Empirical Bayes Shrinkage of Functional Effects, with Application to Analysis of Dynamic eQTLs

Abstract

We introduce functional adaptive shrinkage (FASH), an empirical Bayes method for joint analysis of observation units in which each unit estimates an effect function at several values of a continuous condition variable. The ideas in this paper are motivated by dynamic expression quantitative trait locus (eQTL) studies, which aim to characterize how genetic effects on gene expression vary with time or another continuous condition. FASH integrates a broad family of Gaussian processes defined through linear differential operators into an empirical Bayes shrinkage framework, enabling adaptive smoothing and borrowing of information across units. This provides improved estimation of effect functions and principled hypothesis testing, allowing straightforward computation of significance measures such as local false discovery and false sign rates. To encourage conservative inferences, we propose a simple prior- adjustment method that has theoretical guarantees and can be more broadly used with other empirical Bayes methods. We illustrate the benefits of FASH by reanalyzing dynamic eQTL data on cardiomyocyte differentiation from induced pluripotent stem cells. FASH identified novel dynamic eQTLs, revealed diverse temporal effect patterns, and provided improved power compared with the original analysis. More broadly, FASH offers a flexible statistical framework for joint analysis of functional data, with applications extending beyond genomics. To facilitate use of FASH in dynamic eQTL studies and other settings, we provide an accompanying R package at https: //github.com/stephenslab/fashr.

Figures (16)

• Figure 1: Example illustrating the use of FASH to analyze a dynamic eQTL data set. In all panels, the original eQTL effect estimates are shown as black dots, and vertical error bars depict $\pm 2$ standard errors. The top two panels show the smoothed estimates defined as the posterior means from the $L$-GP method; the different estimates are obtained by different levels of shrinkage toward the constant and linear baseline models (left and right panels, respectively). The bottom two panels summarize the results from the proposed FASH method, with constant and linear baseline models (left and right panels, respectively). The constant and linear baseline models are defined using $L$-GP processes with $L = D^1$ and $L = D^2$, respectively (see \ref{['subsec-ASH']}). The posterior mean effect function is shown as a solid red line; the shaded region shows the 95% credible interval.
• Figure 2: Sample paths randomly drawn from the fitted FASH priors that were estimated from the dynamic eQTL data in \ref{['sec-application']}, for $L = D^1$ (top) and $L = D^2$ (bottom), before (left) and after (right) the BF-based adjustment. For purposes of visualization, we constrained the sample paths to have an initial condition of zero; that is, $\beta(0) = 0$ in the top row, and $\beta(0) = \beta'(0) = 0$ in the bottom row. 5,000 randomly sampled paths are shown in each plot.
• Figure 3: Examples of dynamic eQTLs identified by FASH ($L = D^1$), and comparison with parametric modeling approach strober2019dynamic. In each plot, the posterior mean of $\beta_j$ is shown as a red solid line, and the 95% credible interval is depicted by the shaded region. Observed effect size estimates $\hat{\beta}_j$ are shown as black dots, with vertical error bars representing $\pm 2$ the (adjusted) standard errors $\tilde{s}_{jr}$. The lfdr is the lfdr from the FASH-$\text{IWP}_1$ model after the BF adjustment; the p-values are from the analyses of strober2019dynamic with linear/quadratic interaction models. The inverse-variance weighted least squares estimates for the linear ($G_c \times t$) and quadratic ($G_c \times t^2$) parametric interaction models are shown as dashed lines (green = linear, purple = quadratic).
• Figure 4: Discovery of dynamic eQTLs using FASH vs. linear/quadratic parametric interaction models: (a) genes and (b) gene-variant pairs.
• Figure 5: Examples of nonlinear dynamic eQTLs identified by FASH, with $L = D^2$. In each plot, the posterior mean of $\beta_j$ is shown as a red solid line, and the 95% credible interval is depicted by the shaded region. Observed effect size estimates $\hat{\beta}_j$ are shown as black dots, with vertical error bars representing $\pm 2$ the (adjusted) standard errors $\tilde{s}_{jr}$. The lfdr is the lfdr from the FASH-$\text{IWP}_2$ model after the BF adjustment. The inverse-variance weighted least squares estimates for the linear ($G_c \times t$) and quadratic ($G_c \times t^2$) parametric interaction models are shown as dashed lines (green = linear, purple = quadratic).

Theorems & Definitions (9)

• Lemma 1: BF-Moment
• proof
• Theorem 1: BF-based adjustment gives conservative estimate
• Theorem 1: BF-adjustment gives conservative estimate
• proof
• Lemma S1: Conservative under $H_0$
• proof
• Lemma S2: Conservative under $H_1$
• proof