Why Consistency Is a Separate Assumption
Longitudinal causal questions compare outcomes under treatment strategies, but the data record only the treatments and outcome that occurred. Consistency supplies a specific link between these two descriptions: when a person's observed treatment history matches a specified history, the observed outcome equals the potential outcome indexed by that history. The link is path-specific. It does not say that outcomes under different histories are equal, nor does it establish that observed groups are comparable.
Let there be \(K\) decision times. Write \(A_t\) for treatment at time \(t\), \(L_t\) for covariates available before that decision, and \(\bar A_t=(A_1,\ldots,A_t)\) for the treatment history through time \(t\). The observed history just before decision \(t\) is \(H_t=(L_0,A_1,L_1,\ldots,A_{t-1},L_t)\). For a fixed treatment history \(\bar a_t\), write \(L_t^{\bar a_{t-1}}\) for the covariate value that would occur under the preceding treatments and \(Y^{\bar a_K}\) for the outcome under the full sequence. These potential outcomes must be well-defined for the population and treatment histories under discussion.
The statement is often written compactly as \(L_t=L_t^{\bar A_{t-1}}\) and \(Y=Y^{\bar A_K}\). These are individual-level equalities evaluated at the person's realized treatment history. They do not mean \(Y^{\bar a_K}=Y^{\bar a'_K}\) for two different treatment sequences. Nor does the notation assert that all potential outcomes can be observed for one person.
Fixed Treatment Sequences
For a fixed sequence \(\bar a_K=(a_1,\ldots,a_K)\), consider individuals whose observed treatments satisfy \(\bar A_K=\bar a_K\). The outcome part of consistency directly gives \(Y=Y^{\bar a_K}\) for each such individual. The associated covariate equalities show that their observed covariate history also agrees with the potential covariate history generated by the sequence.
Proof. Fix an individual with \(\bar A_K=\bar a_K\). For every \(t\), the full-history equality implies the prefix equality \(\bar A_{t-1}=\bar a_{t-1}\). The covariate consistency assumption therefore gives \(L_t=L_t^{\bar a_{t-1}}\). Applying the separately assumed outcome-consistency relation to \(\bar A_K=\bar a_K\) gives \(Y=Y^{\bar a_K}\). The two conclusions follow. \(\square\)
Worked Example: A Two-Time Fixed Sequence
Suppose \(K=2\), treatment is binary, and one patient's observed treatment sequence is \((A_1,A_2)=(1,0)\). The potential covariates for this patient are \(L_1^1=7\), \(L_2^{(1,0)}=4\), and the potential outcomes include \(Y^{(1,0)}=12\) and \(Y^{(1,1)}=15\). The observed records are \(L_1=7\), \(L_2=4\), and \(Y=12\).
At the first covariate time, the observed prior treatment is \(A_1=1\), so the covariate consistency relation is \(L_1=L_1^1\), here \(7=7\). At the second covariate time, the observed prior sequence is \((1,0)\), so \(L_2=L_2^{(1,0)}\), here \(4=4\). The full observed treatment sequence is \((1,0)\), and outcome consistency gives \(Y=Y^{(1,0)}\), here \(12=12\).
The value \(Y^{(1,1)}=15\) does not contradict any of these equalities: that potential outcome corresponds to a different treatment sequence. Consistency selects the potential outcome matching the observed sequence; it does not equate outcomes across sequences.
Strategies That Depend on Observed History
A strategy need not prescribe the same treatment to everyone at a given time. A dynamic strategy \(d=(d_1,\ldots,d_K)\) assigns treatment using the history available at each decision: \(A_t=d_t(H_t)\). Its potential covariate and treatment histories are defined recursively. Start with \(L_0^d=L_0\); at time \(t\), set \(A_t^d=d_t(H_t^d)\), where \(H_t^d\) contains the covariate and treatment history generated under the strategy. Then set \(L_{t+1}^d=L_{t+1}^{\bar A_t^d}\), when \(t<K\), and define \(Y^d=Y^{\bar A_K^d}\). This recursive construction requires coherent, well-defined potential covariates and outcomes for the treatment histories it generates.
Proof. We show by induction over decision times that the observed history and the history generated under \(d\) agree. At baseline, \(L_0=L_0^d\) by construction. Suppose the histories agree immediately before time \(t\), so \(H_t=H_t^d\). The individual's adherence assumption gives \(A_t=d_t(H_t)\). Since \(H_t=H_t^d\), this is \(A_t=d_t(H_t^d)=A_t^d\). Thus observed and strategy treatment histories agree through time \(t\): \(\bar A_t=\bar A_t^d\). If \(t<K\), covariate consistency then gives \(L_{t+1}=L_{t+1}^{\bar A_t}=L_{t+1}^{\bar A_t^d}=L_{t+1}^d\). This establishes the induction step, and therefore \(\bar A_K=\bar A_K^d\). Finally, the assumed outcome consistency gives \(Y=Y^{\bar A_K}\), and the equality of treatment histories gives \(Y^{\bar A_K}=Y^{\bar A_K^d}=Y^d\). Hence \(Y=Y^d\). \(\square\)
Worked Example: Following a History-Dependent Rule
Consider two decision times and the rule \(d_1=1\), followed by \(d_2(H_2)=0\) when \(L_1<8\) and \(d_2(H_2)=1\) otherwise. For one individual, suppose \(L_0=5\), \(L_1^1=7\), and the potential outcome under the resulting sequence \((1,0)\) is \(Y^{(1,0)}=12\). The observed record is \(A_1=1,L_1=7,A_2=0,Y=12\).
At time 1, the rule prescribes \(1\), which matches \(A_1\). The observed \(L_1=7\) agrees with \(L_1^1=7\). Since \(7<8\), the rule prescribes \(A_2=0\), matching the observed second treatment. Thus the person's observed sequence is the sequence generated by the rule, \(\bar A_2=\bar A_2^d=(1,0)\). Under the covariate and outcome consistency assumptions, the observed outcome is \(Y=Y^{(1,0)}=12\), and the recursively defined strategy outcome is \(Y^d=Y^{(1,0)}=12\).
The outcome equality depends on the outcome-consistency assumption as well as adherence. Matching observed covariates to their potential covariates is not, by itself, a statement about the observed outcome.
Worked Example: A Nonmatching Sequence Gives No Equality Under the Rule
Use the same strategy as in the previous example. Suppose a different individual has \(L_1^1=9\), observed \(A_1=1,L_1=9,A_2=0\), and potential outcomes \(Y^{(1,0)}=3\) and \(Y^{(1,1)}=8\). Let the observed outcome be \(Y=3\), consistent with the sequence actually received.
At time 2, the strategy sees \(L_1=9\), and \(9\) is not less than \(8\). It therefore prescribes \(A_2^d=1\). The observed \(A_2=0\) does not follow the rule. The strategy-generated sequence is \((1,1)\), whereas the observed sequence is \((1,0)\). Consistency links the observed outcome to \(Y^{(1,0)}\), so \(Y=3=Y^{(1,0)}\). It does not link this observed outcome to \(Y^d=Y^{(1,1)}=8\). The numerical difference \(3\ne8\) makes clear why adherence is essential to the dynamic-strategy result.
What Consistency Does—and Does Not—Establish
Consistency is one component of causal reasoning, not a complete identification argument. It connects observed outcomes to the potential outcomes corresponding to the histories actually received. It does not imply that people who received one history would have had the same potential outcomes as people who received another history. That comparability question concerns exchangeability. Nor does consistency ensure that each treatment choice required by a strategy occurs with positive probability in the relevant histories; that is a positivity question. These are distinct conditions with distinct roles.
This distinction matters in marginal structural models. Such models target contrasts between outcomes under treatment strategies, while the data contain outcomes under realized treatment histories. Consistency provides the individual-level bridge between those quantities for people whose observed histories follow a strategy. It does not, by itself, identify a population mean such as \(E[Y^d]\) from the observed data.
| Question | Role in longitudinal causal analysis |
|---|---|
| Does the observed outcome match the potential outcome for the history actually received? | Consistency |
| Are potential outcomes comparable across observed treatment choices, given the relevant history? | Exchangeability |
| Are the strategy's treatment choices represented in the relevant histories? | Positivity |
A practical diagnostic is to check whether the analysis has a clear mapping between recorded treatment histories and the histories indexing potential outcomes. For each time point, verify that the stated ordering of covariates and treatments matches the actual decision process, and that the outcome equality has been stated rather than silently inferred from covariate equalities. If a strategy is dynamic, verify that its rule uses information available before each treatment decision. These checks can expose a misindexed history or an incoherent strategy definition; they cannot empirically prove the full counterfactual consistency assumption.
Common Pitfalls and Limits
- Deriving outcome consistency from covariate consistency. Agreement between observed and potential covariates does not logically imply \(Y=Y^{\bar A_K}\). State the outcome link explicitly.
- Applying a strategy result without adherence. If even one observed treatment differs from the strategy's prescription, the individual's observed outcome is indexed by the received history, not automatically by the strategy-generated history.
- Using consistency to claim exchangeability. Knowing which potential outcome an observed outcome represents does not make treatment assignment unconfounded.
- Ignoring how histories are indexed. A covariate measured after a treatment decision cannot be treated as if it were part of the history available before that decision. The time ordering must be explicit.
Consistency also has substantive limits. It presumes that the potential outcomes indexed by a treatment history are meaningful for the interventions under study. Whether a treatment category hides different versions that matter for outcomes is a separate issue. Consistency cannot repair an intervention definition that fails to specify a meaningful treatment, and it is not generally testable by comparing observed data alone: for each individual, only one treatment history and its corresponding outcome are observed.
Check Your Understanding
Answer each question, then compare with the concise solutions below.
- For \(K=2\), a person receives \((0,1)\). Which potential outcome does longitudinal consistency equate to the observed outcome?
- A researcher establishes that \(L_1=L_1^{A_1}\) for a patient but gives no outcome-consistency assumption. Can the researcher conclude \(Y=Y^{(A_1,A_2)}\)?
- A dynamic strategy prescribes treatment from the current pre-decision history. What additional condition, beyond well-defined potential histories, is needed to conclude \(Y=Y^d\) for an individual?
- Does consistency alone imply that the mean outcome under a strategy is identified from observed data? Give the reason.
Solutions. (1) It equates \(Y\) to \(Y^{(0,1)}\), because \((0,1)\) is the full history received. (2) No. Covariate consistency does not imply outcome consistency; the link \(Y=Y^{(A_1,A_2)}\) must also be assumed. (3) The individual must follow the strategy at every decision time, and the assumptions must include both covariate-history consistency and the explicit outcome-consistency relation. (4) No. Consistency links observed outcomes to their realized histories, but identification of a strategy mean also requires conditions addressing treatment comparability and treatment support, among other requirements of the chosen identification argument.