TABLE 1.

A selection of causal estimands for exposures A₁ and A₂

Estimand	Definition
Effect of program offer (${\mathfrak{a}}_1$)
${\text{ATE}}_1={\text{ATT}}^{\mathrm{a}}$	Average treatment effect
	$E[Y_{{\mathfrak{a}}_1(1)}]-E[Y_{{\mathfrak{a}}_1(0)}]$
Effect of program uptake (${\mathfrak{a}}_2$)
ATE2	Average treatment effect
	$E[Y_{{\mathfrak{a}}_2(1)}]-E[Y_{{\mathfrak{a}}_2(0)}]$
ATT2	Average treatment effect among the treatedb
	$E[(Y_{{\mathfrak{a}}_2(1)}|A_2=1,A_1=1]-E[Y_{{\mathfrak{a}}_2(0)}|A_2=1,A_1=1]$
ATNT2	Average treatment effect among the nontreatedb
	$E[Y_{{\mathfrak{a}}_2(1)}|A_2=0,A_1=1]-E[Y_{{\mathfrak{a}}_2(0)}|A_2=0,A_1=1]$

^aIntention‐to‐treat.

^bNote that the ATT and ATNT for ${\mathfrak{a}}_2$ can only be derived from the (random) subgroup A₁ = 1 since the program is only available within the randomized trial and to those assigned to it being offered.