High-Dimensional Confounding Adjustment Using Continuous Spike and Slab Priors

Abstract

In observational studies, estimation of a causal effect of a treatment on an outcome relies on proper adjustment for confounding. If the number of the potential confounders (p) is larger than the number of observations (n), then direct control for all potential confounders is infeasible. Existing approaches for dimension reduction and penalization are generally aimed at predicting the outcome, and are less suited for estimation of causal effects. Under standard penalization approaches (e.g. Lasso), if a variable X_j is strongly associated with the treatment T but weakly with the outcome Y, the coefficient β_j will be shrunk towards zero thus leading to confounding bias. Under the assumption of a linear model for the outcome and sparsity, we propose continuous spike and slab priors on the regression coefficients β_j corresponding to the potential confounders X_j . Specifically, we introduce a prior distribution that does not heavily shrink to zero the coefficients (β_j s) of the X_j s that are strongly associated with T but weakly associated with Y. We compare our proposed approach to several state of the art methods proposed in the literature. Our proposed approach has the following features: 1) it reduces confounding bias in high dimensional settings; 2) it shrinks towards zero coefficients of instrumental variables; and 3) it achieves good coverages even in small sample sizes. We apply our approach to the National Health and Nutrition Examination Survey (NHANES) data to estimate the causal effects of persistent pesticide exposure on triglyceride levels.

Keywords: bayesian variable selection; causal inference; high-dimensional data; shrinkage priors.

Figure 1:

The left panel shows $p_{\theta }^{\ast }({\beta }_j)$ for a variety of values of β_j as a function of w_j. The right panel shows Δ_j/n as a function of w_j. Here we fixed λ₁ = 0.1, λ₀ = 30, and θ = 0.05.

Figure 2:

Posterior inclusion probabilities from the homogeneous model for simulations in Sections 4.1 and 4.2.

Figure 3:

Results from analysis of volatile compounds on triglycerides. The upper panel shows the results using the full, n = 177, sample. The lower panel shows the results for just the n = 77 subjects who are over 40 years old.

Figure 4:

The left panel shows a histogram of the ratios of standard errors for the EM-SSL approach and the double post selection approach for the analysis of volatile compounds in the full data. The right panel shows the corresponding histogram for the analysis of subjects over the age of 40. The dashed vertical line is the mean of the ratios that make up the histogram.