Identification Assumptions: Exchangeability, Positivity and Consistency

Every observational causal estimate ends in the same algebra: replace \(\mathbb{E}[Y(t)]\), which nobody observed, with \(\sum_x \mathbb{E}[Y \mid T=t, X=x]\,P(X=x)\), which a spreadsheet can compute. That substitution is three separate assumptions stacked on top of each other, and a failure in any one of them produces a well-defined number that is not the causal effect. The potential outcomes concept introduces the notation and SUTVA; this concept is about the proof those assumptions license and how each one fails in practice (Hernán and Robins, 2020, Causal Inference: What If, Chapman & Hall/CRC).

The proof, one line per assumption

and positivity is what makes the second line meaningful, because \(\mathbb{E}[\,\cdot \mid T=t, X=x]\) is undefined when no unit with \(X = x\) received \(t\).

Conditional exchangeability, \(Y(t) \perp T \mid X\) for each \(t\), says that within levels of \(X\) the treated and untreated are interchangeable: had they swapped treatments, they would have had each other's outcome distributions. It is the formal content of "no unmeasured confounding", and it cannot be tested from the data used to estimate the effect.

Positivity, \(0 < P(T = t \mid X = x)\) for all \(x\) with \(P(x) > 0\), says every covariate stratum has some chance of each treatment.

Consistency, \(T = t \Rightarrow Y = Y(t)\), says the outcome you recorded for a treated unit is the potential outcome under the treatment you are asking about.

Positivity: the one you can partly check

Positivity is the only assumption with a direct empirical diagnostic, and it fails in two distinct ways. Structural violations happen when a rule makes treatment impossible: a feature only shipped to paying users can say nothing about free users. Random violations happen when a stratum is possible but sparse. Both create extreme weights and model-dependent extrapolation (Petersen et al., 2012, Diagnosing and Responding to Violations in the Positivity Assumption, Statistical Methods in Medical Research 21(1)).

The arithmetic is unforgiving. If 2% of users sit in a segment with \(e(x) = 0.01\), each treated unit there carries an inverse-probability weight of 100. With 5,000 users in the segment, you expect 50 treated ones, and their outcomes stand in for all 5,000. An outcome regression avoids the weights but fills the same gap by extrapolating the fitted surface into a region with almost no treated data, which is the same problem hidden more politely. Responses all change the question: trim the segment and redefine the target population, or pick an estimand such as the effect among the treated that needs overlap in one direction only.

Consistency: an assumption about the treatment's definition

Consistency sounds tautological and is not. It fails whenever "treated" covers several versions with different effects. A "discount" that was 5% for some users and 30% for others does not have one potential outcome \(Y(1)\), and an estimate averages over whatever mix of versions happened to be delivered (VanderWeele and Hernán, 2013, Causal Inference under Multiple Versions of Treatment, Journal of Causal Inference 1(1)). The effect then transports only to a future with the same mix.

Whether consistency is even an assumption is disputed. Cole and Frangakis treat it as a substantive assumption linking data to counterfactuals (Cole and Frangakis, 2009, Epidemiology 20(1)). Pearl argues that in a structural causal model it is a theorem that follows from the definition of \(Y(t)\) (Pearl, 2010, On the Consistency Rule in Causal Inference, Epidemiology 21(6)). The practical disagreement is sharper: Hernán and Taubman argue that exposures such as "obesity" lack a well-defined intervention, so their causal effect is ill-posed (Hernán and Taubman, 2008, Does Obesity Shorten Life?, International Journal of Obesity 32), while structural modellers accept such variables if a mechanism can be written down.

SUTVA's no-interference clause is the fourth hidden condition. If one unit's treatment changes another's outcome, \(Y_i(t)\) is not a function of \(t\) alone, and the first line of the proof has nothing to condition on.

When it breaks

Exchangeability is untestable and usually the weakest link. Balance tables and pre-trends check implications on measured variables only. What remains is a sensitivity analysis: how strong would an omitted confounder need to be to change the conclusion.

Adding covariates can repair exchangeability and break positivity. Every variable added to \(X\) splits strata finer and makes empty treatment cells more likely. With 30 binary covariates there are over a billion cells, and most are empty. The standard compromise, modelling across cells, converts a positivity problem into a functional form assumption.

Consistency failures masquerade as heterogeneity. Two studies of "the same" treatment that disagree may have delivered different versions of it. Logging which version a unit actually received is a design decision that no later analysis can recover.

All three are needed for each target. An effect among the treated needs exchangeability only for \(Y(0)\) and positivity only in the direction \(P(T = 0 \mid X) > 0\). Choosing the estimand is partly a choice of which assumptions you can defend.