Instrumental variables as bias amplifiers with general outcome and confounding

Abstract

Drawing causal inference with observational studies is the central pillar of many disciplines. One sufficient condition for identifying the causal effect is that the treatment-outcome relationship is unconfounded conditional on the observed covariates. It is often believed that the more covariates we condition on, the more plausible this unconfoundedness assumption is. This belief has had a huge impact on practical causal inference, suggesting that we should adjust for all pretreatment covariates. However, when there is unmeasured confounding between the treatment and outcome, estimators adjusting for some pretreatment covariate might have greater bias than estimators without adjusting for this covariate. This kind of covariate is called a bias amplifier, and includes instrumental variables that are independent of the confounder, and affect the outcome only through the treatment. Previously, theoretical results for this phenomenon have been established only for linear models. We fill in this gap in the literature by providing a general theory, showing that this phenomenon happens under a wide class of models satisfying certain monotonicity assumptions. We further show that when the treatment follows an additive or multiplicative model conditional on the instrumental variable and the confounder, these monotonicity assumptions can be interpreted as the signs of the arrows of the causal diagrams.

Keywords: Causal inference; Directed acyclic graph; Interaction; Monotonicity; Potential outcome.

Fig. 1.

Two directed acyclic graphs, where is the treatment and is the outcome of interest. (a) Directed acyclic graph for M-bias, where and are unobserved and is observed. (b) Directed acyclic graph for Z-bias, where is an unmeasured confounder and is an instrumental variable for the treatment-outcome relationship.

Fig. 2.

Directed acyclic graph for Z-bias with general instrument and confounder.

Fig. 3.

Biases of the adjusted and unadjusted estimators over random draws of the probabilities. In areas Z-bias arises, and in areas Z-bias does not arise.

Fig. 4.

Directed acyclic graph for Z-bias allowing an arrow from to