Positivity at Each Treatment Decision

Why Positivity Is a Decision-by-Decision Requirement

A treatment regime specifies what treatment is to be assigned at each decision time. To learn the outcome that would result under that regime from observed data, the data must include people who could receive each prescribed treatment at the histories the regime can produce. It is not enough that each treatment occurs somewhere in the study: a treatment may be common overall and absent among people with a particular history.

This requirement is called sequential positivity. It is about the support of treatment assignment at each decision, conditional on the history available then. It is distinct from exchangeability: exchangeability concerns whether treatment assignment is confounded after conditioning on history, whereas positivity concerns whether the required assignment occurs at all.

For a discrete history \(h\), the condition can be checked by inspecting the relevant conditional probability in the population treatment-assignment law. At baseline, check \(P(A_0=a_0\mid L_0=l_0)\) for baseline values that occur under the regime. At a later decision, check \(P(A_t=a_t\mid H_t=h)\) among histories reachable after the regime’s earlier prescribed treatments. Each time has its own condition; a positive probability at one decision does not establish support at another.

This tutorial focuses on a fixed regime and discrete histories. It does not require the prescribed treatment to be equally likely across histories, nor does it require every possible treatment to occur at every history. Only the level prescribed by the regime must have positive probability at histories the regime can produce.

Checking Support in Concrete Examples

Two Consequences of the Condition

The first consequence is that positivity lets the observed assignment process include the prescribed treatment at each reachable decision. More precisely, in a finite discrete setting, positive treatment probabilities at the successive reachable histories ensure positive probability for each regime-compatible observed trajectory that has positive probability under the regime’s covariate process.

To see this, take a trajectory \((l_0,a_0,l_1,a_1,\ldots,l_K,a_K)\) compatible with \(g\). Its observed probability factors by the chain rule as

where the final covariate-transition factor is omitted when there is no \(L_{K+1}\). A positive-probability trajectory under \(g\) has a positive baseline factor and positive covariate-transition factors along that trajectory. Sequential positivity makes each treatment factor positive. The product of these finitely many positive factors is positive, proving the result.

A second consequence explains why a zero probability is more than a technical inconvenience. If a treatment is never observed at a reachable history, the observed data cannot determine the outcome that would have followed that treatment at that history without additional assumptions.

For a direct proof, consider one decision, a history stratum \(H=h\) with positive probability, and binary outcomes. Suppose treatment \(a\) is never given in that stratum. Construct two causal models with exactly the same observed data: in the first, set the unobserved potential outcome \(Y^a=0\) for everyone with \(H=h\); in the second, set \(Y^a=1\) for everyone in that stratum. Keep all other potential outcomes and the treatment assignment mechanism the same. Since no one with \(H=h\) receives \(a\), consistency implies that \(Y^a\) is never observed there. The two models therefore induce the same observed-data distribution, yet their conditional means \(E[Y^a\mid H=h]\) are 0 and 1. The observed data cannot distinguish them, so that mean is not nonparametrically identified. \(\square\)

Practical Assessment and Common Pitfalls

In practice, assess positivity by organizing the check around decision times and the regime’s reachable histories. For each decision, list the prescribed treatment level and the histories that can arise under the earlier regime assignments. Then examine the corresponding conditional assignment probabilities. In a sample, empty cells are a warning sign, but a nonempty cell by itself does not prove that the population probability is positive.

• Overall frequencies are not enough. A treatment can have a substantial marginal frequency and still have probability zero at a particular reachable history.
• Do not infer a specific history from a marginal covariate probability. To establish reachability, verify the relevant conditional probabilities along the history, not merely that each covariate value occurs somewhere.
• Do not test irrelevant histories. The fixed regime requires support at histories it can produce, not at every combination of values one can write down.
• Regression extrapolation is not observed support. A fitted model may return a predicted outcome for an unsupported treatment-history combination, but that prediction relies on modeling assumptions rather than data support there.

The main alternative to a history-specific check is to look only at marginal treatment proportions or to rely on an outcome model to extend across empty treatment-history cells. Neither establishes sequential positivity. The former can miss conditional zero probabilities; the latter may produce estimates only through extrapolation. Positivity is a property of the treatment-assignment distribution relative to the regime, not a guarantee that any particular estimation method will behave well.

Finally, distinguish a structural zero from an apparent sample zero. A population probability may be positive even when a finite study happens to contain no treated individuals in a small history stratum. That sample pattern signals uncertainty about support, but it does not by itself prove population positivity or its failure. Conversely, a genuinely zero population probability violates the condition regardless of sample size. Very small positive probabilities also deserve attention because they can make estimates unstable, even though they do not violate the strict mathematical condition.