Coding Past Treatments in Sequential Models

Why Prior Treatment Needs an Explicit Code

A sequential treatment model estimates the probability of the treatment assigned at a given time from information available before that assignment. When treatment can persist, stop, restart, or switch, a later treatment probability may depend on how the person arrived at the current point, not just on the most recent covariates. The model therefore needs predictors that represent relevant past assignments in a deliberate way.

Let \(A_t\in\{0,1\}\) denote treatment at time \(t\), for \(t=1,\ldots,T\). The prior treatment history at time \(t\) is \(H_t^A=(A_1,\ldots,A_{t-1})\); at \(t=1\), this history is empty. A treatment model may condition on \(H_t^A\) and the measured covariate history. This tutorial focuses on how to encode \(H_t^A\), not on how to construct covariate histories or choose the timing of covariate measurements.

In a pooled logistic model, for example, one might write

where \(\bar L_t\) denotes the covariates included in the model and \(g\) represents other chosen predictors. At the first time point, \(A_{t-1}\) is not observed; it must not be silently set to a meaningful treatment value. One can instead model the first time separately, define the baseline treatment convention explicitly, or use a coding scheme that has a designated first-period value.

Common Ways to Encode Past Assignments

The simplest code is the lagged treatment \(A_{t-1}\). It distinguishes whether treatment was assigned at the immediately preceding time, but ignores earlier assignments once the lag is known. A cumulative count distinguishes the number of prior treated occasions, but generally ignores their order. An ever-treated indicator distinguishes people with any previous treatment from those never treated, but ignores both the number and timing of treated occasions.

A more detailed code uses indicators for particular histories. With binary treatment and \(t=3\), the prior histories are \(00,01,10,11\), where the first digit is \(A_1\) and the second is \(A_2\). Four history indicators can represent a different treatment probability for each pattern. This may be useful when history-specific differences matter, but the number of patterns grows as \(2^{t-1}\), so late-time histories may be sparse.

Initiation and Continuation

For a binary treatment that can be initiated and continued, \(A_{t-1}\) has a direct transition interpretation. Among people with \(A_{t-1}=0\), \(A_t=1\) is initiation. Among people with \(A_{t-1}=1\), \(A_t=1\) is continuation. A model containing \(A_{t-1}\) can give these two groups different treatment probabilities. Whether this is a useful model depends on the application: initiation and continuation may have distinct determinants, and a single linear-logit effect of \(A_{t-1}\) imposes a particular relationship between them.

If treatment is truly absorbing after initiation, then once \(A_{t-1}=1\), the next assignment is known to be 1 by the treatment definition. In that case a continuation probability model is not estimating an uncertain decision for those records; the appropriate risk set and treatment process must reflect the absorbing rule. Do not use an ordinary transition model that estimates positive probabilities of stopping when stopping is impossible by design.

Switching and the Information in History

When treatment can switch on and off, the lag \(A_{t-1}\) identifies the previous state, but not necessarily whether that state was reached by remaining in the same state or by switching. One possible additional predictor is the previous switch indicator \(S_{t-1}=I(A_{t-1}\ne A_{t-2})\), when both past assignments exist. This is a summary of prior treatment history, not the treatment outcome at time \(t\). Other choices include a count of prior switches or indicators for exact treatment sequences. The choice should follow the substantive process being modeled.

What Full History Coding Can and Cannot Guarantee

Proof. Write the model as \(\operatorname{logit}P(A_t=1\mid H_t^A)=\sum_h \eta_h I(H_t^A=h)\). For any observed history \(h^\ast\), exactly one history indicator equals 1: \(I(H_t^A=h^\ast)=1\), while all other indicators equal 0. The linear predictor is therefore \(\eta_{h^\ast}\). Set \(\eta_h=\operatorname{logit}(q_h)=\log(q_h/(1-q_h))\). Applying the inverse logit gives the fitted probability \(q_h\) for each history \(h\), as required. The result assumes each history can be represented in the model; it does not ensure that every history occurs in the sample or that its probability can be estimated precisely. \(\square\)

Full history indicators are flexible but costly in data: at time \(t\), there are \(2^{t-1}\) binary histories. If some histories are absent, their probabilities cannot be estimated from that time-specific data without additional restrictions. If histories are rare, estimates may be highly uncertain or produce near-zero or near-one predicted probabilities.

Derivation. Suppose a summary \(C(H_t^A)\) gives the same value \(c\) for distinct histories \(h_1\) and \(h_2\). Any model whose treatment probability depends on history only through \(C\) must assign the same probability \(r(c)\) to both histories. But a valid history-specific probability function could assign \(q_{h_1}=0.2\) and \(q_{h_2}=0.8\). Since \(0.2\ne0.8\), no single value \(r(c)\) can equal both. Thus the summary cannot represent every possible function of the full history. It may still be appropriate if substantive knowledge or a modeling assumption justifies treating those histories as equivalent.

Choosing and Checking a Coding

There is no universally best history code. A compact code such as \(A_{t-1}\), ever treated, or cumulative treated occasions uses fewer parameters and may be more stable, but it makes distinct histories equivalent unless other predictors separate them. Exact-history indicators avoid that particular restriction but can be impractical when histories are numerous or sparse. Separate models by time can expose history-specific patterns; pooled models can share information across times but impose common features unless time variation is otherwise represented. These are alternative parameterizations, not automatic substitutes for a substantive decision about which past assignments matter.

Before fitting, make a small table of the observed histories and the values of each proposed code. Check whether histories that should have different implications collapse to the same value. After fitting, inspect the number of observations and treatment outcomes within relevant history groups, the predicted treatment probabilities by history, and warnings of separation or unstable estimates. Extreme predicted probabilities can lead to extreme inverse probability weights; they may signal sparse histories, a misspecified treatment model, or limited empirical support. A plausible coding does not by itself guarantee positivity or correct model specification.

A frequent mistake is to use only \(A_{t-1}\) and then claim that the model adjusts for the full treatment history. That conclusion is false whenever earlier assignments can matter after conditioning on the lag. For instance, at \(t=3\), histories \(01\) and \(11\) both have \(A_2=1\), but differ in \(A_1\). The correction is not to include every possible history automatically; it is to state the restriction implied by the chosen code, consider whether it is defensible, and check whether the data support a more detailed representation.