A Hypothetical Estimand for Rescue Medication Use

Why a Hypothetical Estimand for Rescue Medication?

Rescue medication can be clinically necessary, but its use changes what a trial outcome means. If the outcome is symptom improvement at week 12, a participant who takes rescue medication has an observed week-12 outcome that reflects both randomized treatment and rescue medication. A treatment-policy estimand includes that outcome as observed. A hypothetical estimand instead asks what the treatment effect would have been if rescue medication had not been used.

That second question concerns outcomes that were not observed for participants who took rescue medication. Estimating it therefore requires a model for their counterfactual outcomes and assumptions about why rescue was used. Randomization alone does not identify those outcomes. The difference between the two estimates can be clinically informative, but it is not automatically evidence that one estimate is more correct: they address different questions.

Definition: A hypothetical estimand for rescue medication use is a treatment-effect target defined for a specified trial population and outcome under the hypothetical condition that participants do not use rescue medication during a defined period. For a continuous outcome $Y$, a common treatment-effect summary is the difference in mean outcomes under randomized treatment $A=1$ and control $A=0$, with rescue use set aside in both treatment conditions: $$ \Delta_{\mathrm{hyp}} = E\{Y(1,\text{no rescue})\} - E\{Y(0,\text{no rescue})\}. $$ Here $Y(a,\text{no rescue})$ denotes the outcome that would occur under assignment $a$ if rescue medication were not used. This counterfactual outcome is not observed for participants who take rescue medication.

A complete estimand also specifies the population, treatment conditions, outcome variable and time point, and summary measure. For example, the population might be all randomized participants; the treatments might be an active drug and placebo; the outcome might be improvement in symptoms at week 12, measured in points, with higher values better; and the summary might be the active-minus-control difference in mean improvement. The rescue strategy then specifies the hypothetical absence of rescue medication through week 12.

The target is not the mean difference among participants who happened not to use rescue. Rescue use occurs after randomization and may depend on symptom severity, early response, clinician judgment, and other prognostic factors. Comparing only observed nonusers can therefore destroy the balance created by randomization.

What Must Be Assumed to Estimate It?

Suppose $A_i$ is participant $i$'s randomized assignment, $R_i=1$ indicates rescue medication use by week 12, and $X_i$ and $H_i$ denote baseline characteristics and observed history before rescue. A Bayesian analysis can model the hypothetical outcome conditional on treatment and relevant observed information, then use the model to estimate the outcomes that are missing because rescue occurred.

The model does not make the counterfactual observable. Its interpretation depends on assumptions. A typical individual-level analysis requires the following:

Consistency and a well-defined intervention: the observed outcome for a participant who does not use rescue is the outcome under the specified no-rescue condition. The rescue policy, including medication, timing, and period, must be defined clearly.
Conditional exchangeability: after conditioning on the modeled treatment and observed history, rescue users' hypothetical no-rescue outcomes are adequately represented by comparable participants or by the outcome model. In practical terms, relevant predictors of both rescue use and outcome must be measured and included adequately.
Positivity: for histories relevant to the target population, the data must provide information about outcomes without rescue. If nearly everyone with a particular severe history receives rescue, the no-rescue outcome for that history is largely extrapolated.
Model adequacy: the outcome and rescue-use models, including their functional forms and uncertainty, must be suitable for the data. A posterior distribution is conditional on these modeling choices.

These assumptions can be difficult to defend when rescue is triggered by rapidly worsening symptoms, when important clinical judgments are unrecorded, or when rescue is nearly inevitable for some participants. A richly specified Bayesian model can represent uncertainty under stated assumptions; it cannot recover information that the trial did not collect or verify an untestable assumption.

A Bayesian Calculation for a Treatment Effect

For a transparent calculation, suppose an analysis produces an estimated treatment effect $\hat{\Delta}$ with standard error $SE$, and uses a normal likelihood approximation:

\hat{\Delta}\mid\Delta \sim N(\Delta,SE^2), \qquad \Delta\sim N(0,\tau^2).

The first distribution describes sampling uncertainty around the treatment effect; the second is a prior distribution. Under this model, the posterior is normal. Its precision is the sum of the likelihood and prior precisions, and its mean is the precision-weighted average of the observed estimate and prior mean:

\Delta\mid\hat{\Delta}\sim N(m,V),\qquad V=\left(\frac{1}{SE^2}+\frac{1}{\tau^2}\right)^{-1},\qquad m=V\left(\frac{\hat{\Delta}}{SE^2}+\frac{0}{\tau^2}\right).

To justify the result, multiply the likelihood kernel by the prior kernel and collect the terms involving $\Delta$:

-\frac{(\hat{\Delta}-\Delta)^2}{2SE^2}-\frac{\Delta^2}{2\tau^2} = -\frac{1}{2}\left(\frac{1}{SE^2}+\frac{1}{\tau^2}\right)(\Delta-m)^2+C,

where $C$ does not depend on $\Delta$. This is the kernel of a normal distribution with mean $m$ and variance $V$. In an actual hypothetical-estimand analysis, $\hat{\Delta}$ and $SE$ should come from an analysis that accounts for the counterfactual-outcome model and its uncertainty, rather than from treating imputed outcomes as known.

Worked Example: Defining the Target Before Looking at the Estimate

Consider a hypothetical randomized trial of an active symptom treatment versus placebo. The endpoint is improvement in symptoms at week 12, measured in points, with higher values indicating greater improvement. Rescue medication may be taken at any time before week 12.

For the hypothetical estimand, the population is all randomized participants; the treatment conditions are assignment to active treatment and assignment to placebo; the outcome is week-12 improvement in points; and the summary is the difference in mean improvement. The intercurrent-event strategy is to ask what each treatment group's mean would be if no participant used rescue medication through week 12.

Thus the target is $\Delta_{\mathrm{hyp}}=E\{Y(1,\text{no rescue})\}-E\{Y(0,\text{no rescue})\}$. A participant who used rescue still belongs to the target population. Their observed outcome is not automatically their no-rescue outcome; it must be estimated under the model and assumptions. This definition prevents an erroneous change in the target to “the effect among participants who did not use rescue.”

Worked Trial: Treatment Policy and Hypothetical Effects

In the following hypothetical data, higher improvement scores are better. Each group has 40 randomized participants. The treatment-policy outcome uses observed week-12 outcomes for everyone, regardless of rescue use. The hypothetical outcome replaces the observed outcomes of rescue users with model-based predictions of what their week-12 outcomes would have been without rescue. The predictions are illustrative summaries from an individual-level Bayesian outcome model; the analysis also carries their uncertainty into the reported standard error.

Worked Example: Calculating Both Posterior Treatment Effects

The active-treatment group has 8 rescue users and 32 nonusers. Their observed mean improvement is 4.0 points among rescue users and 6.5 points among nonusers. The control group has 16 rescue users and 24 nonusers, with observed means of 3.0 and 5.5 points, respectively. The model predicts mean no-rescue outcomes of 3.5 points for active-treatment rescue users and 1.5 points for control rescue users.

Treatment-policy means. These use the observed outcomes of all randomized participants:

\bar{Y}_{1,\mathrm{TP}}=\frac{32(6.5)+8(4.0)}{40} =\frac{208+32}{40}=6.0\text{ points},

\bar{Y}_{0,\mathrm{TP}}=\frac{24(5.5)+16(3.0)}{40} =\frac{132+48}{40}=4.5\text{ points}.

The treatment-policy difference is $6.0-4.5=1.5$ points. Suppose its standard error is $0.8944$ points. With a prior $\Delta_{\mathrm{TP}}\sim N(0,2^2)$, the posterior variance and mean are

V_{\mathrm{TP}}=\left(\frac{1}{0.8944^2}+\frac{1}{4}\right)^{-1} \approx 0.6667,\qquad m_{\mathrm{TP}}=0.6667\left(\frac{1.5}{0.8944^2}\right)\approx1.250.

The posterior standard deviation is $\sqrt{0.6667}=0.8165$ points. The approximate 95% credible interval is $1.250\pm1.96(0.8165)$, or $(-0.3503,\,2.850)$ points. Given this model and prior, the interval contains 95% of the posterior probability for the treatment-policy effect.

Hypothetical no-rescue means. Replace the observed rescue-user means with the model-predicted no-rescue means, while retaining the observed nonuser outcomes:

\bar{Y}_{1,\mathrm{hyp}}=\frac{32(6.5)+8(3.5)}{40} =\frac{208+28}{40}=5.9\text{ points},

\bar{Y}_{0,\mathrm{hyp}}=\frac{24(5.5)+16(1.5)}{40} =\frac{132+24}{40}=3.9\text{ points}.

The hypothetical difference is $5.9-3.9=2.0$ points. Suppose the model-based standard error, including uncertainty in the counterfactual predictions, is $0.9500$ points. Using the same $N(0,2^2)$ prior gives

V_{\mathrm{hyp}}=\left(\frac{1}{0.9500^2}+\frac{1}{4}\right)^{-1} \approx0.7364,\qquad m_{\mathrm{hyp}}=0.7364\left(\frac{2.0}{0.9500^2}\right)\approx1.632.

The posterior standard deviation is approximately $0.8581$ points, giving the approximate 95% credible interval $1.632\pm1.96(0.8581)=(-0.0501,\,3.314)$ points. Under these illustrative assumptions, the hypothetical posterior mean is $0.382$ points larger than the treatment-policy posterior mean. Both central 95% credible intervals include zero. However, $P(\Delta>0)$ is approximately 0.9371 for the treatment-policy analysis and 0.9714 for the hypothetical analysis.

The difference between the two posterior means is not, by itself, a posterior distribution for the difference between estimands. Both estimates use participants from the same trial and may be statistically dependent. A probability statement about their difference requires a joint model that represents that dependence.

Interpreting the Difference and Checking the Model

In the example, the hypothetical estimate is larger because the model predicts more deterioration without rescue among control rescue users than among active-treatment rescue users. The calculation makes this mechanism explicit. Relative to observed outcomes, the predicted changes are $-0.5$ points for active-treatment rescue users and $-1.5$ points for control rescue users. The resulting changes to the group means are $(8/40)(-0.5)=-0.1$ points for active treatment and $(16/40)(-1.5)=-0.6$ points for control. The active-minus-control difference therefore increases by $-0.1-(-0.6)=0.5$ points, from 1.5 to 2.0.

This is an interpretation of the model's counterfactual predictions, not a direct observation of what would have happened. Useful diagnostics include checking whether rescue users and nonusers overlap in baseline severity and pre-rescue outcome history; examining the timing and frequency of rescue use; checking model fit for observed outcomes and rescue patterns; and testing how results change under plausible alternative predictions for rescue users. A lack of overlap or a large dependence on extrapolation is a warning that the estimate is weakly informed by trial data.

Worked Example: Sensitivity to Counterfactual Assumptions

The treatment-policy difference remains 1.5 points. In the preceding analysis, the model-predicted change from observed to no-rescue outcome among rescue users was $-0.5$ points for active treatment and $-1.5$ points for control. Consider an alternative, still hypothetical sensitivity scenario in which those changes are $-1.5$ points for active treatment and $-1.0$ point for control. The rescue proportions are $8/40=0.20$ and $16/40=0.40$.

The alternative hypothetical treatment effect is the treatment-policy difference plus the rescue proportions times the corresponding changes:

\hat{\Delta}_{\mathrm{hyp,alt}} =1.5+(0.20)(-1.5)-(0.40)(-1.0) =1.5-0.30+0.40=1.60\text{ points}.

For illustration, retain the model-based standard error of $0.9500$ points and prior $N(0,2^2)$. The posterior variance is again approximately $0.7364$, and the posterior mean is $0.7364(1.60/0.9025)\approx1.305$ points. Its approximate 95% credible interval is $1.305\pm1.96(0.8581)=(-0.377,\,2.987)$ points. This sensitivity scenario shifts the estimate from 2.0 to 1.6 points; the interval calculation also shows that this illustrative change does not remove substantial uncertainty. In a real analysis, the standard error should be recomputed under each scenario if the assumptions alter the outcome model or its uncertainty.

How This Estimand Differs From Alternatives

The choice of strategy depends on the clinical question. The treatment-policy estimand after treatment discontinuation, discussed in the earlier material on Treatment-Policy Estimands After Treatment Discontinuation, similarly targets outcomes regardless of the intercurrent event, but rescue use and discontinuation are distinct events and need not have the same implications.

Approach	Question answered	Key caution
Treatment policy	What is the effect of assignment when outcomes after rescue are included?	Rescue use may dilute or otherwise alter the observed treatment contrast; this is part of the target, not necessarily a bias.
Hypothetical no rescue	What would the effect be if rescue medication were not used?	Requires assumptions and modeling for unobserved no-rescue outcomes.
Composite strategy	What is the effect when rescue use is incorporated into a combined outcome definition?	The composite must be clinically interpretable; it answers a different question.
While-on-treatment analysis	What outcomes are seen during a specified treatment period?	Ending follow-up at rescue can create selection and missing-data problems; it does not automatically estimate a no-rescue effect.

A hypothetical estimand is most useful when the no-rescue question is clinically relevant, the trial has rich measurements before rescue, and there is credible overlap to support the counterfactual model. Its limitations should be reported alongside the posterior: dependence on untestable assumptions, possible extrapolation for severe participants, sensitivity to prior and model choices, and the fact that a hypothetical no-rescue condition may not correspond to a feasible or ethical intervention for everyone. The analysis should not be presented as if it were a randomized comparison among rescue users and nonusers.

Common Pitfall

Common mistake: “The participants who did not take rescue were randomized, so their treatment difference estimates the no-rescue effect.”

Correction: Randomization balances treatment assignment at baseline in the full randomized population. Conditioning on post-randomization rescue use can select groups with different prognosis, because symptoms and response may influence rescue. A no-rescue estimand targets the counterfactual outcomes for the specified population, including rescue users; it generally requires modeling and explicit assumptions rather than simply discarding those participants.

Key takeaway: The treatment-policy and hypothetical estimands answer different questions. A Bayesian posterior for the hypothetical effect quantifies uncertainty conditional on the likelihood, prior, and counterfactual model; it does not eliminate the need to justify that model or examine sensitivity to its assumptions.

Check Your Understanding

Answer each question, then compare with the solutions below.

Why is the mean difference among observed nonusers not generally the hypothetical no-rescue treatment effect?
In the trial example, what feature of the counterfactual predictions makes the hypothetical treatment effect larger than the treatment-policy effect?
Name two assumptions or data features that should be examined when assessing a hypothetical no-rescue analysis.
If the prior variance were made smaller while keeping the likelihood unchanged, what would happen to the posterior mean when the prior mean is zero?
Does a 95% posterior credible interval containing zero establish that the null effect is true? Explain.

Solutions. (1) Rescue is post-randomization and can depend on prognosis, so observed nonusers may not represent the full randomized population under no rescue. (2) The model predicts a larger worsening without rescue among control rescue users; because more control participants used rescue, the control mean falls more under the hypothetical scenario, increasing the active-minus-control contrast. (3) Examples include conditional exchangeability given measured history, positivity or overlap, outcome-model adequacy, and whether relevant predictors of rescue and outcome were measured. (4) The posterior mean moves closer to zero, because a smaller prior variance gives the zero-centered prior greater precision. (5) No. It means zero is among the values covered by that interval's posterior probability; the posterior is conditional on the model and prior and does not prove the null effect is true.