Multivariate Causal Steps Model Results

Figure 5.3 represents the type I error and empirical power evaluated at α = 0.05 level for the two steps in the causal mediation analysis under three different causal scenarios for sample size n= 500. The results for n= 200 or 1000 are similar.

Under the null model, all the methods, including the two MVKMR models that test the association between M and G and the additive least square kernel regression that tests the association between M and the phenotype y conditional on G, showed valid type I error. Also, all tests have adequate power under both alternative scenarios, including the partial mediation and the complete mediation scenario. Unsurprising- ly,the MG block test that utilized the block structure of the methylation data have higher power than the corresponding test that treats each CpG marker as individual outcomes. MG Block MG Original M Y Assn.|G n = 500 0.0 0.2 0.4 0.6 0.8 1.0

Null Complete Causal Partial Causal

5.5 Discussion

In this paper, we propose a statistical framework for integrative analysis of genome wide methylation and genotype studies, in which we first test the cumulative genet- ic/epigenetic effects with subsequent mediation analysis to decide their relative roles. Our analysis is based on SNP-set and CpG marker set that are constructed using pri- or biological knowledge, such as proximity to a known gene or other genomic features. This approach unifies the units in analyzing methylation and genotype data. We model the cumulative genetic/epigenetic effect via a flexible, semiparametric kernel machine regression by constructing composite kernels. For genes that show significant associa- tion with the phenotype, we construct mediation analysis to understand their relative roles in affecting the phenotype. The subsequent mediation model tests whether methy- lation is a mediator by first testing the association between genotype and methylation and then testing the association between methylation and the phenotype adjusting for genotypic effect.

We proposed two approaches for the cumulative test, both of which start by con- structing a composite kernel as a weighted average of two kernels based on genotype and methylation respectively. In principle, any valid kernel that measures the similarity between genotype or methylation from different individuals can be used in construct- ing the composite kernel, allowing for flexible modeling with different genotype and methylation effect mechanism. Therefore, the composite kernel approach can be ad- vantageous than the naive model which concatenates the genotype and methylation data to construct a single common kernel (with possible weighting) and tests the join- t effect through this common kernel. By using the single common kernel, the naive model assumes a common genetic and epigenetic effect mechanism which can be unrea- sonable. The composite kernel approach, on the other hand, can incorporate kernels that are specifically designed for SNP data and methylation data, incorporating possible

epistatic SNP effect and other possible interaction effect in methylation data.

Both of our cumulative test approaches rely on the adaptive selection of the opti- mal weights in constructing the composite kernel. The perturbation based approach selects the optimal weight via a grid search of differentw, and adjusts for multiple com- parison via computationally efficient perturbation. In our simulations, we constructed composite kernels as a weighted average of IBS kernels for SNP data and linear ker- nel for methylation data. However, it should be noted that even more flexible model can be conducted by incorporating multiple kernels for genetic and epigenetic data respectively.

The kernel PCA approach relies on the projection of genetic variation onto the epigenetic variation, or reversely. Although the projection enables decomposition of the genetic and epigenetic effect to facilitate analytical p-value computation, it can weaken the association signal. In our simulation experiments, the kernel PCA approach tends to be less powerful than the perturbation approach, especially when the sample size is modest (n = 200, data not shown). The direction of projection can also be important, the relative power of which depends on the relative genetic and epigenetic effect sizes. In our simulation, the power loss can be magnificent for projecting the genetic variation to the epigenetic variation when the genetic effect is dominating, especially when the sample size is modest (n = 200, data not shown). In practice, we suggest using the perturbation approach than the kernel PCA approach in the context of the joint association testing of genetic and epigenetic variation.

For genes that show significant joint genetic/epigenetic effect, we propose a two step process to explore the possible causal relationship between genotype, methylation and phenotype. The multivariate causal steps model extends the classic univariate causal steps model to incorporate multidimensional mediator and independent variable. In each step, we test association via the kernel machine regression framework for flexible

semiparametric modeling of genetic and epigenetic effect. Critical to this process is the correct specification of the causal model prior to analysis that methylation level change is prior to and possibly contributes to variation in the phenotype: i.e, methylation is a potential mediator between genotype on phenotype. This assumption is reasonable in situations when the time order of events can be established, such as in cases when the methylation level is measured at baseline (such as at birth or before disease) or when the phenotype is unlikely to cause methylation level change. Application of the causal model without consideration of the underlying model assumption can result in spurious conclusions.

In summary, we have proposed a statistical framework for genome wide integra- tive analysis of methylation and genotype studies, with first testing the cumulative association of genetic and epigenetic variation at each gene with phenotypic trait and subsequent mediation analysis for causal inference. We employed a semiparametric kernel machine regression framework to allow for flexible modeling of SNP-set and methylation set effect. We show via simulation and real data application the well con- trolled genome wide type I error as well as the superior power compared to competing models.