TABLE 2.

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

Assumption1	Effect2	Lower Bound	Upper Bound
None	Total	−Pr(Y≠X)	Pr(Y=X)
	CDE	−Pr(Y≠X)−Pr(Y=X, Z≠z)	Pr(Y=X)+Pr(Y≠X, Z≠z)
	NDE	−Pr(Y≠X)−Pr(Y=X=1)	Pr(Y=X)+Pr(Y≠X=1)
{1′}	Total	−Pr(Y≠X)	Pr(Y=X)
{2′}	Total	−Pr(Y≠X)	Pr(Y=X)
	CDE	−Pr(Y≠X)−Pr(Y=X=Z≠z)	Pr(Y=X)+Pr(X=z, Y=Z≠z)
	NDE	−Pr(Y≠X)−Pr(Y=X=Z=1)	Pr(Y=X)
{3′}	Total, NDE	0	Pr(Y=X)
	CDE	0	Pr(Y=X)+Pr(X=z, Y=Z≠z)
{4′}	Total, CDE, NDE	0	Pr(Y=X)
{5′}	Total3	Pr(Y=1|X=1)−Pr(Y=1|X=0)	Pr(Y=1|X=1)−Pr(Y=1|X=0)
	CDE	−1+Pr(Y=0,Z=z|X=0) + Pr(Y=1,Z=z|X=1)	1−Pr(Y=1,Z=z|X=0)−Pr(Y=0,Z=z|X=1)
	NDE4	$\begin{array}{cc}\max & \left\{\begin{array}{c}-\mathrm{Pr}(Y=1|X=0),\\ -1+{\mathrm{p}}^{\prime }(0,1|0)-{\mathrm{p}}^{\prime }(1,0|0)+{\mathrm{p}}^{\prime }(1,1|1),\\ -1+{\mathrm{p}}^{\prime }(0,0|0)-{\mathrm{p}}^{\prime }(1,1|0)+{\mathrm{p}}^{\prime }(1,0|1)\end{array}\right\}\end{array}$	$\begin{array}{cc}\min & \left\{\begin{array}{c}\mathrm{Pr}(Y=0|X=0),\\ 1+{\mathrm{p}}^{\prime }(0,1|0)-{\mathrm{p}}^{\prime }(1,0|0)-{\mathrm{p}}^{\prime }(0,0|1),\\ 1+{\mathrm{p}}^{\prime }(0,0|0)-\mathrm{p}\prime (1,1|0)-{\mathrm{p}}^{\prime }(0,1|1)\end{array}\right\}\end{array}$
{1′,2′}, {1′,3′}, {1′,4′}	Total	0	Pr(Y=X=Z)
{2′,5′}	CDE	Pr(Y=1|X=1) − Pr(Y=1|X=0) − Pr(Y=Z≠z|X≠z)	Pr(Y=1|X=1) − Pr(Y=1|X=0) + Pr(Y=Z≠z|X=z)
	NDE	$\begin{array}{cc}\max & \left\{\begin{array}{c}{\mathrm{p}}^{\prime }(1,0|1)-{\mathrm{p}}^{\prime }(0,1|1)+{\mathrm{p}}^{\prime }(0,1|0)-{\mathrm{p}}^{\prime }(1,0|0),\\ {\mathrm{p}}^{\prime }(1,0|1)-{\mathrm{p}}^{\prime }(1,1|0)-{\mathrm{p}}^{\prime }(1,0|0)\end{array}\right\}\end{array}$	Pr(Y=1|X=1) − Pr(Y=1|X=0)
{3′,5′}	CDE	$\begin{array}{cc}\max & \left\{\begin{array}{c}0,\\ \mathrm{Pr}(\mathrm{Y}\ne Z=z|X\ne z)-\mathrm{Pr}(\mathrm{Y}\ne Z=z|X=z)\end{array}\right\}\end{array}$	Pr(Y≠z|X≠z)−Pr(Y≠Z=z|X=z)
	NDE	$\begin{array}{cc}\max & \left\{\begin{array}{c}0,\\ {\mathrm{p}}^{\prime }(0,1|0)-{\mathrm{p}}^{\prime }(0,1|1),\\ {\mathrm{p}}^{\prime }(1,0|1)-{\mathrm{p}}^{\prime }(1,0|0),\\ {\mathrm{p}}^{\prime }(0,1|0)-{\mathrm{p}}^{\prime }(0,1|1)+{\mathrm{p}}^{\prime }(1,0|1)-{\mathrm{p}}^{\prime }(1,0|0)\end{array}\right\}\end{array}$	Pr(Y=1|X=1) − Pr(Y=1|X=0)
{4′,5′}	CDE	$\begin{array}{cc}\max & \left\{\begin{array}{c}0,\\ \mathrm{Pr}(Y\ne z,Z=z|X\ne z)-\mathrm{Pr}(Y\ne z,Z=z|X=z),\\ \mathrm{Pr}(Y=z,Z\ne z|X=z)-\mathrm{Pr}(Y=z,Z\ne z|X\ne z),\\ \mathrm{Pr}(Y\ne z,Z=z|X\ne z)-\mathrm{Pr}(Y\ne z,Z=z|X=z)+\\ \mathrm{Pr}(Y=z,Z\ne z|X=z)-\mathrm{Pr}(Y=z,Z\ne z|X\ne z)\end{array}\right\}\end{array}$	Pr(Y=1|X=1) − Pr(Y=1|X=0)
	NDE	$\begin{array}{cc}\max & \left\{\begin{array}{c}0,\\ {\mathrm{p}}^{\prime }(0,1|0)-{\mathrm{p}}^{\prime }(0,1|1),\\ {\mathrm{p}}^{\prime }(1,0|1)-{\mathrm{p}}^{\prime }(1,0|0),\\ {\mathrm{p}}^{\prime }(0,1|0)-{\mathrm{p}}^{\prime }0,1|1)+{\mathrm{p}}^{\prime }(1,0|1)-{\mathrm{p}}^{\prime }(1,0|0)\end{array}\right\}\end{array}$	Pr(Y=1|X=1) − Pr(Y=1|X=0)

¹

Assumptions are defined in Section 3.3

• {1} No direct X →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 3.1

• Total = Total Average Causal Effect

  • RD_C = Pr(Y=1| SET[X=1]) − Pr(Y=1| SET[X=0])
• CDE = Z-Controlled Direct Effect

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

  • RD_{C|SET[Z=Z(0)]} = Pr(Y=1| SET[X=1], SET[Z=Z₀]) −Pr(Y=1| SET[X=0])

³Causal effect is completely determined (bound width = 0)

⁴

p′ notation:

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