Potential Outcomes Indexed by Treatment Histories

Why Outcomes Need a Treatment-History Index

With a single treatment decision, a potential outcome can be indexed by the treatment assigned at that decision. In longitudinal studies, treatment is assigned repeatedly, and earlier treatment can affect covariates measured later. Those covariates can in turn be relevant to later treatment and the outcome. A potential outcome therefore needs to say which sequence of treatments is being considered, not merely whether treatment was received at one visit.

The purpose here is to establish the notation for those counterfactual quantities. It is important to keep two things separate: the outcome that is observed for a person, and the outcomes that would be observed under other treatment histories. Defining the latter does not make them observed or identify their distributions from the data.

Here \(\mathcal A_t\) is the set of possible treatment values at time \(t\). A history is an ordered sequence: \((1,0)\) and \((0,1)\) are different histories if treatment can be given at two times. The potential outcome is a random variable across the target population; for a particular individual, its value under a specified history is that individual's counterfactual outcome under that history.

We use \(L_0\) for baseline covariates, measured before the treatment decisions. Let \(L_t\) be covariates measured before treatment \(A_t\), for \(t=1,\ldots,K\). Thus \(L_t\) can be affected by treatments before time \(t\), but it precedes the treatment decision at time \(t\). The corresponding potential covariate is written \(L_t^{\bar a_{t-1}}\): it is the value of \(L_t\) under the treatment history up to, but not including, time \(t\). In particular, \(L_1^{\bar a_0}=L_1^{\emptyset}\) has no prior treatment in its index. The endpoint \(Y\), measured after the final decision, is indexed by the full history \(\bar a_K\).

This timing convention makes the indices interpretable. The covariate \(L_t\), measured before \(A_t\), cannot be indexed by \(A_t\) as though that decision had already occurred. By contrast, \(Y\), measured after \(A_K\), can be indexed by all \(K\) treatment decisions. These definitions establish the objects needed to formulate later assumptions; they do not themselves impose those assumptions.

Observed Values and Counterfactual Values

For each person, the data contain the treatment actually received at each time and the outcome actually recorded. The data do not contain that person's outcome under every other treatment history. Potential outcomes are defined for the histories of interest, even though only one treatment path is followed for a person.

A link between an observed outcome and a potential outcome is usually expressed through consistency: when a person's actual treatment history is \(\bar a_K\), the observed endpoint equals \(Y^{\bar a_K}\). The precise conditions and complications surrounding that link are important, but are treated separately. For now, the distinction is essential: \(Y\) is observed, while \(Y^{\bar a_K}\) is a history-specific counterfactual, and the equality is not a consequence of notation alone.

To verify the count, at decision 1 there are \(m_1\) choices. For each such choice, there are \(m_2\) choices at decision 2, producing \(m_1m_2\) histories through time 2. Continuing this multiplication through all \(K\) decisions gives \(\prod_{t=1}^K m_t\). In the binary case each \(m_t=2\), so the count is \(2^K\). This is a count of defined counterfactual quantities, not a claim that the data reveal all of them. For example, with three binary decisions there are \(2^3=8\) histories, but a person follows only one realized history. \(\square\)

Population Targets Are Defined From Potential Outcomes

Once potential outcomes are defined, a population target can be described without referring to a regression coefficient. For a fixed complete history \(\bar a_K\), define the marginal mean \(\mu(\bar a_K)=E[Y^{\bar a_K}]\), provided the potential outcome is integrable, meaning \(E[|Y^{\bar a_K}|]<\infty\). A contrast between two histories, such as \(\mu(\bar a_K)-\mu(\bar a'_K)\), compares their population-average counterfactual outcomes.

“Marginal” means averaged over the target population's distribution of baseline and other characteristics, rather than conditioned on a particular covariate value. This definition identifies the target quantity, not a way to estimate it from observed data. Identification requires additional assumptions, and the equality between an observed-data quantity and this counterfactual target cannot be assumed merely because the target has been written down.

Derivation. The events \(\{X=x\}\) for values with positive probability form a disjoint partition of the population. Decomposing the expectation over that partition gives $$ E[Y^{\bar a_K}] =\sum_x E\!\left[Y^{\bar a_K}\mathbf{1}\{X=x\}\right] =\sum_x E[Y^{\bar a_K}\mid X=x]P(X=x). $$ The integrability condition ensures these expectations are finite. This identity explains how a marginal counterfactual mean relates to subgroup-specific counterfactual means: each subgroup mean is weighted by that subgroup's population prevalence. It does not say that the subgroup means are observed or that they can be recovered from the observed data without further assumptions. \(\square\)

Notation Checks, Comparisons, and Limitations

This framework differs from writing one potential outcome for each treatment level at a single decision: a longitudinal endpoint distinguishes the timing and order of the full sequence. It also differs from a conditional regression coefficient. A coefficient is a feature of a specified model for observed data; \(\mu(\bar a_K)\) is a population mean defined under a specified treatment history. The two quantities need not be the same.

Before using history-indexed notation, check the timing of each measurement and the scope of each index. A covariate measured before decision \(t\) receives only the history through \(t-1\); an endpoint measured after decision \(K\) receives the history through \(K\). The observed record and the potential variables should also be labelled separately. This discipline prevents a notation error from being mistaken for a causal assumption.

• Potential outcomes are not observed outcomes. Writing \(Y^{\bar a_K}\) defines a counterfactual quantity; it does not reveal its value for each person.
• More potential outcomes do not mean more observed information. A person supplies one realized treatment path and one observed endpoint, not a table of endpoints under all paths.
• History order matters. Two sequences with the same number of treated visits can be different interventions and can have different potential outcomes.
• Notation does not guarantee well-defined interventions or identify effects. The meaning of treatment values, consistency, exchangeability, and positivity require separate conditions.

A common mistake is to treat \(Y^{(1,0)}\) as though it were the observed outcome whenever a person's data show “one treated visit.” That reasoning discards time order: a person treated at the first visit and not the second has history \((1,0)\), while someone untreated first and treated second has history \((0,1)\). The correction is to record the complete ordered history and link the observed endpoint to its matching potential outcome only under the appropriate consistency condition.

These definitions also have limitations. They do not specify how treatment versions within a recorded category are handled, which treatment histories are feasible, or which assumptions connect counterfactual distributions to observed-data distributions. Those are substantive and statistical questions, not consequences of the indexing scheme. The notation's value is that it makes the target and the required assumptions explicit enough to discuss without confusing counterfactual definitions with observed evidence.