Per-Protocol Effects Under Sustained Strategies

What Does a Per-Protocol Effect Ask?

A treatment strategy may prescribe more than an initial choice. It can specify what treatment to use at each follow-up time, when to stop, or how to respond to a changing clinical history. A per-protocol effect compares outcomes under sustained adherence to each of the strategies being contrasted. It answers a different question from an intention-to-treat analog, which compares outcomes according to the treatment strategy assigned or initiated at baseline, regardless of later deviations.

The distinction matters when participants switch treatments, stop treatment, or otherwise depart from the protocol. For a per-protocol question, the target is the outcome that would have occurred had the person followed the specified strategy through the relevant follow-up. It is not, in general, the outcome observed among the subset who happened to remain adherent. Defining the estimand therefore requires a clear strategy, a definition of deviation, and a follow-up window.

A strategy can be fixed, such as “use treatment 1 at every scheduled decision,” or history-dependent, such as “use treatment 1 while a measured safety indicator is below a threshold, and otherwise use treatment 0.” At time \(t\), write \(A_t\) for treatment and \(H_t\) for the measured history available before that decision. A rule \(d_t(H_t)\) specifies the treatment required at that time. Sustained adherence means following the rule at each relevant decision while the person remains eligible for that decision.

A protocol deviation is a departure from that operational rule: for example, receiving the other treatment, stopping earlier than permitted, or missing a required treatment decision. The protocol must say which departures count and when they begin. A missed dose may count as a deviation under one protocol but not another. An outcome event that ends the opportunity for later treatment decisions is not automatically a deviation; the protocol and endpoint definition determine how that event is handled.

Specify the Estimand and Follow-Up

The target population is the population eligible at time zero, as defined for the emulated trial. The estimand compares the potential outcomes under specified strategies in that population, over the same protocol-defined follow-up window. It is not restricted to people who would remain adherent in the observed data. Such a restriction would generally define a selected group, and membership in that group can depend on prognosis, treatment response, or events occurring after time zero.

For an emulated trial, state the following before analyzing the data:

• Strategies: the treatment rule at each decision time, including permitted stopping, switching, or exceptions.
• Deviation rule: which observed actions depart from each strategy, and the time at which departure begins.
• Endpoint and horizon: what outcome is measured and when follow-up for that endpoint ends.
• Target population and contrast: who is represented and whether the comparison is a risk difference, risk ratio, or another specified contrast.

Follow-up for the per-protocol estimand continues conceptually to the endpoint or administrative horizon under the counterfactual that the assigned strategy was followed. A participant’s observed deviation does not redefine the scientific question as “what happened up to the deviation.” In an analysis, post-deviation observations cannot simply be treated as outcomes under sustained adherence. How to account for deviations during estimation is a separate implementation question; the estimand must be settled first.

Identification From Longitudinal Data

To connect the per-protocol estimand to observed data, consider decision times \(t=0,\ldots,K-1\). Let \(L_t\) denote measured covariates available before treatment \(A_t\), and let \(H_t=(L_0,A_0,L_1,\ldots,A_{t-1},L_t)\) be the history before that decision. Let \(L_K\) be the covariate history available at the endpoint, and let \(Y\) be the endpoint. For simplicity, the formula below is written for discrete covariates; sums can be replaced by integrals for continuous covariates.

The following assumptions give a standard identification result. Consistency says that for a person who follows strategy \(d\), the observed endpoint and covariate history agree with the corresponding potential outcomes under \(d\). Sequential exchangeability says that, at each decision time, treatment choice is conditionally independent of the potential future covariate and outcome trajectory under \(d\), given the measured history \(H_t\), among people still able to follow the strategy. Positivity requires that, for every history with positive probability under \(d\), the treatment prescribed by \(d_t(H_t)\) has positive probability in the observed data at that history. Assume also that the relevant variables are measured in the required temporal order, and that endpoint ascertainment is complete through the horizon or its missingness is handled under suitable additional assumptions.

In this expression, treatment values in the histories are set to those required by the strategy. The formula averages the conditional endpoint risk over the covariate histories that would arise under that strategy. Thus, if earlier treatment changes later covariates, the distribution of those covariates is allowed to differ between strategies.

Justification. Under strategy \(d\), the treatment at each time is fixed by the rule \(d_t(H_t)\). The joint distribution of the covariate history under that strategy factors into its baseline distribution and successive conditional distributions:

At time \(t\), consistency equates the potential covariate transition under the prescribed treatment with the observed transition among those with that history who receive \(A_t=d_t(H_t)\). Sequential exchangeability makes that observed conditional transition representative of the transition that would occur under the strategy, rather than a selected transition associated with unmeasured prognostic differences. Positivity ensures that the required conditional distribution exists in the observed data. Applying this argument at each decision time gives the product of transition distributions in the displayed formula.

The same reasoning applies to the endpoint: consistency and sequential exchangeability identify its conditional mean under the prescribed treatment history by the observed conditional mean among people with that history and treatment sequence. Averaging that mean over \(f^d(l_0,\ldots,l_K)\), using the law of total expectation, gives the stated expression for \(P(Y^d=1)\). This establishes the result. \(\square\)

Why the Observed Adherers Are Not the Estimand

A tempting shortcut is to keep only people who remained adherent and compare their observed outcomes. That comparison conditions on a post-baseline behavior. If prognosis affects adherence, or if earlier treatment affects both adherence and prognosis, the adherent groups can differ even when the sustained strategies have identical causal effects. This is a selection problem, not a different way of writing the same estimand.

Assumptions, Diagnostics, and Alternatives

The identification assumptions are substantive, not guaranteed by the choice of an analysis method. In particular, sequential exchangeability can fail if important predictors of both adherence or treatment choice and the outcome are unmeasured. Positivity can fail if some histories almost always lead to a treatment inconsistent with the strategy. Consistency can be questionable when a strategy is too vague to distinguish materially different versions of treatment.

Useful diagnostics begin with the protocol and the observed histories:

• Check that treatment, covariates, deviations, and outcomes have dates that establish the required order. A covariate measured after a treatment decision cannot be treated as pre-decision history for that decision.
• Tabulate adherence by important histories and decision times. Sparse or absent observations receiving the prescribed treatment indicate a positivity problem.
• Inspect whether the deviation definition can be applied consistently and whether its timing is recorded precisely.
• Compare the measured histories of people who remain adherent with those who deviate. Large differences are a warning against interpreting adherer-only results causally.
• Check whether the endpoint is observed through the stated horizon and whether loss to follow-up may also select participants.

The main alternative estimand is the intention-to-treat analog: the effect of baseline assignment or policy, allowing later behavior to unfold as observed. It can be appropriate when the question concerns the consequences of adopting that initial policy. An as-treated comparison instead relates outcomes to treatment actually received; without strong control of time-varying confounding, it does not by itself identify the sustained-strategy effect. These approaches answer distinct questions and should not be compared as though they were interchangeable estimators of one quantity.

A per-protocol effect also has limitations. It may rely on extrapolation when adherence to a strategy is rare for some histories, and its interpretation depends on a sufficiently precise intervention definition. Results can be sensitive to unmeasured predictors of deviation. Finally, the hypothetical outcome under sustained adherence may not correspond to a feasible intervention if the strategy requires treatment that could not safely or practically be continued for some participants.

Proof. Let \(Z\) be randomized baseline assignment. Randomization gives \(Z\) independent of the potential outcomes \(Y^{d_1}\) and \(Y^{d_0}\). Under perfect adherence and consistency, \(Y=Y^{d_z}\) for a person assigned \(Z=z\). Therefore,

where the second equality follows from randomization. Subtracting the equation for \(z=0\) from that for \(z=1\) yields the per-protocol risk difference. If participants deviate, the consistency equality with the assigned sustained strategy no longer follows, so the argument does not establish equality. \(\square\)