I previously introduced the problem of unobserved confounding in large scale gene expression analyses and mentioned a few methods that attempt to account such confounding. (You should check that out before reading further.) However, all of these confounder adjustment methods are somewhat classical in that they do not adapt to the distribution of the effects. There is a long history, starting with Stein, of showing that shrinkage and hierarchical modeling can greatly improve estimation and testing. In Gerard and Stephens (2018), I adapted these hierarchical shrinkage methods to the problem of unobserved confounding.

One shrinkage approach (outside of the context of confounder adjustment) that my postdoctoral advisor recently developed is Adaptive SHrinkage (ASH). ASH works in the old normal means problem, suggesting that researchers, when there is one observed covariate, take estimates for the effects $$\mathbf{B}$$ (and their corresponding standard errors, $$\hat{s}_i$$) from an ordinary least squares fit and then assume \begin{align} \label{eq:norm.means} \hat{\mathbf{B}} = \mathbf{B} + \mathbf{e},\ \ \ e_{i} \sim N(0, \hat{s}_i^2),\ \ \ e_1,\ldots,e_p \text{ independent}. \end{align} Though the normal means problem shown above is old, ASH's innovation is to take a non-parameteric empirical Bayes approach to estimation. ASH assumes that the effects (the $$B_i$$'s) follow some unimodal distribution, then estimates this prior by maximum marginal likelihood.

Using simulations that satisfy ASH's independence assumption, ASH was shown to be more powerful and better calibrated than competing procedures. However, when there is unobserved confounding, ASH, like most other methods, can fail drastically. For my next project, I took the ideas from ASH and the confounder adjustment procedures and unified them into a powerful and well-calibrated approach that accounts for unobserved confounding. To describe this method, I note others have shown that many confounder adjustment procedures can be reduced to fitting \begin{align} \label{eq:reduced.model} \hat{\mathbf{B}} = \mathbf{B} + \tilde{\mathbf{\alpha}}^{\intercal}\mathbf{z} + \mathbf{e},\ \ \ e_i \sim N(0, \hat{s}_i^2), \end{align} where $$\tilde{\mathbf{\alpha}}$$ is a known term and $$\mathbf{z} \in \mathbb{R}^q$$ is unknown. To me, this looks just like the normal means model with an extra bias term ($$\tilde{\mathbf{\alpha}}^{\intercal}\mathbf{z}$$). This immediately suggested a strategy to unite ASH with confounder adjustment: (i) Place a unimodal prior over the effects (à la ASH). (ii) Estimate this unimodal prior jointly with $$\mathbf{z}$$. (iii) Return posterior summaries using this estimated unimodal prior. As with all methods, there were many additional practical issues that needed addressing (e.g. variance calibration, robustifying the procedure, dealing with massive datasets), but the result was a well-performing procedure that I think will be quite useful for the scientific community (software available here).

## References

• Gerard, D., & Stephens, M. (2018). Empirical Bayes Shrinkage and False Discovery Rate Estimation, Allowing For Unwanted Variation. To appear in Biostatistics. [Link to Biostatistics] [Link to arXiv]}