TABLE 1.

Three estimation strategies (OR = outcome regression, IPW = inverse probability weighting, AIPW = augmented IPW) for the ATE in the target population, when either the combined sample (generalizability, the ATE is τ) or the S = 0 sample (transportability, the ATE is τ₀) is a random sample of the target population. We present three representation of E [Y (a)] and E [Y (a) | S = 0], a = 0 or 1, to simplify exposition.

	Generalizability (E [Y (a)])	Transportability (E [Y (a) | S = 0])
(S1) Covariates are measured on all individuals in the S = 0 sample, i.e., we have (X, S = 0)
OR	E {ma (X)}	$E\left\{m_a(X)\mid S=0\right\}=E\left\{\frac{\mathbb{1}(S=0)}{P(S=0)}{m}_a(X)\right\}$
IPW	$E\left\{\frac{\mathbb{1}(S=1,A=a)}{P(S=1,A=a\mid X)}Y\right\}$**(1)	$E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)P(S=0)}Y\right\}=\frac{E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)}Y\right\}}{E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)}\right\}}$**(2)
AIPW	$E\left\{\frac{\mathbb{1}(S=1,A=a)}{P(S=1,A=a\mid X)}\left(Y-m_a(X)\right)+m_a(X)\right\}$	$E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)P(S=0)}\left(Y-m_a(X)\right)+\frac{\mathbb{1}(S=0)}{P(S=0)}{m}_a(X)\right\}$
(S2.1) Covariates are measured on all individuals in the S = 0 sample and P (D = 1 | S = 0) is known
OR	$E\left[E\left\{m_a(X)\mid S,D=1\right\}\right]=E\left\{\frac{\mathbb{1}(D=1)}{P(D=1\mid S)}{m}_a(X)\right\}$**(3)	E {ma (X) | S = 0, D = 1} **(4)
IPW	$E\left\{\frac{\mathbb{1}(S=1,A=a)}{P(S=1,A=a\mid X)}Y\right\}$**(5)	$E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)P(S=0)}Y\right\}$**(6)
AIPW	$E\left\{\frac{\mathbb{1}(S=1,A=a)}{P(S=1,A=a\mid X)}\left(Y-m_a(X)\right)+\frac{\mathbb{1}(D=1)}{P(D=1\mid S)}{m}_a(X)\right\}$	$E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)P(S=0)}\left(Y-m_a(X)\right)+\frac{\mathbb{1}(S=0,D=1)}{P(S=0,D=1)}{m}_a(X)\right\}$
(S2.2) Covariates are measured on a subsample of the S = 0 sample and P (D = 1 | S = 0) is unknown
OR	Not identifiable **(7)	E {ma(X) | S = 0, D = 1}
IPW	Not identifiable **(8)	$E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X,D=1)}{P(A=a\mid S=1,X)P(S=1\mid X,D=1)P(S=0,D=1)}Y\right\}$**(9)
AIPW	Not identifiable	$E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X,D=1)}{P(A=a\mid S=1,X)P(S=1\mid X,D=1)P(S=0,D=1)}\left(Y-m_a(X)\right)+\frac{\mathbb{1}(S=0,D=1)}{P(S=0,D=1)}{m}_a(X)\right\}$

⁽¹⁾P (S = 1, A = a | X) = P (A = s | S = 1, X)P (S = 1 | X) where P (A = s | S = 1, X) is designed by the investigator in an RCT and can also be estimated based on the S = 1 sample, and P (S = 1 | X) identified from the combined sample.

⁽²⁾$E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)}\right\}=P(S=0)$, hence $E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)P(S=0)}Y\right\}=E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)}Y\right\}/E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)}\right\}$.

⁽³⁾$P\left(D=1|S\right)=\mathbb{1}\left(S=1\right)+\mathbb{1}\left(s=0\right)P\left(D=1|S=0\right)$.

⁽⁴⁾E {m_a (X) | S = 0} = E {m_a (X) | S = 0, D = 1} because D ⊥ X | S.

⁽⁵⁾P (S = 1 | X) identified by $\frac{P(S=1\mid X)}{P(S=0\mid X)}=\frac{P(S=1\mid X,D=1)}{P(S=0\mid X,D=1)}P(D=1\mid S=0)$. Estimation strategies are proposed in Dahabreh et al. (2019a).

⁽⁶⁾Similar to footnote ⁽⁵⁾, P (S = 1) identified by $\frac{P(S=1)}{P(S=0)}=\frac{P(S=1\mid D=1)}{P(S=0\mid D=1)}P(D=1\mid S=0)$.

⁽⁷⁾Unlike footnote ⁽³⁾, P (D = 1 | S) is not identifiable because P (D = 1 | S = 0) is unknown.

⁽⁸⁾Unlike footnote ⁽⁵⁾, P (S = 1 | X) is not identifiable because P (D = 1 | S = 0) is unknown.

⁽⁹⁾By footnote ⁽¹⁾ and ⁽⁵⁾, $E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X)}{P(S=1,A=a\mid X)P(S=0)}Y\right\}=E\left\{\frac{\mathbb{1}(S=1,A=a)P(S=0\mid X,D=1)}{P(A=a\mid S=1,X)P(S=1\mid X,D=1)P(S=0,D=1)}Y\right\}$. Although P (S = 1 | X) is not identifiable as shown in footnote ⁽⁸⁾, P (S = 1 | X, D = 1) is identifiable.