Stress-Testing Assumptions: A Guide to Bayesian Sensitivity Analyses in Causal Inference

TL;DR

This is the first paper that presents a unified approach with code, examples, and methodology in a three-pronged illustration of sensitivity analyses in Bayesian causal inference, and the goal is for the reader to walk away with implementation-level knowledge.

Abstract

While observational data are routinely used to estimate causal effects of biomedical treatments, doing so requires special methods to adjust for observed confounding. These methods invariably rely on untestable statistical and causal identification assumptions. When these assumptions do not hold, sensitivity analysis methods can be used to characterize how different violations may change our inferences. The Bayesian approach to sensitivity analyses in causal inference has unique advantages as it allows users to encode subjective beliefs about the direction and magnitude of assumption violations via prior distributions and make inferences using the updated posterior. However, uptake of these methods remains low since implementation requires substantial methodological knowledge. Moreover, while implementation with publicly available software is possible, it is not straight-forward. At the same time, there are few papers that provide practical guidance on these fronts. In this paper, we walk through four examples of Bayesian sensitivity analyses: 1) exposure misclassification, 2) unmeasured confounding, and missing not-at-random outcomes with 3) parametric and 4) nonparametric Bayesian models. We show how all of these can be done using a unified Bayesian "missing data" approach. We also cover implementation using Stan, a publicly available open-source software for fitting Bayesian models. To the best of our knowledge, this is the first paper that presents a unified approach with code, examples, and methodology in a three-pronged illustration of sensitivity analyses in Bayesian causal inference. Our goal is for the reader to walk away with implementation-level knowledge.

Figures (4)

• Figure 1: The .R (Left) and .stan (Right) files implementing the causal computation procedure described in Section \ref{['sc:bayesian_causal_standard']} with a binary outcome and $P=1$ binary covariate. Ellipsis indicates lines skipped for compactness; the full code can by found in the companion GitHub repository. We specified $f_{Y|A,L}(y\mid a, l; \eta)$ as a Bernoulli pmf with conditional probability $P_{Y|A,L}(Y=1\mid A=a, L=l; \eta) = \text{expit} ( \eta_0 +\eta_1l + \eta_2a)$. For the covariate model, we specify $f_L(l;\theta)$ as a Bernoulli pmf with probability $0<\theta<1$.
• Figure 2: Left: posterior mean and 95% credible intervals (CrI) for the ATE, $\Psi$, under various degrees of exposure misclassifications. The point at $\xi_1=.99$ on the green curve ($\xi_2=.001$) corresponds to the "best case" scenario where there is nearly no misclassification as sensitivity $=$ specificity $\approx 100\%$. The dashed red line represents the posterior mean of $\Psi$. In contrast, the point at $\xi_1=.8$ on the black curve $\xi_2=.2$ correspond to larger misclassification with sensitivity$=$specificity $\approx 80\%$. Right: tipping point analysis output from Stan . Assuming no unmeasured confounding, the posterior mean and 95% CrI for $\Psi$ in the synthetic data analysis is -.31 [-.39, -.23], indicating lower remission rates due to $A=1$. At approximately values of $\xi_1>1$ and $\xi_2<-1$, however, the upper limit of the 95% CrI for $\Psi$ surpasses zero.
• Figure 3: Results from synthetic example described in Section \ref{['sc:ex_mnar_parametric']}. A value of $\xi_3=0$ represents strong prior belief in MAR - leading to a credible interval for $\Psi$ that contains zero. However, as we allow deviations from MAR, posterior inferences shift in favor of a positive effect for group $A=1$ versus $A=0$ at about $\xi_3=.75$.
• Figure 4: Posterior Inference with TSB model and missing outcomes. Red/blue points depict treated/untreated subjects. Treated/Untreated subjects with missing outcomes are represented as a tick mark on the x-axis at the location of their observed $L$ value. Top Row: Bayesian inference under the model described in Section \ref{['sc:tsb_stan_example']} with $\xi_3=0$ - i.e. a strong prior belief that outcome missingness is at-random (MAR). The left panel shows, at each tick, several posterior draws of the missing outcome for that subject. Draws for treated/untreated subjects are shown in light red/light blue. The spread reflects posterior uncertainty about that subject's outcome value. The right panel shows the posterior mean regression line for treated/untreated subjects in dark red/blue. Several posterior draws of the regression line are overlaid in lighter shades to visualize posterior uncertainty. Bottom Row: Same analysis as in the top row but with a prior belief in MNAR with $\xi_3=-1$. This is a prior belief that treated patients with missing outcomes had lower than average outcomes. Notice that the posterior draws of the missing outcome values for treated subjects are now shifted down in the right panel. Consequently, the regression line for treated subjects now has a dip to reflect this prior belief in MNAR outcomes.