Consistency Under a Longitudinal Intervention

Why Longitudinal Consistency Matters

A longitudinal intervention specifies treatment decisions over time. To interpret its effect, we need to connect the outcome that would occur under that intervention to what is observed when a person actually follows it. Consistency provides this link: when the observed treatment history agrees with a well-defined regime, the observed outcome is the outcome under that regime.

The link must account for treatment and covariate histories, not just a final treatment label. A treatment decision at one time may depend on information measured earlier, and that information may itself have been affected by prior treatment. Consistency describes how the observed trajectory corresponds to a counterfactual trajectory; it does not by itself say who follows the regime or how to estimate its average effect.

Definition: Let \(t=1,\ldots,K\) index treatment times. Let \(A_t\) be the treatment received at time \(t\), \(L_t\) the covariates measured before that treatment, and \(Y\) the final outcome. Write \(\bar A_t=(A_1,\ldots,A_t)\) and \(\bar L_t=(L_1,\ldots,L_t)\) for histories through time \(t\). A treatment regime \(d\) specifies a treatment decision \(d_t\) at each time. A static regime specifies treatment values in advance; a dynamic regime specifies decisions as functions of measured history. Let \(L_t^d\) and \(Y^d\) denote the covariate at time \(t\) and final outcome, respectively, under regime \(d\).

For a static regime, such as “give treatment at both times,” the regime specifies a fixed sequence \((a_1,a_2)\). For a dynamic regime, the next treatment may depend on covariates observed so far. For example, it could specify treatment at time 2 only when a measured biomarker exceeds a threshold. In either case, the counterfactual trajectory under \(d\) is generated by applying its decisions at each treatment time and allowing subsequent covariates and the final outcome to take their values under that intervention.

The notation \(Y^d\) names a counterfactual outcome, not an additional observed outcome. For one person, we observe the outcome under the treatment actually received, not outcomes under every possible regime. Consistency tells us when the observed outcome is equal to a particular regime-specific counterfactual.

The Consistency Link

For a static treatment sequence \(\bar a_K=(a_1,\ldots,a_K)\), consistency says that if a person’s observed treatment history is exactly \(\bar a_K\), then the person’s observed covariate trajectory and outcome agree with the corresponding counterfactual trajectory under that sequence. In particular, when \(\bar A_K=\bar a_K\), \(Y=Y^{\bar a_K}\). The same idea applies to intermediate covariates: if the observed treatments through time \(t-1\) match \(\bar a_{t-1}\), then the observed \(L_t\) equals \(L_t^{\bar a_{t-1}}\).

For a dynamic regime, treatment values are determined using history, so matching means that the observed treatment at each time is the decision prescribed by the regime for the history that has actually occurred. If the person follows those decisions at every time, the person’s observed trajectory is the trajectory under \(d\), and the observed final outcome equals \(Y^d\).

Consistency for a dynamic regime: Suppose that, at each time \(t\), the observed treatment satisfies \(A_t=d_t(\bar L_t,\bar A_{t-1})\), using the measured history available to the regime at that time. If this holds at every treatment time, then the observed covariates follow the regime-specific covariate trajectory and \(Y=Y^d\). For a static regime, this reduces to equality between the observed treatment sequence and the sequence specified by the regime.

This statement presumes that the regime’s treatment decisions and the treatment actually received refer to the same well-defined intervention. If a recorded treatment label groups together materially different versions, matching the labels may not be enough. Treatment versions are considered below.

Worked Example: Matching a Static Two-Time Regime

Consider a hypothetical two-time study. The static regime \(d\) specifies treatment at both times, so its sequence is \((1,1)\), where 1 denotes the specified treatment. One participant has observed history \(A_1=1,A_2=1\), an observed covariate value \(L_2=6\) units, and final outcome \(Y=14\) points. Suppose the regime-specific counterfactual values for this participant are \(L_2^d=6\) units and \(Y^d=14\) points.

The treatment sequence matches the static regime: \((A_1,A_2)=(1,1)\). Consistency therefore links the observed trajectory to the regime trajectory. The observed intermediate covariate agrees with its regime-specific value, \(L_2=L_2^d=6\) units, and the observed outcome agrees with its regime-specific counterfactual, \(Y=Y^d=14\) points.

The example illustrates the implication of consistency; it does not show that the counterfactual value is separately observed or empirically verified. For a person whose observed sequence were \((1,0)\), consistency would instead link the observed outcome to the counterfactual for \((1,0)\), not to \(Y^d\) for \((1,1)\).

Worked Example: Matching a Biomarker-Guided Dynamic Regime

Suppose a hypothetical regime assigns treatment at time 1 and then assigns treatment at time 2 if the measured biomarker \(L_2\) is at least 5 units; otherwise it does not. Thus \(d_1=1\), and \(d_2(L_2)=1\) when \(L_2\geq 5\), with \(d_2(L_2)=0\) when \(L_2<5\). One participant receives \(A_1=1\), has observed \(L_2=6\) units, and receives \(A_2=1\). The final observed outcome is \(Y=14\) points.

At time 1, the observed treatment agrees with the rule: \(A_1=d_1=1\). At time 2, the rule is evaluated using the observed biomarker: \(d_2(6)=1\), so \(A_2=d_2(L_2)=1\). The observed decisions therefore follow the regime at both times. By longitudinal consistency, this participant’s observed trajectory is the trajectory under \(d\), and \(Y=Y^d=14\) points.

Now consider a second hypothetical participant with the same first treatment but \(L_2=4\) units and \(A_2=0\). The rule prescribes \(d_2(4)=0\), so this participant also follows the regime. The two participants receive different time-2 treatment values, yet both match the same dynamic regime because its decision depends on the biomarker.

If the second participant instead had \(L_2=4\) units and \(A_2=1\), the observed treatment would not match the regime at time 2. Consistency would not equate that participant’s observed outcome with \(Y^d\). It would still relate the observed outcome to the counterfactual corresponding to the treatment actually received, under the relevant well-defined treatment specification.

Why the Link Follows Over Time

The longitudinal link can be understood recursively. For clarity, suppose the regime uses covariates measured before each treatment and a person’s observed decisions match \(d\) at every time. At the first time, the observed treatment agrees with the regime’s first decision. Consistency then identifies the observed covariates measured after that decision with the corresponding covariates under the regime. Given the same history at the next decision point, the regime prescribes the same next treatment. Repeating this reasoning carries the match forward through the final treatment and outcome.

\begin{aligned} A_1 &= d_1(L_1),\\ L_2 &= L_2^d,\\ A_2 &= d_2(L_1,L_2,A_1),\\ &\ \vdots\\ A_K &= d_K(\bar L_K,\bar A_{K-1}) \quad\Longrightarrow\quad Y=Y^d. \end{aligned}

The intermediate equalities express the recursive consistency link: once the observed history agrees with the regime history, the next covariate is the value under that regime; that covariate then supplies the history for the next decision. Inductively, the complete observed trajectory agrees with the regime trajectory, so the final outcome is \(Y^d\). This is a statement about the relation between observed and counterfactual outcomes, not a claim that treatment assignment was unconfounded.

Worked Example: Checking the Match at Every Decision Time

A hypothetical three-time regime specifies \(A_1=0\), specifies \(A_2=1\) if \(L_2\geq 3\) units and \(A_2=0\) otherwise, and specifies \(A_3=1\) if \(L_3\geq 8\) units and \(A_3=0\) otherwise. A participant’s observed trajectory is \(A_1=0\), \(L_2=4\) units, \(A_2=1\), \(L_3=7\) units, \(A_3=0\), and \(Y=9\) points.

At time 1, \(A_1=0\) matches the rule. At time 2, \(L_2=4\) units meets the threshold, so \(d_2(4)=1=A_2\). At time 3, \(L_3=7\) units is below the threshold, so \(d_3(7)=0=A_3\). Every observed treatment decision matches the dynamic rule, so consistency gives \(Y=Y^d=9\) points for this participant.

If instead \(A_3=1\), the participant would depart from the rule at the last decision. Even if the first two decisions matched, consistency would not identify the observed final outcome with \(Y^d\), because the full observed treatment path did not follow the regime.

Treatment Versions Must Be Specified

Consistency requires more than agreement on a coarse treatment label. Suppose \(A_t=1\) means “received the drug,” but the label includes both a 5-mg dose and a 20-mg dose. If those versions can produce different outcomes, a regime that merely says \(A_t=1\) does not uniquely specify the intervention. It is then unclear which counterfactual outcome \(Y^d\) is intended, and observed receipt of one version cannot automatically be treated as receipt of the other.

There are two standard ways to make the causal question precise. The intervention can specify the relevant version, such as a particular dose and schedule; alternatively, the treatment can be defined so that its versions are interchangeable for the outcome and covariate process of interest. The required specification depends on the scientific question. If dose is part of the intervention, it should be represented in the treatment definition or regime rather than hidden inside a broad label.

Worked Example: Refining a Treatment Label

Imagine a hypothetical two-time regime described only as “give treatment at both times.” In the study, the label \(A_t=1\) includes either a 5-mg dose or a 20-mg dose. One participant receives 5 mg at both times and has an observed outcome of 12 points. The regime’s intended dose schedule has not been stated.

It is not justified to conclude from the two recorded values \(A_1=A_2=1\) alone that the observed outcome equals the counterfactual under a particular 20-mg-at-both-times regime. The observed labels match, but the treatment versions do not. The relevant comparison is undefined until the intervention is clarified.

If the scientific target is the 5-mg dose at both times, define that version explicitly and check whether the participant received it at both times. If the target is a 20-mg dose at both times, this participant’s observed history does not match it. Alternatively, if the scientific and clinical context supports treating both doses as equivalent versions for the question being asked, the broader label may be appropriate. That equivalence is a substantive assumption, not a consequence of the matching treatment codes.

What Consistency Does—and Does Not—Establish

Consistency is one ingredient in causal reasoning, not a complete identification argument. It does not assert that participants who follow a regime are comparable to those who do not. Sequential exchangeability concerns that separate issue: whether treatment decisions are conditionally independent of relevant future counterfactuals given measured histories. Positivity concerns whether the treatments required by a regime occur with adequate probability in the relevant histories. Neither follows from consistency, and consistency does not follow from either.

This distinction matters because seeing that some participants followed a regime does not, on its own, reveal the average outcome that would occur if everyone followed it. Consistency links an individual’s observed outcome to their counterfactual when their treatment path matches the regime. Additional conditions and methods are needed to use observed data to learn population-level regime effects.

Common pitfall: “This participant received treatment at every visit, so the observed outcome is the outcome under the sustained-treatment regime.” This reasoning is incomplete if the regime specifies different doses, timing, or other treatment versions, or if “treatment at every visit” does not match the regime’s decisions for that participant’s history. First specify the intervention precisely; then check every decision in the observed history against it. Only a complete match supports the consistency link to \(Y^d\).

A careful consistency check therefore asks three questions: Is the regime fully specified at each time? Does the observed treatment decision match the regime given the history available at that time? Do the treatment labels distinguish versions that matter for the intervention? A “yes” to all three makes the individual-level link clear. It still leaves separate questions about whether observed data can identify or estimate an average causal effect.

Check Your Understanding

Use the definitions and examples above to answer these questions.

For a static regime specifying \(A_1=1,A_2=0\), what observed treatment history is required for consistency to link the final observed outcome to \(Y^d\)?
A dynamic rule assigns treatment at time 2 only when \(L_2\geq 5\). Explain why two participants with different time-2 treatments could both follow the same regime.
What does consistency imply when a participant follows a regime at every treatment time? What does it not imply about participants who do not follow it?
Why can matching recorded treatment labels fail to establish consistency when one label includes multiple doses that may matter?
How do consistency, sequential exchangeability, and positivity address different parts of longitudinal causal reasoning?