Application of quantile regression to recent genetic and -omic studies

Abstract

This paper provides a review of recent applications of quantile regression to the fields of genetic and the emerging -omic studies. It begins with a general background about this statistical approach following the seminal paper of Koenker and Bassett (Econometrica 46:33–50, 1978). Applications are described, as diverse as genetic association studies, penetrance estimation, gene expression, CGH array experiments, RNAseq experiments, methylation data and proteomics. This paper also introduces recent extensions of quantile regression with a particular focus on the Copula-quantile regression, an approach we recently proposed for sib-pair analysis. A real data example from eQTL analysis is then presented and the $R$ codes, which run the analyses are provided. Finally, we conclude with some statistical software presentation and some general statements about the potential and interests of quantile regression in modern biological experiments.

Appendix

Supplemental material: code R for the real data example

The code illustrates the application of quantile regression and Copula quantile regression to the CEPH data introduced in Sect. 4. Using Haseman and Elston (1972)’s approach, the goal is to regress the squared difference between two sibs’ trait values, where the trait corresponds to the gene expression for each of the three transcripts studied in Table 1, on the proportion of alleles shared identical-by descent at the SNP marker considered.

The database including the variables is ‘bb’. We used the variables: ‘trait1$X217225’ and ‘trait2$X217225’, which are the gene expression trait values of sib1 and sib2, respectively. The variable ‘IBD’ is the probability of alleles shared identical by descent for the pair of relatives at the SNP marker.

The following line of code runs QR treating the observations as independent (se=‘iid’). The value of the quantile is given by ‘tau’. The case ‘tau=0.5’ corresponds the the median regression. The command ‘summary’ prints the parameter estimates, standard errors and p values.

The following line of code runs QR using a Huber sandwich variance estimate for the parameters of interest (se=‘nid’).

The code below is the function for the Copula quantile regression using a Gaussian Copula (See Eq. 7 of “Appendix 2”). In this function, ‘ecdf’ computes an empirical cumulative distribution function, ‘pnorm’ is the normal density function, ‘qnorm’ is the normal cumulative distribution function and ‘rho’ is the correlation coefficient of the Copula function.

The following line of code runs the Copula quantile regression function given above. The function ‘nlrq’ corresponds to the non-linear quantile regression and ‘start’ gives the starting value of the parameter ‘rho’, i.e. the correlation, of the Gaussian Copula

Supplemental material: recent extensions of QR

Beyond the more standard approaches for QR presented in the previous sections, recent extensions of the methodology have also been proposed including support vector machine (SVM) quantile regression, the Copula quantile regression and nonlinear quantile regression.

Appendix 1: Support vector machine quantile regression

SVM algorithm developed by Vapnik (1998) is based on statistical learning theory. In classification problems, the objective is to determine an optimal hyperplane that separates classes. This is equivalent to maximizing the margin between classes. In support vector machine regression (SVR) (Scholkopf and Smola 2002; Sohn et al. 2008a, b), the goal is to determine a hyperplane that fits the data very well. The basic idea of SVR is to find a model function $f({\varvec{x}})$ representing the relationship between several characteristics associated with environmental and genomic information and a target such as microarray gene expression. Sohn et al. (2008a) applied SVM quantile regression (SVMQR) to cDNA microarray expression data to identify genes differentially expressed. They use three microarray data sets including experiments on a diet-induced obese (DIO) mouse model, data on E.coli and a last one on high-density lipoprotein (HDL)-deficient mouse. For each of this data set, the authors compared two treatment groups (e.g. high- vs. low-fat diet in the first experiment, drug A vs. drug B. in the second experiment and apoAI knockout mice vs. wild-type mice in the third experiment). The authors concluded that the SVMQR approach was superior to classical methods for identifying differentially expressed genes. In a companion paper, Sohn et al. (2008b) proposed to use SVMQR for print-tip normalization of the expression data. Their goal was to adjust systematic variations due to dye biases, where those biases depend on the spot overall intensity and/or spatial location within the array. Using the same microarray data sets, the authors showed that their novel approach outperformed methods based on Loess adjustment.

Appendix 2: Copula quantile regression

The problem of characterizing the dependence between random variables at a given quantile is an important issue, especially if the distributions of the variables are heavy-tailed. QR modeling approach based on a Copula function can handle the dependency between two variables, e.g. $X$ and $Y$. The idea is that the joint distribution $F_{X,Y}$ can be decomposed into the marginals distribution of $X$ and $Y$ denoted, respectively, by $F_X$ and $F_Y$ and the dependence function is specified by the Copula function (Bouyé and Salmon 2002). The form of the linear quantile relationship implied by the copula is deduced. According to the Sklar’s theorem (Sklar 1959), there exists a unique bivariate copula ${\varvec{C}}: [0,1]^2 \rightarrow [0,1]$ satisfying

$$\begin{aligned} F_{X,Y}(x, y)={\varvec{C}}\left( F_X(x), F_Y(y)\right) , \end{aligned}$$

where $F_X$ and $F_Y$ are continuous and represent the marginal distribution function of $X$ and $Y$ respectively. Different families of Copula correspond to different types of dependence structure (see Nelsen (1998), for a general introduction to Copulas). With Gaussian Copulas, the dependence is measured by a correlation $\rho$. The bivariate Gaussian Copula has the form

$$\begin{aligned} C(x,y; \rho )= \Phi _2 \left( \Phi ^{[-1]}(x), \Phi ^{[-1]}(y); \rho \right) \end{aligned}$$

where $\Phi _2$ corresponds to the bivariate Gaussian distribution, $\Phi$ to the univariate distribution and $\Phi ^{[-1]}$ to the pseudo-inverse of $\Phi$. So, by Sklar’s theorem, for marginal distribution functions $F_X$ and $F_Y$, the joint distribution

$$\begin{aligned} F _{X,Y}(x, y) = \Phi _2 \left( \Phi ^{[-1]}(F_X(x)), \Phi ^{[-1]}(F_Y(y)); \rho \right) \end{aligned}$$

is a bivariate distribution function with marginals $F_X$ and $F_Y$ and the Copula that connects $F_{X,Y}(x,y)$ to $F_X(x)$ and $F_Y(y)$ is the Gaussian Copula. For a parametric copula ${\varvec{C}}(.,.;\rho )$, the $\theta$-th copula quantile curve of $y$ conditional on $x$ is defined by

$$\begin{aligned} \theta = C^{\star }(F_X(x),F_Y(y) ; \rho ) \end{aligned}$$

(5)

where $C^{\star }(x,y;\rho ) = \frac{\partial }{\partial x} {\varvec{C}}(x,y; \rho )$. The relationship between $X$ and $Y$ can be expressed using (5) by

$$\begin{aligned} y = q(x,\theta ; \rho ) \end{aligned}$$

(6)

where $q(x,\theta ; \rho ) = F_Y^{[-1]}\left( D(F_X(x),\theta ; \rho ))\right)$ with $D$ the partial inverse in the second argument of $C^{\star }$ and $F_Y^{[-1]}$ the pseudo-inverse of $F_Y$. The relationship (6) can be also expressed using uniform margins as

$$\begin{aligned} v = r(u,\theta ; \rho ) \end{aligned}$$

(7)

with $u=F_X(x)$ and $v=F_Y(y)$. By (5) and (7), the $\theta$-th Gaussian copula quantile is defined by

$$\begin{aligned} \theta = \Phi \left( \frac{\Phi ^{[-1]}(v) -\rho \phi ^{[-1]} (u)}{\sqrt{1-\rho ^2}} \right) , \end{aligned}$$

and the relationship between $y$ and $x$ is

$$\begin{aligned} y=F_Y^{[-1]} \left( \Phi \left( \rho \,\Phi ^{[-1]}(F_X(x)) + \sqrt{1-\rho ^2}\,\Phi ^{[-1]}(\theta ) \right) \right) . \end{aligned}$$

(8)

The $\theta$-th copula QR $q(x_i,\theta ,\rho )$ is defined as any solution to the minimization problem

$$\begin{aligned} \min _{\rho } \left( \sum _{i\in {\mathcal F}_{\theta }} \theta |y_i - q(x_i,\theta ; \rho )| + \sum _{i \in {\mathcal F}_{1-\theta }} (1-\theta ) |y_i - q(x_i,\theta ; \rho )|\right) \end{aligned}$$

with ${\mathcal F}_{\theta } = \{i : y_i \ge q(x_i,\theta ;\rho )\}$ and ${\mathcal F}_{1-\theta }$ its complement.

Using Copula quantile regression, the dependence $\rho$ can be estimated at various quantiles. Besides, depending on the choice of the Copula function, the relationship between the two random variables can be nonlinear, offering some flexibility in the modeling of the dependence between these two variables. An application of the this method is given in Sect. 5.

Appendix 3: Nonlinear quantile regression

Different types of nonlinear QR approaches have been proposed. Bouyé and Salmon (2002) extended Koenker and Bassett (1978)’s work and proposed a nonlinear QR based on Copula function. QR has also been applied to censored survival (duration) data, which offers a more flexible alternative to the Cox proportional hazard model for some applications. Developments on censored QR were presented in Portnoy (2003) and Peng and Huang (2008). In particular, the Portnoy and Peng-Huang estimators can be both considered as regression-based generalizations of Kaplan-Meier and Nelson-Aalen estimator of the cumulative hazard function for randomly censored observations. Software implementations of the previous censored QR estimators for the $R$ language are available in the quantreg package of Koenker (2008) using the function “crq”. Koenker and Park (1996) proposed a procedure to compute QR estimates for problems in which the response function is nonlinear in parameters using interior point methods. As an example, Ho et al. (2009) presented a model selection approach to discover genes with linear or nonlinear age-dependent gene expression patterns from microarray data.