Table 3.

Examples of graphs for Z, A, U relationships and for W, Y, U relationships. The two pieces of graphs can be combined in to a directed acyclic graph that encodes the negative control assumptions. Gray-colored graphs are invalid because of violation of key assumptions

Examples of graphs for Z, A, U relationships
	Z → A (pre-treatment)	A → Z (post-treatment)	Z ⫫ A
No arrow between U and Z (may violate Assumption 5 and 7)
U → Z
Z → U	May violate Assumption 4 if there is W → U
Examples of graphs for W, Y, U relationships
	W → Y (a)	Y (a) → W (violate Assumptions 3 and 4)	Y (a) ⫫ W | (U, X)
No arrow between U and W (violate Assumption 5 and 7)
U → W
	May violate Assumption 4 if there is Z → U
W → U