Artificial Censoring for Strategy Deviations

Why a Strategy Deviation Changes Follow-Up

In a target trial emulation, a treatment strategy is an operational rule for which actions are compatible with the assigned strategy at each decision time. A participant’s observed treatment can cease to follow that rule. To estimate the effect of following the strategy, the analysis then stops counting that participant’s follow-up for that strategy at the first incompatible action. This is artificial censoring: an analysis rule, not necessarily a real-world loss of observation.

After cloning participants to represent different strategies, the same person may contribute a record to more than one strategy. Each clone is evaluated against its assigned rule. A person can therefore remain uncensored for one clone and be artificially censored for another. The per-protocol estimand and the strategy rules determine which actions count as deviations; censoring is not a general label for treatment change.

For a deterministic strategy, the allowed set may contain just one action. For example, “continue treatment at every visit” requires \(A_{it}=1\) at each specified visit. A rule that permits either of two clinically acceptable doses has a larger allowed set. The operational definition must be explicit about visit windows, grace periods, and whether switching or stopping is allowed.

The timing convention matters. Suppose the outcome is assessed before a treatment decision at time \(t\). An event assessed then belongs to follow-up even if the participant makes an incompatible choice afterward. If treatment is chosen first and the next outcome interval follows, an incompatible choice censors that clone before the interval begins. The protocol must establish this ordering; otherwise the same event can be wrongly included or excluded.

Artificial Censoring Is Not Loss to Follow-Up

Ordinary loss to follow-up occurs when the study can no longer ascertain a participant’s outcome or subsequent history. Artificial censoring occurs when the data may still be available, but a clone’s observed actions no longer satisfy its assigned strategy. One person can experience either event or both, and the two indicators should be recorded separately.

If a participant both deviates and later becomes unreachable, the strategy-specific clone is already artificially censored at the deviation. Conversely, a participant who remains adherent but becomes unreachable has not deviated; the missing follow-up is ordinary loss to follow-up. Administrative end at the planned endpoint is not a strategy deviation. Nor should a clinical event automatically be called censoring: whether it is an outcome, a competing event, or a censoring event depends on the estimand and protocol.

Formal Rule and Its Immediate Consequence

The recursive definition has a direct interpretation: a clone is retained at time \(t\) exactly when every action it has taken so far was allowed. This makes artificial censoring a first-deviation rule rather than a judgment based on a later treatment history.

Proof. At \(t=1\), the recursion gives \(C_{i1}^d=C_{i0}^d I\{A_{i0}\in\mathcal{A}_0^d(H_{i0})\}\), and \(C_{i0}^d=1\), which is the stated product. Suppose the product expression holds at time \(t\). Then $$ C_{i,t+1}^d=C_{it}^d I\{A_{it}\in\mathcal{A}_t^d(H_{it})\} =\prod_{s=0}^{t} I\{A_{is}\in\mathcal{A}_s^d(H_{is})\}. $$ By induction, the expression holds at every decision time. A product of indicators equals one exactly when every factor equals one. \(\square\)

Why Artificial Censoring Can Bias a Per-Protocol Estimate

The clones remaining uncensored may differ from the target population. For example, a time-varying health measure may predict both whether a participant continues treatment and the outcome. Restricting analysis to adherent clones can then select a group with a different outcome risk. The strategy deviation itself is not a reason to discard the data without further thought: the selection it creates may need to be addressed using inverse-probability weighting.

For a deterministic strategy, define the conditional probability of remaining compatible at decision time \(t\), among clones still compatible just before that decision, by $$ q_{it}^d=P\{A_{it}\in\mathcal{A}_t^d(H_{it})\mid H_{it},C_{it}^d=1\}. $$ For a clone that remains compatible through \(K\) decisions, its artificial-censoring weight is $$ W_{iK}^d=\prod_{t=0}^{K-1}\frac{1}{q_{it}^d}. $$ A clone’s weight is updated only while it remains compatible. Histories \(H_{it}\) must be measured before the decision at \(t\); using information recorded after that decision to predict compatibility reverses the time order.

Derivation. Consider a history through the final outcome, and write \(I_{it}^d=I\{A_{it}\in\mathcal{A}_t^d(H_{it})\}\). Under the observed treatment process, the conditional probability contribution of a compatible action at time \(t\) is \(q_{it}^d\). Under the intervention that follows \(d\), that action is required. Consequently, the ratio of the strategy’s probability for a compatible history to the observed probability of that history contributes \(I_{it}^d/q_{it}^d\) at time \(t\). Multiplying over decisions gives $$ \prod_{t=0}^{K-1}\frac{I_{it}^d}{q_{it}^d} = C_{iK}^d W_{iK}^d. $$ The conditional exchangeability and consistency assumptions allow the observed conditional distributions along compatible histories to represent those under the strategy. Positivity makes each probability ratio defined. Thus, for the outcome \(Y\) at the end of follow-up, the weighted observed mean among compatible clones represents the mean under the strategy: $$ E\!\left[C_{iK}^d W_{iK}^dY\right]=E[Y^d], $$ with expectations taken over the target baseline population and with no additional loss to follow-up in this identity. Taking the corresponding weighted means for each strategy gives the per-protocol contrast. This derivation does not make the assumptions true: unmeasured predictors of treatment and outcome, incorrect timing, or a zero probability of following the strategy can invalidate the interpretation. \(\square\)

Diagnostics, Alternatives, and Limitations

Before interpreting a weighted estimate, check that the artificial-censoring rule has been implemented as written. Tabulate the number of clones censored at each decision and the recorded reason. Compare histories just before deviation with histories among clones that remain compatible. Inspect estimated \(q_{it}^d\) values and the resulting weights; very small probabilities produce large weights, so a few clones can dominate the estimate. Also check that outcome measurement continues to be distinguished from strategy compatibility.

A restriction to observed adherers without weighting is simpler, but it generally estimates an outcome among a selected group rather than the effect of following the strategy in the target population. An intention-to-treat analog instead compares initial assignments or strategies without censoring later deviations for this purpose; it answers a different causal question. Inverse-probability weighting can target a per-protocol effect, but requires measured histories that suffice for conditional exchangeability and adequate positivity. Weight truncation may reduce instability, but it changes the estimator and can introduce bias; it does not create support for histories where strategy adherence is impossible.

A final common mistake is to censor a clone whenever treatment differs from the baseline assignment, even when the strategy permits a grace period or allows switching. That rule does not implement the specified strategy. Define the allowed actions at each decision first, apply the rule at the correct time, and keep administrative end and ordinary loss to follow-up distinct. If no participants with some history can follow a strategy, the data cannot identify that strategy’s effect for that history by weighting alone.