  We introduce the concept of pattern graphs--directed acyclic graphs
representing how response patterns are associated. A pattern graph represents
an identifying restriction that is nonparametrically identified/saturated and
is often a missing not at random restriction. We introduce a selection model
and a pattern mixture model formulations using the pattern graphs and show that
they are equivalent. A pattern graph leads to an inverse probability weighting
estimator as well as an imputation-based estimator. We also study the
semi-parametric efficiency theory and derive a multiply-robust estimator using
pattern graphs.
