A Powerful Test for SNP Effects on Multivariate Binary Outcomes Using Kernel Machine Regression

Abstract

Evaluating multiple binary outcomes is common in genetic studies of complex diseases. These outcomes are often correlated because they are collected from the same individual and they may share common marker effects. In this paper, we propose a procedure to test for effect of a single nucleotide polymorphism-set on multiple, possibly correlated, binary responses. We develop a score-based test using a non-parametric modeling framework that jointly models the global effect of the marker set. We account for the non-linear effects and potentially complicated interaction between markers using reproducing kernels. Our testing procedure only requires estimation under the null hypothesis and we use multivariate generalized estimating equations to estimate the model components to account for the correlation among the outcomes. We evaluate finite sample performance of our test via simulation study and demonstrate our methods using the Clinical Antipsychotic Trials of Intervention Effectiveness antibody study data and the CoLaus study data.

Appendix

From Sect. 2.1, the parameters ${\varvec{\beta }}$ and h in (1) can be estimated by maximizing the penalized log-likelihood using a Fisher scoring or a Newton–Raphson algorithm. [24] show that, by treating ${\varvec{\beta }}$ as a vector of fixed effects and $\mathbf{h } = (h_{1},\ldots , h_{n})^\mathrm{T}$ as a vector of random effects, the logistic KM estimator is the same as the penalized quasi-likelihood estimator from a logistic mixed model $\text {logit}(p_{i}) = \mathbf{x }_{i}^\mathrm{T} {\varvec{\beta }} + h_{i},$ where $\mathbf{h } \sim N(\mathbf 0 ,\, \tau \mathbf{K }),\, \tau = 1/ \lambda ,\,\lambda$ is the penalty parameter from the penalized likelihood, and $\mathbf{K }$ is a square matrix whose $(i,\,j)$th element is (2). The normal equations given in (5) of [24] coincide with iteratively fitting a working linear mixed model $\widetilde{\mathbf{y }} = \mathbf{X } {\varvec{\beta }} + \mathbf{h } + \varvec{\varepsilon }$ until convergence, where ${\varvec{\beta }}$ and $\mathbf{h }$ are estimated using BLUE and BLUP, respectively and $\varvec{\varepsilon } \sim \text {N}(\mathbf 0 ,\, \mathbf{D })$ where $\mathbf{D } = \text {diag}\{p_{i}(1-p_{i})\}.$ The regularization parameter $\tau$ can be estimated by treating it as a variance component and maximizing the REML criterion

$$\begin{aligned} \ell \approx {-}\frac{1}{2} \text {log }\left| \mathbf{V }_{u}\right| - \frac{1}{2} \text {log }\left| \mathbf{X }^\mathrm{T} \mathbf{V }_{u}^{-1} \mathbf{X }\right| - \frac{1}{2} (\widetilde{\mathbf{y }} - \mathbf{X } \widehat{{\varvec{\beta }}})^\mathrm{T} \mathbf{V }_{u}^{-1} (\widetilde{\mathbf{y }} - \mathbf{X } \widehat{{\varvec{\beta }}}), \end{aligned}$$

(8)

where $\mathbf{V }_{u} = \mathbf{D }^{-1} + \tau \mathbf{K }$ and $\widetilde{\mathbf{y }} = \mathbf{X } {\varvec{\beta }} + \mathbf{h } + \mathbf{D }^{-1}(\mathbf{y } - \mathbf p ).$ We refer to [24] for full details.

Testing the overall genetic effect $H_{0}{\text {:}}\,h(\mathbf{z }) = 0$ for UV responses is equivalent to testing $H_{0}{\text {:}}\, \tau = 0.$ Liu et al. [24] propose the following score test statistic based on the derivative of (8) with respect to $\tau$

$$\begin{aligned} S = \frac{Q(\widehat{{\varvec{\beta }}}_{0}) - p_{Q}}{\sigma _{Q}}, \end{aligned}$$

(9)

where $Q(\widehat{{\varvec{\beta }}}_{0}) = (\widetilde{\mathbf{y }} - \mathbf{X } \widehat{{\varvec{\beta }}}_{0})^\mathrm{T} \mathbf{D } \mathbf{K } \mathbf{D } (\widetilde{\mathbf{y }} - \mathbf{X } \widehat{{\varvec{\beta }}}_{0}) = (\mathbf{y } - \widehat{\mathbf{p }}_{0})^\mathrm{T} \mathbf{K } (\mathbf{y } - \widehat{\mathbf{p }}_{0}),\, \text {logit} (\widehat{\mathbf{p }}_{0}) = \mathbf{X } \widehat{{\varvec{\beta }}}_{0},\, \widehat{{\varvec{\beta }}}_{0}$ is the MLE of ${\varvec{\beta }}$ under the null logistic model, $p_{Q} = \text {tr}\{\mathbf{P }_{0} \mathbf{K } \},\, \sigma _{Q} = 2 \text {tr}\{\mathbf{P }_{0} \mathbf{K } \mathbf{P }_{0} \mathbf{K } \},$ and $\mathbf{P }_{0} = \mathbf{D }_{0} - \mathbf{D }_{0} \mathbf{X } (\mathbf{X }^\mathrm{T}\mathbf{D }_{0} \mathbf{X })^{-1} \mathbf{X }^\mathrm{T} \mathbf{D }_{0}$ where $\mathbf{D }_{0} = \text {diag}\{\hat{p}_{i0}(1-\hat{p}_{i0}) \}.$

From Sect. 2.2.1, we modify the working model in (5). Define $\mathbf{D }$ as the block diagonal matrix with blocks $\mathbf{D }_{1}, \ldots , \mathbf{D }_{t}.$ Then the variance–covariance matrix of $\varvec{\varepsilon }$ is $\mathbf{D }^{-1} \mathbf{D }^{1/2} \mathbf{S } \mathbf{D }^{1/2} \mathbf{D }^{-1} = \mathbf{D }^{-1/2} \mathbf{S } \mathbf{D }^{-1/2}$ and the modified working model will have the same form as (5) but with $\varvec{\varepsilon } \sim \text {MVN}(\mathbf 0 ,\, \mathbf{D }^{-1/2} \mathbf{S } \mathbf{D }^{-1/2}).$ The parameters ${\varvec{\beta }}_{j}$ and $\mathbf{h }_{j}$ can now be estimated using BLUE and BLUP, respectively, and the variance components $\tau _{j}$ can be estimated by maximizing the restricted quasi-likelihood criterion

$$\begin{aligned} \ell \approx {-}\frac{1}{2} \text {log }\left| \mathbf{V }_{m}\right| - \frac{1}{2} \text {log }\left| \mathbf{X }^\mathrm{T} \mathbf{V }_{m}^{-1} \mathbf{X }\right| - \frac{1}{2} \left( \begin{array}{c} \widetilde{\mathbf{y }}_{1} - \mathbf{X } \widehat{{\varvec{\beta }}}_{1} \\ \vdots \\ \widetilde{\mathbf{y }}_{t} - \mathbf{X } \widehat{{\varvec{\beta }}}_{t} \end{array}\right) ^\mathrm{T} \mathbf{V }_{m}^{-1} \left( \begin{array}{c} \widetilde{\mathbf{y }}_{1} - \mathbf{X } \widehat{{\varvec{\beta }}}_{1} \\ \vdots \\ \widetilde{\mathbf{y }}_{t} - \mathbf{X } \widehat{{\varvec{\beta }}}_{t} \end{array}\right) , \end{aligned}$$

(10)

where $\mathbf{V }_{m} = \mathbf{D }^{-1/2} \mathbf{S } \mathbf{D }^{-1/2} + \mathbf{V }_{h}.$

The main goal is to test for genetic pathway effects $H_{0}{\text {:}}\, \mathbf{h }(\cdot ) = \mathbf 0$ which is equivalent to testing $H_{0}{\text {:}}\, \tau _{1} = \cdots = \tau _{t} = 0.$ To do this, we propose a score-type test statistic based on the derivative of the quasi-likelihood like that in (10). Taking the derivative of the criterion in (10) with respect to $\tau _{j}$ for $j=1,\ldots , t$ and then setting $\tau _{j} = 0,$ the score function for $\tau _{j}$ is $S_{j} = Q_{j}({\varvec{\beta }},\,\varvec{\theta }) - p_{jQ},$ where

$$\begin{aligned} Q_{j}({\varvec{\beta }},\, \varvec{\theta }) = (\mathbf{y }-\widehat{\mathbf{p }})^\mathrm{T} \left( \mathbf{S } \mathbf{D }^{1/2}\right) ^{-1} \mathbf{D }^{1/2} \mathbf{K }_{j}^{*} \mathbf{D }^{1/2} \left( \mathbf{D }^{1/2} \mathbf{S }\right) ^{-1} (\mathbf{y }-\widehat{\mathbf{p }} ), \end{aligned}$$

and $p_{jQ} = \text {tr}\{ \mathbf{P } \mathbf{K }_{j}^{*} \}.$ Because the $\tau _{j}$’s are considered as variance components and thus are non-negative, testing $H_{0}{\text {:}}\,\tau _{1} = \cdots = \tau _{t} = 0$ is equivalent to testing $H_{0}{\text {:}}\,\tau _{1} + \cdots + \tau _{t} = 0$ and we adopt a similar technique to [25].

From Sect. 2.2.2, in order to evaluate Q in (6), we estimate ${\varvec{\beta }}$ and $\varvec{\theta }$ under the null using GEEs. We posit the GEEs under $H_{0}$

$$\begin{aligned} \widetilde{\mathbf{X }}^\mathrm{T} \mathbf{D } \mathbf{V }_{m0}^{-1} (\mathbf{y }-\mathbf{p }) = \mathbf 0 , \end{aligned}$$

(11)

where $\widetilde{\mathbf{X }}$ is an $nt \times nt$ block diagonal matrix with elements $\mathbf{X }$ and $\mathbf{V }_{m0} = \mathbf{D }^{-1/2} \mathbf{S } \mathbf{D }^{-1/2}.$ If there is no genetic pathway effect ($H_{0}{\text {:}}\,\mathbf{h }(\cdot ) = \mathbf 0$ is true), then $\mathbf {V} _{m0}$ is the working variance–covariance matrix of $\mathbf{y }.$ To solve (11), Liang and Zeger [19] suggest using a modified Fisher scoring algorithm to find ${\varvec{\beta }}$ and a method of moments estimation for $\varvec{\theta }.$ The updating equation is

$$\begin{aligned} \widehat{{\varvec{\beta }}}^{(k+1)} = \widehat{{\varvec{\beta }}}^{(k)} + \left( \widetilde{\mathbf{X }}^\mathrm{T} \widehat{\mathbf{D }} \widehat{\mathbf{V }}_{m0}^{-1} \widehat{\mathbf{D }} \widetilde{\mathbf{X }}\right) ^{-1}\widetilde{\mathbf{X }}^\mathrm{T} \widehat{\mathbf{D }} \widehat{\mathbf{V }}_{m0}^{-1} (\mathbf{y } - \widehat{\mathbf{p }}). \end{aligned}$$

The initial estimates in the first iteration come from fitting a GLM assuming independence.

We then use a simulation-based technique to get the p-values of Q. It can be shown that, under $H_0,$ var$(\mathbf{y }_{j} - \widehat{\mathbf{p }}_{j})$ can be approximated as $\widehat{\mathbf{D }}_{j} - \widehat{\mathbf{D }}_{j} \mathbf{X } (\mathbf{X }^\mathrm{T} \widehat{\mathbf{D }}_{j} \mathbf{X })^{-1} \mathbf{X }^\mathrm{T} \widehat{\mathbf{D }}_{j} = \widehat{\mathbf{P }}_{j}.$ This follows from the fact that under $H_0,\,\mathbf{y }_{j} - \widehat{\mathbf{p }}_{j} = \mathbf{D }_{j}( \widetilde{\mathbf{y }}_{j} - \mathbf{X } \widehat{{\varvec{\beta }}}_{j}),$ and that $\text {Cov}(\widetilde{\mathbf{y }}_{j} - \mathbf{X } \widehat{{\varvec{\beta }}}_{j}) = \mathbf{D }_{j}^{-1} - {\mathbf{D }}_{j}^{-1} \mathbf{X } (\mathbf{X }^\mathrm{T} {\mathbf{D }}_{j} \mathbf{X })^{-1} \mathbf{X }^\mathrm{T} {\mathbf{D }}_{j}^{-1}$ from linear model theory. If the multiple outcomes were independent, then the variance–covariance matrix of $\mathbf{y } - \widehat{\mathbf{p }}$ would be $\widehat{\mathbf{P }},$ a block diagonal matrix with elements $\widehat{\mathbf{P }}_{j},$ but since the outcomes are correlated, the variance–covariance matrix of $\mathbf{y } - \widehat{\mathbf{p }}$ is $\widehat{\mathbf{P }}^{1/2} \widehat{\mathbf{S }} \widehat{\mathbf{P }}^{1/2},$ which is no longer block diagonal. By defining $\widehat{\mathbf{M }} = \widehat{\mathbf{P }}^{1/2} \widehat{\mathbf{S }} \widehat{\mathbf{P }}^{1/2},$ (6) can be rewritten as $Q = \{ \widehat{\mathbf{M }}^{-1/2}({y}- \widehat{\mathbf{p }})\}^\mathrm{T} \widehat{\mathbf{B }} \{ \widehat{\mathbf{M }} ^{1/2} (\mathbf{y }-\widehat{\mathbf{p }}) \}$ where $\widehat{\mathbf{B }} = \widehat{\mathbf{M }}^{1/2} ( \widehat{\mathbf{S }} \widehat{\mathbf{D }}^{1/2} )^{-1} \widehat{\mathbf{D }}^{1/2} {K} \widehat{\mathbf{D }}^{1/2} ( \widehat{\mathbf{D }}^{1/2} \widehat{\mathbf{S }})^{-1} \widehat{\mathbf{M }}^{1/2}.$ Using eigenvalue decomposition, we can write $\widehat{\mathbf{B }} = \mathbf{U } {\varvec{\Lambda }} \mathbf{U }^\mathrm{T}$, where $\mathbf{U }$ is the matrix of orthogonal eigenvectors of $\widehat{\mathbf{B }}$ and ${\varvec{\Lambda }}$ is a diagonal matrix whose elements are the corresponding eigenvalues. Thus $Q = \widehat{\mathbf{R }}^\mathrm{T} {\varvec{\Lambda }} \widehat{\mathbf{R }},$ where $\widehat{\mathbf{R }} = \mathbf{U }^\mathrm{T} \widehat{\mathbf{M }}^{-1/2} (\mathbf{y } - \widehat{\mathbf{p }}).$