Table 2 Sobel et al. identifiability conditions

A1. Extended stable unit treatment value assumption (eSUTVA): For all possible assignments t and allocations s, Y_i(s; t) = Y_i(s_i; t_i) ≡ Y_i(s; t)

A2. Study sampling assumption: For all subjects i in study s, s = 1,…, m, the random vectors Y_i; X_i | S_i = s; T_i = t are independent and identically distributed Y; X | S = s; T = t

A3a. Strong response consistency assumption for treatment t: For all s; s’ and subjects i, Y_i(s; t) = Y_i(s’; t)

A3b. Weak response consistency assumption for treatment t: For all s, s’ and X:

F (y(s; t) | S = s’; X = x) = F (y(s’; t) | S = s’; X = x)

A4. Weak consistency of effects of treatment t versus t’: For all s, s’ and X, the causal estimands:

H (F (y(s; t) | S = s’; X = x); F (y(s; t’) | S = s’; X = x)) = H (F (y(s’; t) | S = s’; X = x); F (y(s’; t’) | S = s’; X = x))

A5a. Strong equivalence of treatments t₁ and t₂ in study s: For all i: Y_i(s; t₁) = Y_i(s; t₂)

A5b. Weak equivalence of treatments t₁ and t₂ in study s:

F (y(s; t₁) | S = s; X = x) = F (y(s; t₂) | S = s; X = x)

A6. Unconfounded treatment assignment given observed covariates: for every s, and treatment t ∈ T _s, F (y(s; t) | T = t; S = s; X₁ = x₁) = F (y(s; t) | S = s; X₁ = x₁)

A7. Unconfounded study selection, given observed covariates: For all studies s, s’ and treatments t, F (y(s; t) | S = s; X₂ = x₂) = F (y(s; t) | S = s’; X₂ = x₂)