A Bayesian view of doubly robust causal inference: Table 1.

Abstract

In causal inference confounding may be controlled either through regression adjustment in an outcome model, or through propensity score adjustment or inverse probability of treatment weighting, or both. The latter approaches, which are based on modelling of the treatment assignment mechanism and their doubly robust extensions have been difficult to motivate using formal Bayesian arguments; in principle, for likelihood-based inferences, the treatment assignment model can play no part in inferences concerning the expected outcomes if the models are assumed to be correctly specified. On the other hand, forcing dependency between the outcome and treatment assignment models by allowing the former to be misspecified results in loss of the balancing property of the propensity scores and the loss of any double robustness. In this paper, we explain in the framework of misspecified models why doubly robust inferences cannot arise from purely likelihood-based arguments, and demonstrate this through simulations. As an alternative to Bayesian propensity score analysis, we propose a Bayesian posterior predictive approach for constructing doubly robust estimation procedures. Our approach appropriately decouples the outcome and treatment assignment models by incorporating the inverse treatment assignment probabilities in Bayesian causal inferences as importance sampling weights in Monte Carlo integration.A revised version of this article has been accepted for publication in Biometrika, published by Oxford University Press. Saarela, O., Belzile, L. R. and D. A. Stephens. A Bayesian view of doubly robust causal inference,Interest then lies in the identifiability of the average potential outcomes based on the observed data.

A review of methods for comparing treatments evaluated in studies that form disconnected networks of evidence

“…Another source of uncertainty is structural uncertainty arising from model misspecification that produces biased estimates of relative effect, although it can be reduced using doubly robust estimation. Alternatively, it might be possible to incorporate external information to mitigate this or take a Bayesian perspective . Generating joint posterior distributions about parameters should be seen as an important aim in HTA to properly represent uncertainty about inputs to decision analytic models.…”

“…Alternatively, it might be possible to incorporate external information to mitigate this or take a Bayesian perspective. 78 Generating joint posterior distributions about parameters should be seen as an important aim in HTA to properly represent uncertainty about inputs to decision analytic models.…”

Constructing inverse probability weights for institutional comparisons in healthcare

“…The standardized proportions of partial nephrectomies shown here were calculated using the truncated and normalized weights, with the standard errors estimated using the bootstrap. Since the bootstrap can easily produce further positivity violations when refitting the multinomial logistic assignment model in the bootstrap resamples, we made use of the Bayesian analogue that uses continuous‐valued Dirichlet distributed sampling weights instead 46 . The reference level of care in both plots is the overall proportion of partial nephrectomies in the province.…”

Bayesian estimation of the average treatment effect on the treated using inverse weighting

“…The second equality above follows from the connection between the predictive distributions under the experimental and observational representations. This connection arises through the importance sampling identity …”

“…Thus, this particular two‐step approach is neither proper Bayesian nor does it result in good frequentist properties. Second, it was shown that the point estimator does not possess good small‐sample properties . Kaplan and Chen also examined the differences in the causal estimate when incorporating noninformative versus informative priors in a model averaging stage .…”