A Functional Approach to Testing Overall Effect of Interaction Between DNA Methylation and SNPs

Technical Summary: A Functional Approach to Testing Overall Effect of Interaction Between DNA Methylation and SNPs

Problem Statement
Current methodologies for investigating the interplay between DNA methylation and Single Nucleotide Polymorphisms (SNPs) typically rely on ad-hoc, pairwise testing procedures. These approaches involve selecting specific pairs of CpG sites and SNPs based on a fixed genomic distance threshold, fitting separate linear models for each pair, and applying multiple testing corrections (e.g., Bonferroni or FDR). The authors identify two primary limitations in this standard practice:

1. Statistical Power and Error Control: The necessity of performing a massive number of individual tests complicates the control of Type I error rates and often leads to a significant loss of statistical power.
2. Ignoring Spatial Structure: These methods fail to account for the inherent spatial correlation structure of DNA methylation levels, where measurements at neighboring genomic positions tend to be highly correlated. By treating methylation as discrete points rather than a continuous profile, existing methods may miss broader interaction patterns.

Methodology
The paper proposes a functional data analysis (FDA) framework to test the overall significance of the interaction between DNA methylation profiles and a set of SNPs on a quantitative phenotype.

• Data Representation:

  • DNA Methylation: Instead of treating methylation at discrete CpG sites as scalar variables, the authors model the methylation profile as a smooth, continuous curve $\Pi_i(t)$ over a genomic region $G$. This is achieved by applying kernel smoothing (specifically using a standard normal density kernel with an adaptive bandwidth) to raw methylation estimates.
  • SNPs: SNP data are treated as scalar predictors $G_{id}$ located at specific genomic positions $u_d$.

• The Proposed Model:
  The authors formulate a scalar-on-function regression model that includes an interaction term between the functional methylation curve and the scalar SNP variables. The model is defined as:
  $$Y_i = \zeta_0 + \sum_{s=1}^S \zeta_s W_{is} + \sum_{d=1}^D \alpha_d G_{id} + \int_0^1 \delta(t)\Pi_i(t)dt + \sum_{d=1}^D \eta_d G_{id} \int_0^1 \psi_\rho(|t - u_d|)\Pi_i(t)dt + \varepsilon_i$$
  Key components include:

  • $\delta(t)$: A functional coefficient representing the main effect of methylation.
  • $\eta_d$: Regression coefficients quantifying the interaction effect for the $d$-th SNP.
  • $\psi_\rho(|t - u_d|)$: A non-increasing weight function (e.g., exponential, concave, or linear) that modulates the interaction based on the distance between the SNP position $u_d$ and the methylation position $t$. This allows the model to consider the entire methylation curve while weighting regions closer to the SNP more heavily. The parameter $\rho$ controls the extent of this interaction.

• Estimation and Inference:

  • Estimation: The model is estimated using a penalized spline approach (B-Splines) for the functional coefficient $\delta(t)$. The smoothing parameter is treated as a variance component, and the model is fitted as a linear mixed model using Restricted Maximum Likelihood (REML).
  • Hypothesis Testing: To test the overall significance of the interaction, the authors propose a Wald-type statistic ($T_D$) to test the null hypothesis $H_0: \eta_d = 0$ for all $d$. Under the null, this statistic asymptotically follows an F-distribution.

Key Contributions

1. Functional Modeling of Interactions: The paper introduces a novel framework that treats DNA methylation as a functional predictor, thereby naturally incorporating the spatial correlation structure of the data.
2. Global Interaction Test: Unlike pairwise approaches, this method provides a single test for the overall effect of interactions between a set of SNPs and the methylation profile, avoiding the multiple testing burden associated with testing thousands of individual CpG-SNP pairs.
3. Flexible Weighting: The inclusion of a tunable weight function $\psi_\rho$ allows researchers to specify the spatial scale of the interaction (local vs. regional) without arbitrarily discarding data points outside a fixed threshold.

Results

• Simulation Studies:

  • Type I Error: Extensive simulations under the null hypothesis (no interaction) demonstrated that the proposed test effectively controls Type I error rates across various sample sizes ($N=200, 400$) and different specifications of the weight function $\psi_\rho$ and parameter $\rho$.
  • Statistical Power: Under the alternative hypothesis, the method showed increased empirical power compared to existing pairwise methods, particularly when multiple interactions were present. Power increased with larger sample sizes and stronger interaction coefficients.
  • Robustness: The method proved robust to misspecification of the parameter $\rho$ and the parametric form of $\psi_\rho$, with only minor losses in power observed when the true parameter value differed from the test value.
  • Real Data Simulation: Simulations using real data residuals (which followed a mixture of normal distributions) confirmed that Type I error control is maintained even when error terms deviate from a single normal distribution.

• Real Data Application:
  The method was applied to an obesity dataset (GSE73103) involving 355 individuals, 52 obesity-associated SNPs, and DNA methylation data from the Illumina 450K array.

  • The analysis focused on Chromosome 1, utilizing 5 known obesity-associated SNPs and ~24,000 CpG sites.
  • The proposed test detected a significant interaction between obesity-associated SNPs and DNA methylation profiles, regardless of the parametric form of $\psi_\rho$ or the value of $\rho$.
  • Comparison: When compared to the existing pairwise method (LintestSNPCpG) in simulation scenarios based on this real data, the proposed functional approach demonstrated superior empirical power, particularly in scenarios involving interactions between one SNP and multiple CpGs (20, 50, or 100 sites).

Significance and Claims
The authors claim that their functional approach offers a flexible and powerful alternative to traditional pairwise testing in epigenetic studies. By modeling methylation as a continuous curve, the method captures the biological reality of spatial correlation and reduces the dimensionality of the problem, leading to better control of Type I errors and higher statistical power.

The paper modestly notes that the method requires the specification of the weight function form and the parameter $\rho$. However, simulation results suggest that the method is robust to these choices, providing guidelines for selection (small $\rho$ for large genomic regions, large $\rho$ for local interactions).

The authors acknowledge limitations regarding scalability to genome-wide analyses where the number of SNPs exceeds the sample size, suggesting that the method is best suited for candidate-gene or regional analyses. They propose that future work should focus on incorporating dimension-reduction techniques or regularization to handle high-dimensional SNP sets. The method is implemented in the R package funInterMethSNP, facilitating reproducibility and application in diverse research settings.