Adjustments and their ConsequencesuCollapsibility Analysis using Graphical Models

We review probabilistic and graphical rules for detecting situations in which a dependence of one variable on another is altered by adjusting for a third variable (i.e., non-collapsibility or non-invariance under adjustment), whether that dependence is causal or purely predictive. We focus on distinguishing situations in which adjustment will reduce, increase, or leave unchanged the degree of bias in an association that is taken to represent a causal effect of one variable on the other. We then consider situations in which adjustment may partially remove or introduce a bias in estimating causal effects, and some additional special cases useful for case-control studies, cohort studies with loss, and trials with non-compliance (non-adherence).