Assumption Audit for a Specified Treatment Regime

From a Treatment Question to an Assumption Audit

An assumption audit starts with a particular intervention question, not with a generic list of conditions. The question specifies who is eligible, when treatment decisions occur, what information is available at each decision, and which rule assigns treatment. Those details determine the counterfactual outcome of interest and the histories on which positivity must hold.

Use the following timeline. Let $L_0$ denote baseline covariates. At each time $t=1,\ldots,T$, treatment $A_t$ is assigned using information available before that decision; $L_t$ is then measured after treatment and before the next decision. The outcome $Y$ is measured after $A_T$. Define the observed pre-treatment history as $H_t=(L_0,A_1,L_1,\ldots,A_{t-1},L_{t-1})$. A specified regime $g$ gives a treatment $g_t(h)$ for each possible history $h$ at time $t$. A static regime sets treatment to a fixed value at each time; a dynamic regime can depend on the observed covariate history.

Definition: The potential outcome $Y^g$ is the outcome that would be observed if, at each decision time $t$, treatment were assigned according to $g_t(H_t)$, with the history evolving under that same regime. The regime-specific mean is $E[Y^g]$. Define $L_t^g$ and $H_t^g$ similarly as the covariate and pre-treatment history that would occur under $g$.

This definition makes the rule part of the estimand. “Treat if blood pressure is high” is incomplete until “high,” the measurement time, and any other decision inputs are specified. If a rule depends on information unavailable at the decision, it does not define an implementable treatment regime.

The Four Parts of the Audit

Define the target.
State the population, outcome, follow-up period, and complete treatment rule $g$. Specify whether the target is a mean, a risk, or a contrast between regimes.

Check consistency.
Ask whether the recorded treatment that matches $g$ corresponds to the intervention represented by $g$, and whether observed outcomes then equal the appropriate potential outcomes.

Assess sequential exchangeability.
For each treatment decision, ask whether the measured pre-treatment history suffices to control common causes of that decision and subsequent potential outcomes.

Assess regime-specific positivity.
For histories the regime can generate, check whether the observed treatment mechanism gives positive probability to the treatment that the rule prescribes.

Precise Assumptions and a Regime-Specific Result

Consistency: For any treatment sequence $\bar a_T=(a_1,\ldots,a_T)$, if a person’s observed treatments through time $t$ equal $\bar a_t$, their observed covariate $L_t$ equals $L_t^{\bar a_t}$; if the full observed sequence equals $\bar a_T$, their observed outcome equals $Y^{\bar a_T}$. This requires that the treatment represented by $a_t$ have a sufficiently well-defined version: materially different versions cannot be silently treated as the same intervention.

Under regime consistency, if a person’s observed treatment at every time matches the regime applied to their observed history, then their observed outcome is $Y^g$. This is not merely a statement that the treatment labels agree: it also requires the intervention to be sufficiently well specified that the observed treatment implements the intervention being studied.

Sequential exchangeability: At each time $t$, conditional on the observed pre-treatment history $H_t$, the treatment assignment $A_t$ is independent of the potential future covariates and outcomes under each possible continuation of treatment. The condition is required at histories relevant to the regime. In words, after conditioning on the recorded past, treatment assignment must not carry further information about future potential outcomes.

The assumption concerns information available before the decision. Conditioning on a variable measured after $A_t$ does not establish exchangeability for that decision. Whether the recorded history is sufficient is a scientific claim about confounding, not a fact verified just by fitting a model.

Regime-specific positivity: For discrete treatments, for almost every history $h$ that occurs under $g$, the observed conditional probability satisfies $P(A_t=g_t(h)\mid H_t=h)>0$. For continuous treatments, the regime’s assigned value must lie in the supported treatment region, with the appropriate local density or support condition. Positivity is required at each decision along histories the regime can generate.

Positivity is therefore target-specific. A treatment can be absent at some histories that the regime never reaches without obstructing identification of that regime’s mean. Conversely, a high overall treatment frequency does not resolve a zero probability at a history the regime does reach.

Proposition: Under consistency and sequential exchangeability, at any regime-relevant history $h$ where the prescribed treatment has positive probability, the distribution of the next potential covariate under that treatment agrees with the observed conditional distribution: $$ P(L_t^{\bar a_t}=l\mid H_t=h) = P(L_t=l\mid H_t=h,A_t=a_t), \qquad a_t=g_t(h), $$ where $\bar a_t$ extends the treatment history represented in $h$ by $a_t$.

Proof. Sequential exchangeability states that, conditional on $H_t=h$, assignment $A_t$ is independent of the potential covariate $L_t^{\bar a_t}$. Thus its conditional distribution does not change when we also condition on $A_t=a_t$. Consistency says that among people with $H_t=h$ and $A_t=a_t$, the observed $L_t$ equals $L_t^{\bar a_t}$. Positive treatment probability ensures this conditional observed group is defined. Combining these statements gives the displayed equality. $\square$

The equality explains why the observed conditional covariate distribution can be used to build the covariate history under the regime. It does not say that the regime’s histories equal observed histories for each individual; it identifies their distributions using the assumptions.

Theorem: Suppose treatment and covariate variables have finite support, $E[|Y|]<\infty$, consistency and sequential exchangeability hold, and regime-specific positivity holds at every decision. Then the mean under regime $g$ is identified by $$ E[Y^g] = \sum_{l_0,\ldots,l_T} E[Y\mid L_0=l_0,\bar A_T=\bar g(\bar l_T),\bar L_T=\bar l_T]\, P(L_0=l_0) \prod_{t=1}^{T} P\!\left(L_t=l_t\mid H_t=h_t,\ A_t=g_t(h_t)\right), $$ where $h_t$ is the history formed from $l_0$, the earlier covariate values, and the regime-assigned earlier treatments; $\bar g(\bar l_T)$ denotes the resulting treatment sequence. Terms involving histories with zero probability under $g$ contribute zero.

Proof. The baseline distribution under the regime is $P(L_0=l_0)$, since treatment has not yet been assigned. Suppose the regime’s history distribution has been identified through the history just before time $t$. Under $g$, treatment at that history is $g_t(h_t)$. By the proposition, the conditional distribution of the next covariate $L_t^g$, given that history, is the observed distribution of $L_t$ given $H_t=h_t$ and $A_t=g_t(h_t)$. Multiplying this conditional distribution by the already identified history distribution gives the regime history distribution through $L_t$. Repeating this step for $t=1,\ldots,T$ gives the product of conditional covariate probabilities in the theorem. Finally, consistency and sequential exchangeability identify the conditional mean of $Y^g$ given the full regime history with the observed conditional mean of $Y$ among people whose treatments match the regime. Summing that conditional mean over the identified regime history distribution yields the displayed expression. Positivity ensures every conditional distribution used in this argument is observed at the required histories. $\square$

For continuous variables, the corresponding expression uses integrals and conditional densities or probability kernels. The finite-support statement makes the identification logic explicit without needing additional measure-theoretic conditions for those integrals.

Worked Audits

Worked Example: A Dynamic Blood-Pressure Rule

A study follows adults for two annual treatment decisions. At each visit $t$, $L_t$ includes the most recent systolic blood pressure and renal status, measured before the treatment decision. Define $g_t(H_t)=1$ if systolic pressure is at least 140 mmHg and $g_t(H_t)=0$ otherwise. The outcome $Y$ is a cardiovascular event by the end of follow-up. The target is $E[Y^g]$.

Potential outcome. $Y^g$ is the event outcome if treatment is assigned at each annual visit according to the pressure measured at that visit. The rule must specify what treatment $1$ means, the measurement protocol, and whether treatment begins or continues when the threshold is met.

Consistency. If a participant’s observed decisions match the threshold rule at both visits, the observed event outcome must equal $Y^g$. This requires, for example, that “treatment $1$” not combine materially different drug regimens with different effects unless that variation is part of the intervention definition.

Exchangeability. At each visit, the audit asks whether $H_t$ includes the relevant pre-decision causes of treatment and subsequent outcomes. For instance, if clinician assessments of frailty affect both treatment and event risk but are not recorded in $H_t$, sequential exchangeability is doubtful. This cannot be settled by observing that the pressure threshold is recorded accurately.

Positivity. Suppose that at a severe-pressure history $h$, the observed probability of treatment is $0.92$, and at a lower-pressure history it is $0.18$. The rule assigns treatment at the first history and no treatment at the second. The required probabilities are therefore $0.92>0$ and $1-0.18=0.82>0$. If both histories occur under the rule and these probabilities apply at each decision, this part of the audit passes for those histories. It does not establish exchangeability or consistency.

Worked Example: A Regime With a Positivity Failure

Suppose a two-decision regime assigns treatment at every visit. At the first visit, $P(A_1=1\mid L_0)=0.40$. Among people whose first treatment was $1$, a measured high-risk history $L_1=1$ can occur. At that second-visit history, however, the observed treatment mechanism gives $P(A_2=1\mid H_2=h)=0$, because the clinical protocol always withholds treatment from this group.

The regime can generate $H_2=h$: the first treatment has positive probability, and the high-risk covariate value can occur after it. Yet the regime prescribes $g_2(h)=1$, while the observed probability of that treatment is zero. Thus $P(A_2=g_2(h)\mid H_2=h)=0$, violating regime-specific positivity. A large sample cannot provide observed outcomes for the missing treatment-history combination, so the regime mean is not identified by the stated assumptions.

The first-decision probability $0.40$ is positive, and treatment may be common overall. Neither fact repairs the second-decision support failure. The audit locates the failure at the particular regime-reachable history and decision where the prescribed treatment is absent.

Worked Example: A Randomized Decision and an Unmeasured Predictor

At a treatment visit, a trial randomizes $A_t$ with probability $0.5$ to each arm within every recorded history $H_t$. The next outcome is $Y$. Assume the treatment is well defined and follow-up is complete. Under the randomization design, $A_t$ is independent of potential outcomes given $H_t$, so sequential exchangeability holds at this decision. Positivity also holds because the probability of either assigned treatment is $0.5>0$.

Now consider the same probabilities in an observational study, where clinicians choose treatment and an unrecorded measure of illness severity affects both that choice and $Y$. Seeing conditional probabilities near $0.5$ does not establish exchangeability: assignment may still depend on severity after conditioning on recorded $H_t$. The numeric positivity check passes, but the exchangeability audit does not. The example separates two assumptions that are often mistakenly treated as interchangeable.

Diagnostics, Alternatives, and Limits

Some parts of an audit have useful empirical diagnostics. Treatment frequencies by observed history can reveal structural zeros or sparse support; fitted probabilities near zero can flag practical instability. For a dynamic rule, tabulate or model treatment support in histories the rule actually reaches, not just in the full study population. If continuous treatment is involved, inspect local support around the prescribed values rather than interpreting a point probability as though it were positive.

These checks cannot prove positivity in the population, and finite data cannot prove exchangeability. A zero observed count may reflect a true structural rule or a rare event in a finite sample. A high fitted treatment probability is not evidence that all confounders were measured. Consistency also needs substantive knowledge about intervention versions and measurement; it is not diagnosed by a good model fit.

An audit focused on one specified regime differs from a global positivity audit: it asks about treatment support along the regime’s own histories, which can be less demanding than support for every possible treatment sequence. A randomized trial provides a useful reference for exchangeability and positivity, but it does not automatically solve consistency problems, and its results may not apply to every target population. Alternative adjustment strategies still require a defensible target, treatment definition, and assumptions that connect observed data to the potential outcome.

The main limitation is that the audit organizes assumptions; it does not make them true. If an important confounder is unmeasured, treatment versions are ambiguous, or the regime prescribes unsupported decisions, an estimated regime mean may be misleading even when the computation is correct. These conditions should be reported explicitly rather than hidden behind a single claim that “the assumptions hold.”

Common Mistake

Pitfall: “Treatment is common overall, so positivity holds for my rule.” This reasoning averages over histories and can conceal a zero treatment probability at a history the regime reaches. The correction is to evaluate $P(A_t=g_t(h)\mid H_t=h)$ decision by decision on regime-relevant histories. Also keep the assumptions separate: positive support does not imply exchangeability, and neither condition resolves an ill-defined treatment version.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

What information must be specified to define a dynamic treatment regime and its potential outcome?
A regime reaches history $h$ with positive probability and prescribes $A_t=0$, but $P(A_t=0\mid H_t=h)=0$. Which assumption fails, and why does a large sample not fix it?
Why does a treatment probability of $0.5$ in each recorded history not establish sequential exchangeability in an observational study?
Suppose observed treatment matches a rule, but the same treatment label includes two materially different interventions. Which audit condition is threatened?

Solutions. (1) Specify the target population, decision times, available pre-treatment history, treatment assigned for each history, outcome, and follow-up period; $Y^g$ is then the outcome under that complete rule. (2) Regime-specific positivity fails because the prescribed treatment has zero probability at a history the regime can reach. More observations cannot create support where the treatment mechanism assigns probability zero. (3) Equal conditional treatment probabilities address support, not whether treatment remains associated with potential outcomes through unmeasured causes; exchangeability requires no such residual association given the history. (4) Consistency is threatened because the treatment version is not sufficiently well defined to connect observed outcomes to the specified intervention.