Random Effects Model for Multiple Pathway Analysis with Applications to Type II Diabetes Microarray Data

Abstract

Close to three percent of the world’s population suffer from diabetes. Despite the range of treatment options available for diabetes patients, not all patients benefit from them. Investigating how different pathways correlate with phenotype of interest may help unravel novel drug targets and discover a possible cure. Many pathway-based methods have been developed to incorporate biological knowledge into the study of microarray data. Most of these methods can only analyze individual pathways but cannot deal with two or more pathways in a model based framework. This represents a serious limitation because, like genes, individual pathways do not work in isolation, and joint modeling may enable researchers to uncover patterns not seen in individual pathway-based analysis. In this paper, we propose a random effects model to analyze two or more pathways. We also derive score test statistics for significance of pathway effects. We apply our method to a microarray study of Type II diabetes. Our method may eludicate how pathways crosstalk with each other and facilitate the investigation of pathway crosstalks. Further hypothesis on the biological mechanisms underlying the disease and traits of interest may be generated and tested based on this method.

Appendix

1.1 Asymptotic Distribution of Score Test for Individual Null

Regularity conditions:

• 1. $\Sigma _{-j}^{-1} \Sigma _{\tau _j}$ is of full rank.

• 2. As $n \rightarrow \infty$, the number of observations at any level of any random effect is bounded by a constant which follows from that fact that each pathway effect is a vector of size $n$.

• 3. The sequences $\{ \xi _i \psi ^{-1}_{i} y_i \}$ and $\{\xi _i\}$ are uniformly bounded $\forall i=1,...,n$.

• 4. There exists a positive definite matrix $I^{0}$ such that $\lim _{n\rightarrow \infty } \frac{I}{n} = I^{0}$, which is reasonable given conditions 1 and 2.

• 5. For any given $q$ by 1 constant vector $\eta$, let $\Phi _{\eta } = \sum _{l=1}^{q} \frac{1}{2} \eta \Sigma _{-j}^{-1} \Sigma _{\tau _j} \Sigma _{-j}^{-1}$. Then $y^{*}_{\eta } = \Phi _{\eta } y = (y^{*}_{\eta ,1},...,y^{*}_{\eta ,n})^{T}$ forms an $m$-dependent sequence for some constant $m$.

• 6. The usual asymptotic behavior of the maximum likelihood estimates of parameter vector $\tau _1, ..., \tau _{j-1}, \tau _{j+1}, ..., \tau _{q}$ holds, including consistency and efficiency.

Proof. Let $\tau ^{*}_{-j}$ be the true value of $\tau _{-j}$. First, let’s proof the asymptotic normality of $n^{\frac{1}{2}} \eta U(\tau ^{*}_{-j})$. Let $\eta$ be a any constant vector of size $q$. The score test in Sect. 3.1 under $H_0$ can be written as:

$$\begin{aligned} \eta U_{\tau _j} = y^{T} \Phi y - trace\left( \sum _{l=1}^{q} \frac{1}{2} \Sigma _{-j}^{-1} \Sigma _{\tau _j}\right) , \end{aligned}$$

where $\Phi = \sum _{l=1}^{m} \frac{1}{2} \eta \Sigma _{-j}^{-1} \Sigma _{\tau _j} \Sigma _{-j}^{-1}$. Under conditions 1 and 2, the above equation can be rewritten as:

$$\begin{aligned} \eta U_{\tau _j} = \sum _{i=1}^{n} U_{\eta , i} = \sum _{i=1}^{n} (y^{*}_{\eta ,i} - \Gamma _{\eta ,i}), \end{aligned}$$

where $y^{*}_{\eta ,i} = \Phi _{\eta } y = (y^{*}_{\eta ,1},...,y^{*}_{\eta ,n})^{T}$ is a weighted sum of the $Y_i$s and $\Gamma _{\eta ,i} = \sum _{l=1}^{m} \frac{1}{2} (\eta _{l}) \xi _{i}$.

Under condition 3, the sequence $\{\xi _i \psi ^{-1}_{i} \xi _{i'} \psi ^{-1}_{i'} y_i y_{i'}\}$ for ($i,i' = 1,...,n$) are uniformly bounded. This implies that $U_{\eta , i}$s are also uniformly bounded for any given $\eta$.

Under conditions 4, 5, and an application of Theorem 7.3.1 of Chung [10], it follows that:

$$\begin{aligned} n^{-\frac{1}{2}} \eta ^{T} U(\tau ^{*}_{-j}) \rightarrow N(E(U_{\tau _j}), \eta ^{T} I^{0}(\tau ^{*}_j) \eta ), \end{aligned}$$

in distribution as $n\rightarrow \infty$. Using Cramer-Wald theorem, we have $n^{-\frac{1}{2}} U(\tau ^{*}_{-j}) \rightarrow N(E(U_{\tau _j}), I^{0}(\tau ^{*}_j))$

Under condition 6, we have that $n^{-\frac{1}{2}} U(\hat{\tau }_{-j}) \rightarrow N(E(U_{\hat{\tau }_j}), I^{0}(\hat{\tau }_j))$ and it follows from Slutsky’s theorem that:

$$\begin{aligned} S_{\tau _j}(\hat{\upsilon }^{T}) = U_{\tau _j}(\hat{\upsilon })/(\tilde{\vartheta }^{-1}_{jj} (\hat{\upsilon }))^{\frac{1}{2}}, \end{aligned}$$

where $\tilde{\vartheta }_{jj} = \vartheta _{\tau _j \tau _j} - \vartheta _{\upsilon \tau _j}^{T}\vartheta _{\upsilon \upsilon }^{-1}\vartheta _{\upsilon \tau _j}$.

The above score test follows an asymptotically normal distribution with mean $E(U_{\tau _j})$ and variance 1. And $E(U_{\tau _j}) = \sum _{ii} R_{ii}^{*} \mu _{2i}^{T} + 2 \sum _{i \ne j} R_{ij}^{*T} \mu _{1i}^{T} \mu _{1j}^{T}$.