TABLE 1.

Analytic Bounds on Risk Difference Causal Effects in the Pretreatment-covariate DAG [Z →X →Y, Z →Y] under Various Assumptions

Assumption1	Effect Measure2	Lower Bound	Upper Bound
None	RDC	−Pr(Y≠X)	Pr(Y=X)
	RDC|Z=z	−Pr(Y≠X|Z=z)	Pr(Y=X|Z=z)
	RDC|SET[Z=z]	−Pr(Y≠X)−Pr(Y=X,Z≠z)	Pr(Y=X)+Pr(Y≠X, Z≠z)
{1}	RDC, RDC|SET[Z=z]	−Pr(Y≠X)	Pr(Y=X)
	RDC|Z=z	−Pr(Y≠X |Z=z)	Pr(Y=X|Z=z)
{2}	RDC	0	Pr(Y=X)
	RDC|Z=z	0	Pr(Y=X|Z=z)
	RDC|SET[Z=z]	0	Pr(Y=X)+Pr(Y≠X, Z≠z)
{3}	RDC	0	Pr(Y=X)
	RDC|Z=z	0	Pr(Y=X|Z=z)
	RDC|SET[Z=z]	0	Pr(Y=X)+Pr(Y≠z, X=z, Z≠z)
{4}	All	0	min{Pr(Y=X|Z=0), Pr(Y=X|Z=1)}
{5}	RDC	−Pr(Y≠X)	Pr(Y=X)
	RDC|Z=z, RDC|SET[Z=z]	−Pr(Y≠X|Z=z)	Pr(Y=X|Z=z)
{1,5}	All	max{a1,…, a8}3	min{b1,…, b8}3
{2,5}	RDC	0	Pr(Y=X)
	RDC|Z=z, RDC|SET[Z=z]	0	Pr(Y=X|Z=z)
{3,5}	RDC	0	$\begin{array}{cc}\min & \left\{\begin{array}{l}\mathrm{Pr}(Y=X),\\ 1-\mathrm{p}(1,0|0)-\mathrm{p}(0,1|1),\\ \mathrm{Pr}(Y=X)+\mathrm{p}(1,0,1)-\mathrm{p}(1,0,0),\\ \mathrm{Pr}(Y=X|Z=1)+\mathrm{p}(1,0,1)-\mathrm{p}(1,0,0),\end{array}\right\}\end{array}$
	RDC|Z=z, RDC|SET[Z=z]	0	$\begin{array}{cc}\min & \left\{\begin{array}{l}\mathrm{Pr}(Y=X|Z=z),\\ 1-\mathrm{p}(1,0|0)-\mathrm{p}(0,1|1)\end{array}\right\}\end{array}$
{4,5}	All	0	$\begin{array}{cc}\min & \left\{\begin{array}{l}\mathrm{Pr}(Y=X|Z=0),\\ \mathrm{Pr}(Y=X|Z=1),\\ \mathrm{Pr}(Y=X|Z=0)+\mathrm{p}(1,0|1)-\mathrm{p}(1,0|0),\\ \mathrm{Pr}(Y=X|Z=1)+\mathrm{p}(1,0|0)-\mathrm{p}(1,0|1)\end{array}\right\}\end{array}$
{1,2}, {1,3}, {1,4}	RDC, RDC|SET[Z=z]	0	Pr(Y=X)
	RDC|Z=z	0	Pr(Y=X|Z=z)
{1,2,5}, {1,3,5}, {1,4,5}	All	Pr(Y=1|Z=1) − Pr(Y=1|Z=0)	Pr(Y=1|Z=1) − Pr(Y=1|Z=0) + p(1,1| 0) + p(0,0| 1)

¹

Assumptions are defined in Section 2.3

• {1} No direct Z →Y effect
• {2} Partial monotonicity
• {3} Full monotonicity
• {4} Full monotonicity and no interaction between Z and X
• {5} Exogenous (e.g. randomized)

²

Effect Measures are defined in Section 2.2

• RD_C = Pr(Y=1| SET[X=1]) − Pr(Y=1| SET[X=0])
• RD_C|Z=z = Pr(Y=1| SET[X=1], Z=z) − Pr(Y=1| SET[X=0], Z=z)
• RD_C|SET[Z=z] = Pr(Y=1| SET[X=1], SET[Z=z]) − Pr(Y=1| SET[X=0], SET[Z=z])

³Balke & Pearl bounds (1997)

⁴

p notation:

• p(y,x,z) = Pr(Y=y, X=x, Z=z)
• p(y,x | z)= Pr(Y=y, X=x | Z=z)