  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.
