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Drawing causal inference with observational studies is the central pillar ofmany disciplines. One sufficient condition for identifying the causal effect isthat the treatment-outcome relationship is unconfounded conditional on theobserved covariates. It is often believed that the more covariates we conditionon, the more plausible this unconfoundedness assumption is. This belief has hada huge impact on practical causal inference, suggesting that we should adjustfor all pretreatment covariates. However, when there is unmeasured confoundingbetween the treatment and outcome, estimators adjusting for some pretreatmentcovariate might have greater bias than estimators without adjusting for thiscovariate. This kind of covariate is called a bias amplifier, and includesinstrumental variables that are independent of the confounder, and affect theoutcome only through the treatment. Previously, theoretical results for thisphenomenon have been established only for linear models. We fill in this gap inthe literature by providing a general theory, showing that this phenomenonhappens under a wide class of models satisfying certain monotonicityassumptions. We further show that when the treatment follows an additive ormultiplicative model conditional on the instrumental variable and theconfounder, these monotonicity assumptions can be interpreted as the signs ofthe arrows of the causal diagrams.

Instrumental variables as bias amplifiers with general outcome and confounding