_i=0,\text{and}{\sum }_{i=1}^n{Z}_{il}{\theta }_i=0,\text{for}1\le l\le q$.

The proof of the theorem is presented in the Supplementary Materials. After we obtain the solution θ̂ from (2.6), we plug the solution into (2.5) and solve for β and β₀ in

$$\underset{\mathit{\beta },{\beta }_0}{min}{\sum }_{i=1}^n{\rho }_{\tau }(y_i^{\prime }-{\beta }_0-Z_i^{\top }\mathit{\beta }),$$

where $y_i^{\prime }=y_i-\frac{1}{\lambda }{\sum }_{i\prime =1}^n{\widehat{\theta }}_{i\prime }K({\mathbf{X}}_i,{\mathbf{X}}_{i\prime })$. For the nonparametric function f(·), we plug in the estimate θ̂ into (2.3) and get the estimate f̂(·). We used quantreg package in R to solve the above regression problem and the quadratic problem in (2.6).

The regularization parameter λ plays an important role in controlling the smoothness of the function f(·). We discuss the selection of the tuning parameter in Web Appendix C.

3. Testing for the Joint Effect

In biomedical studies, evidence of association between a gene and a response is as valuable as estimation of the actual effect. Our goal is to test the hypothesis H₀ : f(·) = 0. Note that we have assumed that E(f(·)) = 0 for identifiability, this hypothesis is equivalent to testing whether X_i has a constant effect or not. Using LSKM, Liu et al. (2007) tested the whole genetic effects using the score test, where they assume ε_i ∼ N(0, σ²). Since the check function is nonsmooth, we can not directly apply the score test under the least squares case. Instead, we propose a score type of test statistic using the subgradient of the check function. Recall that we have n independent triples (Y_i, X_i, Z_i). We fit a linear quantile regression using only (Y_i, Z_i), that is to fit the null model. Let ${\widehat{u}}_i=Y_i-{\widehat{\beta }}_0-{\mathbf{Z}}_i^{\top }\widehat{\mathit{\beta }}(1\le i\le n)$, and define w_i = τ if û_i > 0, and w_i = τ – 1 if û_i < 0. For those û_i = 0, we assign the corresponding w_i = τ with probability 1 – τ, and w_i = τ – 1 with probability τ. Our proposed test statistic is

$$T={\mathbf{w}}^{\top }\mathbf{K}\mathbf{w},$$

where K is the aforementioned kernel matrix and w = (w₁, …, w_n)^⊤.

To get the p-value and perform hypothesis testing, we need to get the null distribution of the test statistic T. As our test statistic T depends on the binary random variable w_i, the distribution of T is no longer mixture of chi-square distribution as the least squares case. We apply a permutation based procedure to empirically obtain the distribution of T. We first fit our quantile regression kernel machine procedure on the data {(Y_i, X_i, Z_i), i = 1, …, n} and get the residuals ${\widehat{\upsilon }}_i=Y_i-\widehat{f}({\mathbf{X}}_i)-{\mathbf{Z}}_i^{\top }\widehat{\mathit{\beta }}-{\widehat{\beta }}_0(1\le i\le n)$.

Step 1: Permute υ̂_i (1 ≤ i ≤ n) without replacement and a new permutation ${\widehat{\upsilon }}_i^{\ast }$.

Step 2: Add ${\widehat{\beta }}_0+{\mathbf{Z}}_i^{\top }\widehat{\mathit{\beta }}\text{to}{\widehat{\upsilon }}_i^{\ast }$ and get the mimic data $Y_i^{\ast }$.

Step 3: Use $(Y_i^{\ast },{\mathbf{Z}}_i)$ to do a linear quantile regression and get the new residuals ${\widehat{u}}_i^{\ast }$. Obtain the $w_i^{\ast }$ using the same rule as w_i, where we replace û_i by ${\widehat{u}}_i^{\ast }$. Denote ${\mathbf{w}}^{\ast }={(w_1^{\ast },\dots ,w_n^{\ast })}^{\top }$.

Step 4: Calculate the mimic test statistic, T* = w*^⊤Kw*.

Repeat the above steps for N times, and we obtain the $T_1^{\ast },\dots ,T_N^{\ast }$, which mimics the null distribution of the test statistic. Finally, the p-value can be calculated as $p–\text{value}={\sum }_{i=1}^{N}1(T_i^{\ast }\ge T)/N$, where 1(·) denotes the indicator function.

4. Simulation Study

We conduct a simulation study to evaluate the performance of our proposed procedure. We investigate two aspects of the model, namely the estimation of the effects of both X_i and Z_i, and testing for the effect of X_i.

4.1 Simulation for Estimation

We generate data from the model in (2.1) where Z_i = (Z_i1, Z_i2)^⊤ were generated from a standard bivariate normal distribution. We generate X_i using the same frequency distribution of the SNPs as found on the MTR gene in the real data application (p = 20 SNPs). We set the true value of β = (1, 1)^⊤, and β₀ = 0. Define $g({\mathbf{X}}_i)=1+{\sum }_{k=1}^p{X}_{ik}{\eta }_k+{\sum }_{k=2}^p{X}_{i1}{X}_{ik}{\gamma }_k$. with η₁ = 0.4, η₂ = … = η_p = 0.7, γ₂ = … γ_P = 0.2. We consider f₁(·) = g(·) and f₂(·) = sign{g(·)}|g(·)|^1/2 as our true functions. For ε_i, we consider the standard normal, t (with degrees of freedom 3) and ${\chi }_1^2-1$ distributions. We consider the sample size n = 200. For the quantile, we use τ = 0.1, 0.5, and 0.8. We use the identity-by-state (IBS) kernel (Wessel and Schork, 2006) in our simulation. We use LSKM as a benchmark approach with five fold cross validation to tune the regularization parameter.

We run 1000 Monte Carlo repetitions and report the mean and standard deviation of the β estimates, which are vectors of length 2. We also record the bootstrap standard deviation, which is a byproduct of the tuning process to compare with the numerical study. For LSKM, since we do not use bootstrap tuning, we do not report this quantity, and we present the result using “NA”. We also record the mean absolute deviation (MAD) as $\frac{1}{n}{\sum }_{i=1}^n|\overline{f}({\mathbf{X}}_i)-\overline{\widehat{f}}({\mathbf{X}}_i)|$, where f̄ is the centered function for f and $\overline{\widehat{f}}$ is the centered estimated function.

The results are in Table 1. We can see that our method performs well in estimating the parameters for different quantiles and different functions. The bootstrap estimate of the standard deviation is close to the standard deviation obtained from 1000 Monte Carlo estimates, which suggests that we can obtain reasonable variance estimates of the parameter by our bootstrap tuning method. We can also see that MADs are quite different for f₁(·) and f₂(·) because the true f₁(·) and f₂(·) are not in the same magnitude. For normal error, we have found that LSKM can achieve smaller MAD as well as smaller standard deviation of β than QRKM. This is expected because the error ε_i's are generated from normal distribution, and least squares estimates are more efficient than quantile regression method under this scenario. For different error distributions, i.e. t-distribution with degrees of freedom 3 and ${\chi }_1^2-1$, the findings are similar as the normal error case, except that sometimes our QRKM may have smaller MAD and standard deviation of β at certain quantiles than LSKM. The possible reason is that when the error deviates from normal, least squares estimates may be less efficient than quantile regression under certain scenarios.

Table 1.

Simulation results of our estimation procedure compared with LSKM for categorical covariates using the same frequency distribution as that on the MTR gene. Displayed are average estimates for β (column 4), empirical and bootstrap estimated standard error (columns 5-6), mean absolute deviation (MAD) (column 7) for different error distributions (normal, t and chi-squares), different functions f₁(·) and f₂(·) and different quantiles τ = 0.1, 0.5, 0.8. The true value of β is (1,1)^T, and the simulation is based on 1000 Monte Carlo repetitions.

Method	Distribution	f	mean of β	sd of β	bootstrap sd	MAD
QRKM τ = 0.1	normal	f1	(1.01,1.00)	(0.18,0.17)	(0.17,0.17)	1.03
QRKM τ = 0.5	normal	f1	(1.00,0.99)	(0.11,0.12)	(0.14,0.14)	0.57
QRKM τ = 0.8	normal	f1	(1.00,1.00)	(0.15,0.16)	(0.16,0.16)	0.60
LSKM	normal	f1	(1.00,1.00)	(0.08,0.08)	NA	0.35
QRKM τ = 0.1	normal	f2	(1.01,1.00)	(0.13,0.13)	(0.12,0.12)	0.26
QRKM τ = 0.5	normal	f2	(1.00,1.00)	(0.09,0.09)	(0.09,0.10)	0.22
QRKM τ = 0.8	normal	f2	(1.00,1.00)	(0.11,0.11)	(0.10,0.10)	0.24
LSKM	normal	f2	(1.00,1.00)	(0.08,0.07)	NA	0.21
QRKM τ = 0.1	t	f1	(1.01,1.01)	(0.26,0.25)	(0.24,0.24)	1.33
QRKM τ = 0.5	t	f1	(1.00,1.00)	(0.14,0.15)	(0.16,0.16)	0.68
QRKM τ = 0.8	t	f1	(1.00,1.00)	(0.21,0.22)	(0.22,0.22)	0.82
LSKM	t	f1	(1.01,1.00)	(0.15,0.14)	NA	0.49
QRKM τ = 0.1	t	f2	(1.01,1.00)	(0.20,0.20)	(0.19,0.19)	0.30
QRKM τ = 0.5	t	f2	(1.00,1.00)	(0.10,0.10)	(0.11,0.11)	0.23
QRKM τ = 0.8	t	f2	(1.00,1.00)	(0.14,0.14)	(0.14,0.14)	0.26
LSKM	t	f2	(1.00,1.00)	(0.14,0.14)	NA	0.35
QRKM τ = 0.1	${\chi }_1^2-1$	f1	(1.00,1.00)	(0.09,0.10)	(0.10,0.10)	0.61
QRKM τ = 0.5	${\chi }_1^2-1$	f1	(1.00,1.00)	(0.11,0.11)	(0.13,0.13)	0.46
QRKM τ = 0.8	${\chi }_1^2-1$	f1	(1.00,1.01)	(0.25,0.24)	(0.25,0.24)	0.89
LSKM	${\chi }_1^2-1$	f1	(1.01,1.00)	(0.11,0.12)	NA	0.43
QRKM τ = 0.1	${\chi }_1^2-1$	f2	(1.00,1.00)	(0.03,0.03)	(0.04,0.04)	0.09
QRKM τ = 0.5	${\chi }_1^2-1$	f2	(1.00,1.00)	(0.07,0.07)	(0.08,0.08)	0.19
QRKM τ = 0.8	${\chi }_1^2-1$	f2	(1.01,1.01)	(0.20,0.19)	(0.18,0.18)	0.30
LSKM	${\chi }_1^2-1$	f2	(1.00,1.00)	(0.10,0.11)	NA	0.29

We have considered two other cases, namely, (1) a different gene, CBS (p = 10) is used to simulate X_i and IBS kernel was used, and (2) X_i are generated from a continuous distribution and gaussian kernel was used. Detailed descriptions and the results are in Section B in the Supplementary Materials. The findings of both cases are similar as the MTR case.

4.2 Simulation for Testing

The simulation settings are the same as in Section 4.1 with the modification that the data are generated from the model

$$Y_i=c\ast h({\mathbf{X}}_i)+{\mathbf{Z}}_i^{\top }\mathit{\beta }+{\epsilon }_i,$$

where h(·) is either f₁(·) or f₂(·), and c quantifies departure from H₀. For each generated data set, we fit our quantile regression model in (2.1) and test for f(·) = 0. We investigate our testing procedure for different quantile levels τ = 0.1, 0.5, 0.8. Note that when c = 0, it corresponds to the null hypothesis of no effect of X. We set different critical values α = 0.001, 0.01, 0.05 to check the type-I error using our permutation type of test. When c = 0, the true model does not involve X, so the type-I error would be the same for f₁ and f₂ functions. To examine power, we consider different c values for f₁ (c = 0.1, 0.2, 0.3, 0.4, 0.5) and f₂ (c = 0.6, 1.2, 1.8, 2.4, 3). We set different c values for f₁ and f₂ respectively because the magnitudes of f₁ and f₂ are different. We compare our test with the score test in LSKM.

Table 2 shows the type I error analysis results based on 10000 replicates. It is clear that our testing procedure has reasonable type-I error rate under different settings. This is true when the error distribution changes from standard normal to t and chi-squared as well. However, for the LSKM test, we can see that the type-I error rate is at the nominal level when the error distribution is normal, however, the type-I error rates are inflated when the errors deviate from normal distribution, especially for nominal level 0.001 and 0.01. This indicates that we should use LSKM with caution when the error is not normal. For our QRKM test, it is robust to the error distributions.

Table 2.

Displayed are the Type-I error of our proposed test for different quantiles compared with LSKM test for different nominal levels and different error distributions (normal, t and chi-squares). We consider categorical covariates using the same frequency distribution as that on the MTR gene. The simulation is based on 10000 Monte Carlo repetitions.

Method	distribution	0.001	0.01	0.05
QRKM τ = 0.1	normal	0.0003	0.0089	0.0498
QRKM τ = 0.5	normal	0.0008	0.0113	0.0496
QRKM τ = 0.8	normal	0.0007	0.0096	0.0473
LSKM	normal	0.0015	0.0103	0.0514
QRKM τ = 0.1	t	0.0005	0.0098	0.0499
QRKM τ = 0.5	t	0.0005	0.0102	0.0501
QRKM τ = 0.8	t	0.0009	0.0093	0.0471
LSKM	t	0.0056	0.0175	0.0582
QRKM τ = 0.1	${\chi }_1^2-1$	0.0009	0.0107	0.0492
QRKM τ = 0.5	${\chi }_1^2-1$	0.0010	0.0096	0.0475
QRKM τ = 0.8	${\chi }_1^2-1$	0.0007	0.0112	0.0483
LSKM	${\chi }_1^2-1$	0.0058	0.0208	0.0546

We have also plotted the power curves for nominal level 0.05 in Figure 1. From the results, we can see that for symmetric error distributions (normal and t), quantile 0.5 has the largest power, followed by quantile 0.8. For quantile 0.1 and LSKM, their powers are comparable. For chi-square error distribution, since it is right skewed, the quantile 0.1 and quantile 0.5 have comparable larger power, followed by quantile 0.8, and the LSKM has the lowest power. From the results, we can see that our testing procedure is generally more powerful than the LSKM test for the categorical covariates.

Figure 1.

Results from simulation study: the categorical covariates using the same frequency distribution as that on the MTR gene. Displayed are the power curves of QRKM for quantiles τ = 0.1, 0.5, 0.8 and the LSKM comparison test for different functions f₁ and f₂ and different error distributions (normal, t and chi-squares) based on 500 Monte Carlo repetitions. The first row corresponds to function f₁, the second row corresponds to function f₂. The three columns from left to right correspond to normal, t and chi-squared random errors. The dashed, dotted, dash dotted and solid lines correspond to the power curves of QRKM for quantiles τ = 0.1, τ = 0.5, τ = 0.8 and LSKM test respectively.

We repeated the above testing simulations using a different gene, CBS gene (p = 10), to simulate the categorical covariates X_i. The results are in Section B.1 in the Supplementary Materials, and the findings are similar as the MTR case. We have also considered the continuous covariates X_i for our testing procedure. The detailed description as well as the results are in Section B.2 in the Supplementary Materials. We found that, for continuous covariates, our testing procedure has reasonable type-I error rates as expected; however, the type-I error rates for LSKM are more stable for continuous covariates (using gaussian kernel) compared the categorical covariates (using IBS kernel). From the power results, our QRKM test may achieve larger or smaller power than LSKM test for continuous covariates under different scenarios. Since LSKM is testing the association in terms of mean, while our method is testing association at lower or higher quantiles, sometimes LSKM may have larger power because the two methods are testing different hypothesis.

5. Application to the Vitamin Intervention for Stroke Prevention Trial

To illustrate the proposed methodology, we applied the QRKM on samples collected from the VISP trial. The trial enrolled patients who were 35 or older with a nondisabling cerebral infarction within 120 days of randomization and Hcy levels in the top quartile for the U.S. population. Subjects were randomly assigned to receive daily doses of either a high-dose formulation (containing 25 mg vitamin B6, 0.4 mg vitamin B12, and 2.5 mg folic acid) or a low-dose formulation (containing 200 mg vitamin B6, 6 mg vitamin B12, and 20 mg folic acid). The patients were followed up for a maximum of 2 years. The average follow-up time was 1.7 years. A total of 2100 VISP participants were enrolled in VISP genetic study and genotype information was collected from 9 candidate genes that are involved in homocysteine metabolism. Our analysis here focused on the genetic influence on the Hcy level obtained from a 2 hr methionine load test measured at baseline. After deleting the subjects with missingness, we obtain 1587 subjects. We assess the joint effect of each of the 9 genes, adjusting for the age, sex and the population stratification (the first 10 PCs of all genes) as did in the original study (Hsu et al., 2011). We used the IBS kernel to summarize the multi-locus information. In QRKM, we separately considered the Hcy level at the quantiles of 0.1, 0.5 and 0.8 as our response variable in Table 3.

Table 3.

Results from the real data application. The number of SNPs on each gene is reported in column 2. Columns 3-5 display the p-values of our proposed test for 9 genes on different quantiles 0.1, 0.5, 0.8. The last column report the p-values of the LSKM test.

Genename	SNP number	τ = 0.1	τ = 0.5	τ = 0.8	LSKM
BHMT	5	0.072	0.108	0.263	0.413
BHMT2	4	0.222	0.371	0.748	0.514
CBS	10	0.013	0.0004	0.0004	0.046
CTH	10	0.559	0.432	0.820	0.039
MTHFR	9	0.745	0.478	0.429	0.509
MTR	20	0.785	0.864	0.572	0.446
MTRR	5	0.020	0.070	0.860	0.170
TCN1	3	0.808	0.218	0.0006	0.073
TCN2	20	0.455	0.622	0.835	0.314

From the results, we see that the gene CBS is significant at the quantiles 0.5 and 0.8, and gene TCN1 is significant at the quantile 0.8 after Bonferroni correction (i.e., 0.05/(9 × 3) = 0.00185) using our QRKM method. For the LSKM method, after Bonferroni correction (i.e., 0.05/9 = 0.0056), none of the genes are significant. Gene TCN1 encodes a member of the vitamin B12-binding protein family. It is associated with vitamin B12 deficiency. Several genome-wide association studies have also shown that SNPs in gene TCN1 were associated with vitamin B12 (Grarup et al., 2013; Tanaka et al., 2009). Low intake of B12 level may be a risk factor for elevated Hcy level (Kalita et al., 2009). Additionally, gene TCN1 may influence transcobalamin blood level and is a weak determinant of Hcy level (Namour et al., 2001). Further, gene CBS is a member of the folate one-carbon metabolism pathway. This pathway may mediate many biological processes in the cell, such as methionine metabolism and Hcy synthesis. Gene CBS provides instructions for making an enzyme which is responsible for using vitamin B6 to convert Hcy to cystathionine (Yi et al., 2000). This gene was also shown to be associated with Hcy level (Lievers et al., 2001; Zinck et al., 2015). As the LSKM test is built based on the conditional mean while our QRKM focuses on different quantiles, in practical situations, it might happen that the association is not through the mean but through the quantiles. Our QRKM test can detect such associations while the mean based LSKM test might not. Ideally, one could employ all these tests to cover various types of associations.

For estimation, we pick up the TCN1 gene. Out of the 3 SNPs, there are 14 distinct multi-locus genotypes. We plot the estimates of the effects across different quantiles 0.1, 0.5 and 0.8 for different genotypes, see Figure 2(a). We then proceed with effect estimation of the multi-locus genotypes on the Hcy quantile of 0.8, which is found to be significant in our testing procedure. In Figure 2(b), we report the effect estimates of each genotype and their 95% confidence intervals constructed using the standard error obtained by the bootstrap.

Figure 2.

Effect estimation of TCN1 gene in VISP data analysis. Panel (a) plots the estimated effects for each of the 3-SNP genotypes across different quantiles 0.1, 0.5 and 0.8, represented by dashed, dotted and dash dotted lines respectively. Panel (b) plots the estimated effects and confidence intervals for different genotypes when quantile level is 0.8.

We have included the frequencies of all the genotypes in Web Table 6 in the Supplementary Materials, and they maintain the same order as those in Figure 2. The genotypes are displayed by their order of appearance in the VISP data. We observe several genotypes with the effect confidence intervals that do not intersect with 0 for quantile 0.8, and we mark them bold. The genotype specific information on the effect sizes may facilitate the comprehension of the possible mechanisms that lead to the global significance of TCN1.

6. Discussion

We develop estimation and testing procedures for semiparametric quantile regression under the kernel machine framework. The usual least squares based LSKM test provides association testing based on conditional mean response. In contrast, our proposed procedure investigates association at pre-specified quantile level(s). In practical situations, it is possible that the marker set has no significant association in terms of mean response, and the actual association is at a lower or higher quantile of the response. Thus these two tests (LSKM and quantile based test) are detecting two different types of association. Ideally, in practical situations, one could employ both these tests to capture a wide variety of association rather than just focusing on mean based tests.