Sequentially Randomized Treatment as a Reference Case

Why Sequential Randomization Is a Reference Case

In a longitudinal study, treatment may be assigned more than once. At each decision, the assignment can depend on information already recorded, such as a participant’s current health. A trial that randomizes treatment conditional on that recorded history is a useful reference case: it shows precisely what sequential exchangeability means when assignment is controlled by design rather than inferred from observational data.

Randomization at each decision does not by itself guarantee that every causal comparison is identified. The treatments must have well-defined versions, observed outcomes must correspond to the potential outcomes for the treatments actually received, and the design must give relevant treatment options positive probability at the histories where they are needed. These are the roles of consistency and positivity.

The relevant potential outcomes include \(Y^{\bar a}\) and, when covariates evolve over time, future covariates under the specified past treatment sequence. This matters because a later treatment may be randomized conditional on a covariate affected by earlier treatment. The randomization condition is therefore local to each decision: treatment is independent of relevant future potential outcomes after conditioning on what is known at that decision.

How the Design Implies Exchangeability

Suppose that at each decision the trial uses a randomization mechanism whose probabilities may depend on \(H_t\), but whose random draw is independent of the relevant potential outcomes conditional on \(H_t\) and the past treatment history. In particular, an adaptive design can assign different probabilities to different recorded histories. It remains randomized as long as its random draw does not use unrecorded information about the potential outcomes.

Proof. Fix a decision time \(t\), a history \(h_t\), and a past treatment sequence \(\bar a_{t-1}\). Let \(Q_t\) denote the vector of relevant potential outcomes. The randomization assumption says that, conditional on this history and past treatment, the assignment draw is independent of \(Q_t\). Since \(A_t\) is generated from that draw and the recorded history, its conditional distribution does not depend on \(Q_t\). Thus, for every treatment value \(a_t\) and every measurable set \(B\),

This factorization is conditional independence, which proves the required exchangeability statement. The argument applies at each decision, not just at baseline. \(\square\)

Consistency and Positivity Are Additional Conditions

Sequential exchangeability concerns how treatment is assigned relative to potential outcomes. Two further conditions connect that assignment process to causal quantities defined for treatment regimes.

For randomized studies, positivity is often built into the design by assigning every relevant option a nonzero probability at each eligible history. But a design that deterministically assigns treatment for some histories does not meet positivity for a regime that requests the opposite treatment there. Consistency can also fail even in a randomized study if a treatment label combines versions that have different effects and are not specified by the intervention.

Deriving a Regime Mean

The connection between the assumptions and identification can be seen directly in a discrete-data setting. Suppose the covariates and treatments have finite support, there is no loss to follow-up, and the regime \(\bar a\) has sequential positivity. Let \(p(l_0)\) denote the baseline covariate mass function and \(p(l_{t+1}\mid \bar l_t,\bar a_t)\) the observed covariate transition distribution under the indicated treatment history.

Derivation. First condition on the covariate history that would occur under the regime. By the law of total expectation,

At each time, exchangeability says that among people with the specified observed history, treatment assignment does not select people on the relevant potential covariates or outcome. Consistency equates those potential values with observed values for people who followed the corresponding treatment history. Positivity ensures that the conditional observed distributions used in this substitution exist for each regime-relevant history. Applying these facts successively, the regime covariate-history probability becomes

and the conditional mean of the final potential outcome becomes \(E[Y\mid \bar L_K=\bar l_K,\bar A_K=\bar a]\). Substitution yields the displayed result. This derivation uses the finite-support assumption to write sums; with continuous covariates, the corresponding expression uses integrals and conditional densities. \(\square\)

Common Pitfalls and Practical Checks

A randomized design is a reference case, not a label that automatically validates every causal analysis. The design must randomize the treatment actually used in the causal question, and its assignment probabilities must support the regime of interest.

• Confusing conditional and marginal probabilities. A product of assignment probabilities along one specified history is a path-specific probability. The sequence probability conditional only on baseline averages over intermediate histories, as in the first worked example.
• Calling baseline randomization sufficient for every later decision. Later assignments can be associated with potential outcomes if clinicians choose treatment using evolving prognostic information. Exchangeability must hold at each decision, conditional on the recorded history.
• Ignoring treatment versions. If “treatment” includes materially different doses or delivery methods, consistency for that broad label may not hold. Define the intervention precisely enough that the potential outcome is well specified.
• Checking support only in the whole sample. A treatment may occur often overall but never occur for a particular relevant history. That is a local positivity failure for regimes requiring that treatment at that history.

In practice, inspect the design protocol and the recorded history used at every randomization, verify that the treatment definition matches the causal question, and tabulate or model assignment probabilities by relevant history. In observational studies, these same assumptions must be defended from substantive knowledge and data; they do not follow from an observed association pattern. Positivity diagnostics can reveal sparse or unsupported histories, but no diagnostic alone establishes exchangeability or consistency.