We consider the problem of identifying a conditional causal effect through
covariate adjustment. We focus on the setting where the causal graph is known
up to one of two types of graphs: a maximally oriented partially directed
acyclic graph (MPDAG) or a partial ancestral graph (PAG). Both MPDAGs and PAGs
represent equivalence classes of possible underlying causal models. After
defining adjustment sets in this setting, we provide a necessary and sufficient
graphical criterion -- the conditional adjustment criterion -- for finding
these sets under conditioning on variables unaffected by treatment. We further
provide explicit sets from the graph that satisfy the conditional adjustment
criterion, and therefore, can be used as adjustment sets for conditional causal
effect identification.Comment: 29 pages, 6 figure