Introduction

GWAS leveraging measures collected at multiple time points have found novel genetic associations. We hypothesize that a subset of these variants may be involved in gene-by-gene (GxG) or gene-by-environment (GxE) interaction. We propose an analytic approach that concurrently models SNPs’ phenotype level and heteroskedastic effect longitudinally.

Methods

Population and within-family longitudinal (and delta) GWAS in the form of a 3-level hierarchical generalized linear mixed effects model (HGLM) (Cao, Maxwell, and Wei 2015; Rönnegård, Shen, and Alam 2010):

\[Y_{ijk} = \textbf{X}_{ijk} \beta + G_{1ijk} \gamma_1 + G_{2ijk} \gamma_2 + \mu_{ijk} + \delta_{ij} + \epsilon_i, \] where \(\textbf{X}\) is a matrix of covariates (i.e., time, sex, batch, 10 PCs) including the intercept with \(\beta\) as their respective effect size, \(G_1\)/\(G_2\) are the copies of the effect allele (expressed as deviation from the family mean) with \(\gamma_1\)/\(\gamma_2\) as their effect on phenotype mean, \(\mu_{ijk}\) is the random intercept of individual, \(\delta_{ij}\) is the genetic relatedness matrix (GRM) and \[ \epsilon_i \sim N (0, G_{0i} \sigma_0^2 + G_{1i} \sigma_1^2 + G_{2i} \sigma_2^2) \]

We assume uncorrelated variance components: \(\mu_{ijk} \sim N(0, \sigma^2_{\mu})\), \(\delta_{ij} \sim N(0, \sigma^2_{F}\Phi)\), and \(\epsilon_i \sim N(0, \textbf{V}_\textbf{E})\) where \(\textbf{V}_\textbf{E}\) is an \(n \times n\) matrix with the \(i^{th}\) diagonal equal to \(G_{0i} \sigma_0^2 + G_{1i} \sigma_1^2 + G_{2i} \sigma_2^2\) and \(\Phi\) is \(2\times\) the kinship matrix.

Simulation Results

SNP mean vs. heteroskedastic effect visualized affecting phenotype variance and kurtosis. # A tibble: 3 × 3 dat$snp var mean 1 0 0.358 0.00604 2 1 0.357 0.803
3 2 0.362 1.60
# A tibble: 3 × 3 dat$snp var mean 1 0 0.358 0.00192 2 1 1.01 0.000113 3 2 2.91 0.0332

Mean and variance effects are inherently correlated.

This mean-variance relationship must be decorrelated (Young, Wauthier, and Donnelly 2018): \(d_l = \alpha_{vl} - r_{av} \alpha_{vl}\).

Non-additive effects are more difficult to detect.