Artificial Censoring at Protocol Deviations

Why Deviations Create an Artificial Censoring Problem

A protocol may specify a treatment strategy over time: for example, use treatment $1$ at each visit, or choose treatment according to a rule that depends on the patient’s current history. A participant who departs from that strategy no longer contributes an observed outcome trajectory that follows the protocol. One way to estimate outcomes under sustained adherence is to artificially censor that participant at the first departure, then weight the remaining adherent histories to account for who was selected to remain adherent.

This is not the same as claiming that a deviation makes later outcomes irrelevant. The target is the outcome that would have been observed had participants followed the specified strategy. Artificial censoring creates missing outcomes for that target among people who deviate. Inverse-probability weighting can recover the target only under assumptions about why people adhere or deviate.

Definition: Let decision times be $t=0,\ldots,K-1$. Let $A_t$ be the treatment received at time $t$, $L_t$ the measured history available before that decision, and $d_t(H_t)$ the treatment prescribed by a regime $d$, where $H_t=(L_0,A_0,L_1,\ldots,A_{t-1},L_t)$. Define $I_t=1$ when $A_t=d_t(H_t)$, and $I_t=0$ otherwise. The artificial at-risk indicator through decision $t$ is $R_t=\prod_{j=0}^{t} I_j$, with $R_{-1}=1$. A participant is artificially censored at the first $t$ for which $I_t=0$.

The outcome $Y$ is measured after the final treatment decision. A participant with $R_{K-1}=1$ has followed the regime at every decision and contributes an observed outcome consistent with that regime. For someone who deviates, later observed outcomes may be valuable for other questions, but they are not outcomes under sustained adherence to $d$.

Constructing the Adherence Weights

Among participants still adherent through $t-1$, define the conditional probability of following the regime at decision $t$ as $q_t(H_t)=P(I_t=1\mid H_t,R_{t-1}=1)$. This is a conditional probability under the observed treatment process. It can be estimated with a model for whether the received treatment matches the prescribed treatment, using information available before the decision. For a deterministic regime, this is the probability of receiving the particular treatment $d_t(H_t)$.

Definition: The unstabilized artificial-censoring weight through decision $t$, for a participant adherent through $t$, is $$ W_t^D=\prod_{j=0}^{t}\frac{1}{q_j(H_j)}. $$ A participant who deviates at $t$ is censored then and does not contribute to later adherent-risk sets. The weight is used for the outcomes and histories observed while the participant remains adherent.

Each factor reverses the selection associated with passing one more adherence decision. A person with a history for which adherence is uncommon receives a larger weight if they do adhere. The probability model must respect time ordering: predictors in $H_t$ must be known before the treatment decision at $t$, and the model is fit among participants with $R_{t-1}=1$.

Worked Example: Calculating a Two-Decision Weight

Suppose the regime prescribes treatment $1$ at both visits. For a participant who is adherent at visit 0, the estimated probability of receiving treatment $1$ is $q_0=0.80$. Given that participant’s updated history and continued adherence, the probability of receiving treatment $1$ at visit 1 is $q_1=0.50$. The participant follows the regime at both visits.

The weight after the first decision is $W_0^D=1/0.80=1.25$. After the second decision it is $$ W_1^D=\frac{1}{0.80}\frac{1}{0.50}=1.25(2)=2.50. $$

Thus this adherent participant represents 2.5 histories in the weighted analysis, rather than one unweighted history. If the participant had received treatment $0$ at visit 1, they would be artificially censored at that decision and would not receive a weight for contributing an adherent outcome after visit 1.

Worked Example: A Weighted Mean in Two History Groups

Consider a one-decision regime and 100 participants, split equally between history groups $L_0=0$ and $L_0=1$. Suppose 40 of the 50 participants with $L_0=0$ adhere, with outcome sum 8 among those adherers. In the other group, 25 of 50 adhere, with outcome sum 15. The adherence probabilities are therefore $q_0(0)=40/50=0.80$ and $q_0(1)=25/50=0.50$.

The adherers in the first group each receive weight $1/0.80=1.25$. Their weighted count is $40(1.25)=50$, and their weighted outcome total is $8(1.25)=10$. In the second group, each weight is $1/0.50=2$, giving weighted count $25(2)=50$ and weighted outcome total $15(2)=30$. The combined weighted mean is $$ \frac{10+30}{50+50}=0.40. $$

The unweighted mean among adherers is $(8+15)/(40+25)=23/65$, approximately $0.354$. That mean describes the selected group of observed adherers. The weighted calculation restores the two history groups to their original sizes in this example; the difference illustrates why simply restricting the analysis to adherers can be misleading.

Assumptions and the Identification Result

Weighting does not recover unobserved outcomes by arithmetic alone. Let $Y^d$ denote the outcome that would be observed if the participant followed regime $d$, and let $X^d$ denote the corresponding full history and outcome trajectory. The following assumptions connect observed adherers to that counterfactual trajectory.

Consistency and no interference: if a participant follows $d$ through the outcome time, the observed trajectory equals $X^d$; one participant’s treatment does not change another participant’s outcome.
Sequential adherence exchangeability: at each decision, conditional on the measured pre-decision history among those adherent so far, adherence is independent of the future counterfactual trajectory under $d$. This rules out unmeasured predictors of both deviation and the regime-specific outcome, after conditioning on $H_t$.
Positivity: $q_t(H_t)>0$ for histories that occur with positive probability under regime $d$. A strategy that is never followed for a relevant history cannot be recovered from the observed data by weighting.
Correct probability specification: the probabilities $q_t(H_t)$ used in the weights equal the relevant conditional adherence probabilities, or are consistently estimated under an appropriate model.

Theorem: Under consistency, sequential adherence exchangeability, positivity, and correctly specified adherence probabilities, for any integrable measurable function $g$ of the full trajectory, $$ E\left[\frac{R_{K-1}}{\prod_{t=0}^{K-1}q_t(H_t)}g(X)\right]=E[g(X^d)]. $$ On the left, $g(X)$ is evaluated on the observed trajectory among participants adherent at every decision. In particular, taking $g(X)=Y$ identifies the mean outcome under sustained adherence to $d$.

A derivation can be stated using conditional transition distributions, so it does not require a density for the covariates. Start with the baseline history distribution. At decision $t$, conditional on a history $H_t$ and prior adherence, the probability of adhering is $q_t(H_t)$. Conditional on adherence, the distribution of the next history is the distribution under regime $d$: exchangeability makes adherers comparable, conditional on $H_t$, to those who would have followed $d$; consistency equates their observed next history to that regime-specific history.

Consequently, the joint contribution of adherence and the next history has the form $q_t(H_t)$ times the regime-specific transition distribution. Multiplying by the weight factor $1/q_t(H_t)$ cancels the adherence probability at that step. Repeating this argument for $t=0,\ldots,K-1$ cancels every adherence-selection factor. What remains is the baseline distribution followed by the transition distributions under $d$, which is precisely the distribution of $X^d$. Integrating $g$ against these equal trajectory distributions proves the identity. $\square$

Worked Example: Estimating Adherence Probabilities From a Model

Suppose treatment is binary, and the regime prescribes $A_t=1$. A fitted logistic model among participants adherent before decision $t$ gives $q_t(H_t)=\operatorname{expit}(-0.4+0.8L_t)$, where $L_t$ is a binary pre-decision predictor and $\operatorname{expit}(x)=1/(1+e^{-x})$.

For $L_t=0$, $q_t=\operatorname{expit}(-0.4)=1/(1+e^{0.4})$, approximately $0.4013$, so an adherer’s one-step factor is $1/0.4013$, approximately $2.49$. For $L_t=1$, $q_t=\operatorname{expit}(0.4)=1/(1+e^{-0.4})$, approximately $0.5987$, giving a factor of approximately $1.67$. The probabilities sum to one across the two possible values of $L_t$ only in the sense that they are probabilities at different histories; they are not complementary treatment probabilities for one person.

If a participant with $L_0=0$ adheres at the first decision and later has $L_1=1$ and adheres again, the cumulative weight is approximately $2.49(1.67)=4.16$. The larger contribution reflects that this particular adherent history was relatively unlikely at its first decision.

What the Assumption Means—and What It Does Not

Sequential adherence exchangeability is an assumption about the reasons people deviate. It is plausible only when the measured pre-decision history captures relevant common causes of adherence and future outcomes. The earlier material on modeling dropout driven by prior outcomes is relevant here: prior outcomes can be important predictors of subsequent adherence, and leaving them out can make the censoring model inadequate for the target.

A common mistake is to argue, “The protocol was specified in advance, so adherers represent everyone under the protocol.” A prespecified rule does not make adherence random. If worsening health both makes deviation more likely and predicts the outcome, adherers may have a more favorable prognosis even within the measured history. The correction is to model adherence conditional on sufficiently rich, correctly timed history and to state the exchangeability assumption explicitly; weighting cannot adjust for unmeasured predictors.

Another failure occurs when some histories have adherence probabilities near zero. Their inverse probabilities become large, and a few observed adherers can dominate the estimate. A probability of exactly zero violates positivity and makes the target non-identifiable from these data. Estimated probabilities and the resulting weights should therefore be inspected for extreme values and for history strata with little or no adherence. Such checks can reveal instability or lack of empirical support, but cannot prove exchangeability or establish that the model is correct.

Artificial-censoring weights also do not automatically adjust for natural loss to follow-up. If participants can disappear from observation for reasons other than protocol deviation, that is a separate censoring process requiring its own assumptions and adjustment. Nor does adherence weighting fix baseline confounding merely because a regime has been defined. The target and required adjustment depend on the study design and the sources of treatment selection.

The main alternatives answer different questions or rely on different machinery. An unweighted analysis restricted to observed adherers describes that selected group, not generally the regime-specific population. A treatment-policy analysis that retains outcomes after deviation targets outcomes under the treatment practices that actually occurred, rather than sustained adherence. The g-formula can also identify regime-specific outcomes under suitable assumptions, by modeling outcome and covariate transitions instead of reweighting adherent histories. In every approach, the causal assumptions and support in the data remain essential.

Key takeaway: Artificial censoring defines which observed histories follow the protocol; inverse adherence-probability weights address the resulting selection. Their interpretation as outcomes under continued adherence requires consistency, measured-history exchangeability, positivity, and adequate probability estimation.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

At decision times 0 and 1, an adherent participant has probabilities $q_0=0.5$ and $q_1=0.8$ of following the regime. What is the cumulative weight after both decisions?
Why must the history used to predict adherence at time $t$ be measured before the treatment decision at $t$?
What does positivity require for a history that occurs under the target regime?
Give one reason a weighted analysis might fail even if the adherence model fits the observed adherence data well.
Does restricting the analysis to observed adherers generally identify the mean outcome under sustained adherence? Explain.

Solutions. (1) The weight is $(1/0.5)(1/0.8)=2(1.25)=2.5$. (2) Later information cannot be used to predict a decision made earlier; using it can condition on post-decision variables and misrepresent the probability of adherence given information actually available at that decision. (3) The conditional probability of following the prescribed treatment must be positive, $q_t(H_t)>0$, on relevant histories. (4) Exchangeability may fail because an unmeasured factor predicts both deviation and the counterfactual outcome; model fit alone cannot detect or remove that bias. (5) Generally not. Adherers can be selected by health history and prognosis, so their unweighted outcomes describe the adherer group unless additional assumptions make that group representative of the target population.

Artificial Censoring at Protocol Deviations

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