Sequential Trial Emulation with Repeated Eligibility

Why Emulate a Sequence of Trials?

A treatment decision may arise repeatedly. For example, a patient who remains untreated and clinically eligible this month may still be eligible at the next monthly assessment. Emulating only the first opportunity can discard later eligible treatment decisions; treating every opportunity as an independent person, on the other hand, changes the target population and can understate uncertainty. A sequential design makes each decision date an explicit trial time zero, then states how those trials combine.

The key question is not simply whether the same person can appear more than once. It is what population is being averaged over. If the target is the collection of eligible treatment opportunities, repeat appearances are part of the target. If the target is a population of unique people, the design must instead define how each person is represented.

Defining Successive Trials and Their Pooled Target

Let \(k=1,\ldots,K\) index prespecified eligibility dates. Let \(E_{ik}=1\) indicate that person \(i\) meets the trial's eligibility criteria at date \(k\), assessed using information available by that date. Each trial has its own time zero at \(k\), the same clearly specified treatment strategies, and a follow-up window defined from that time zero. A person may have \(E_{ik}=1\) for multiple values of \(k\), provided the protocol permits re-entry.

For example, a protocol might include people who are alive, untreated, and meet a clinical threshold at each monthly date. The criteria must be applied anew at each date. Prior treatment, changed clinical status, or other history can make someone ineligible at a later date. Treatment classification and follow-up for trial \(k\) must be anchored at that trial's own time zero—not assigned using future information.

Let \(Y_{ik}^{a}\) denote the outcome that person \(i\) would have during trial \(k\)'s follow-up under strategy \(a\). For a binary outcome, define the trial-specific target risk as \(R_k^a=E[Y_{ik}^{a}\mid E_{ik}=1]\). A pooled risk under date weights \(q_k\) is

Here \(\Delta(q)\) is a pooled risk difference. The weights define the estimand, not just a computational convenience. Equal weights \(q_k=1/K\) give equal importance to each trial date. Weights proportional to the number of eligible person-trial records give each record equal importance. Those choices need not answer the same question.

Repeated Entry Changes What Is Being Averaged

When weights are proportional to eligible record counts, the natural target is a randomly selected eligible person-trial record. A person eligible on four dates can then contribute four times as much as a person eligible on one date. This is not automatically a defect: it is exactly the target when each eligible treatment opportunity is the unit of interest. It is not an equal-person target.

The distinction can be seen even without outcomes. Suppose person A is eligible at all three dates and person B at only the first. There are four person-trial records. Record weighting assigns A \(3/4\) of the target weight and B \(1/4\). Equal weighting of the two unique people instead assigns each \(1/2\). A pooled result must state which interpretation is intended.

Identification Within Trials and After Pooling

A pooled contrast is identifiable when its trial-specific risks are identifiable under the assumptions for each trial, and the chosen weights are defined. For a concise identification result, let \(L_{ik}\) be measured preassignment covariates in trial \(k\), and let \(A_{ik}\) denote the assigned strategy. Assume consistency, conditional exchangeability, and positivity within each trial: among eligible people, \(Y_{ik}^{a}\) equals the observed outcome when \(A_{ik}=a\); \(Y_{ik}^{a}\) is independent of \(A_{ik}\) conditional on \(L_{ik}\) and \(E_{ik}=1\); and \(P(A_{ik}=a\mid L_{ik}=l,E_{ik}=1)>0\) for covariate values in the trial's target population. Also require that the outcome and covariates are well-defined and measured for the stated follow-up and that the treatment strategies do not have ambiguous versions relevant to the outcome.

Proof. By consistency, \(E[Y_{ik}^{a}\mid L_{ik}=l,E_{ik}=1]\) equals \(E[Y_{ik}\mid A_{ik}=a,L_{ik}=l,E_{ik}=1]\) after using conditional exchangeability to replace the potential-outcome mean among those assigned \(a\) by the mean among all eligible people with \(L_{ik}=l\). Average that conditional mean over the covariate distribution among eligible people. The law of total expectation gives

This identifies \(R_k^a\). Since \(R^a(q)\) is the defined finite weighted sum of these risks, substituting their identified expressions identifies the pooled risk as well. The same reasoning applies to a continuous covariate distribution using conditional expectations and integration. \(\square\)

The assumptions apply at each date and to its eligible population. Pooling does not repair a date where a strategy is absent for a relevant covariate pattern, nor does it make unmeasured confounding disappear. A common treatment label across dates also needs to represent a sufficiently comparable intervention; otherwise the pooled quantity combines different strategies.

Inference, Diagnostics, and Alternative Designs

Repeated records from the same person are generally dependent. Even if each record is correctly constructed, standard errors that treat every record as independent can be too small. The dependence arises because outcomes and covariate histories for one person are related across their eligible dates. Inference should account for clustering by person, for example through a participant-clustered variance estimator or a resampling procedure that resamples people and retains all of each selected person's records.

The reason can be stated directly. If a pooled estimator is a mean of \(M\) record contributions \(Z_{ik}\), its variance depends on covariances between records from the same person:

The second sum includes within-person pairs. Dropping those terms is justified only if the relevant covariances are zero, which repeated observations do not generally guarantee. This variance identity follows by expanding the variance of a sum; it explains why the unit used for uncertainty calculation must reflect repeated contribution, even when the target itself is a person-trial record.

Before analysis, inspect the number of eligible records by date, the number of unique people, the distribution of each person's number of entries, and the treatment and covariate patterns within dates. A sudden change in eligibility counts can reflect a genuine population change or inconsistent implementation. Few treated or untreated records in a date-specific covariate group warn of weak positivity. A participant with many records may dominate record-weighted summaries. Also verify that eligibility uses only information available at that date and that outcomes are measured over comparable follow-up windows.

A first-eligible-date design avoids repeat entries and targets people at their first qualifying opportunity, but it answers a different question and can discard later treatment opportunities. A sequence of separate trial-specific analyses preserves date-specific results but does not by itself provide one pooled target. A pooled sequential analysis supplies that summary, provided its date weights are prespecified and reported. None of these choices is universally preferable: the right design follows the scientific population and decision question.

Common Pitfall

• Wrong reasoning: “The same person appears several times, so the pooled target is the population of unique people.” Repeated records do not become unique-person records merely because they share an identifier.
• Correction: State whether the target is eligible person-trial records, equally weighted trial dates, or unique people. Then choose weights that implement that target and use inference that accounts for within-person dependence.
• Another warning: Do not decide at date \(k\) that a person belongs to a treatment group based on whether they initiate treatment later. Define eligibility, assignment, and time zero using the information and rules specified for that trial date.