Table 3 Dahabreh et al. identifiability conditions when pooling trials

B1. Consistency of potential outcomes: If T_i = t; then Y_i^t = Y_i, for every individual i in the target population or the populations underlying the trials in S

B2. Conditional exchangeability over treatment assignment T:

E[Y^t ∣ X = x; S = s; T = t] = E[Y^t ∣ X = x; S = s], for every trial s ∈ S, each treatment t ∈ T, and every x with f(x; S = s) > 0

B3. Positivity of the treatment assignment probability in the trials:

For every treatment t ∈ T, Pr[T = t ∣ X = x; S = s] > 0 for every trial s ∈ S and every x with f (x; S = s) > 0

B4. Conditional exchangeability in measure between the trial and the target population: For every pair of treatments t and t′ in T, E[Y^t – Y^{t ′}∣ X = x; S = 0] = E[Y^t – Y^{t ′}∣ X = x; S = s] for every trial s ∈ S and every x with f (x; S = 0) > 0

B5. Positivity of the probability of participation in the trials: Pr[S = s ∣ X = x] > 0 for every s ∈ S and every x with f (x; S = 0) > 0

Under conditions B4 and B5 the conditional mean difference of each trial is equal to the conditional causal effect of the target population:

E[Y^t – Y^{t ′}∣ X; S = 1] = E[Y^t – Y^{t ′}∣ X; S = m] = E[Y^t – Y^{t ′}∣ X; S = 0]

Under conditions B1–B3, the common conditional mean difference is giver from the formula:

τ(t; t ′; X) ≡ E[Y ∣ X; S = 1; T = t] – E[Y ∣ X; S = 1; T = t′] = … = E[Y ∣ X; S = m; T = t] – E[Y ∣ X; S = m; T = t ′]

Finally, the ATE for the target population is:

E[Y^t – Y^{t ′}∣ X; S = 0] ≡ E[τ(t; t ′; X) ∣ S = 0]