Why Event Handling Is Part of the Estimand
In a phase 1/2 trial, a participant may discontinue study treatment, receive rescue therapy, or have the assigned dose reduced. Each event can affect the efficacy and toxicity measured later. A joint dose model therefore needs more than a dose and an outcome definition: it needs a rule for what outcome is being estimated when an event occurs. Different rules answer different clinical questions and can lead to different dose recommendations.
The strategy must be specified before analyzing outcomes. For example, an estimate that includes outcomes after rescue therapy describes the consequence of assigning a dose within a care pathway that permits rescue. It does not, by itself, estimate what that dose would have achieved without rescue. The latter is a different, counterfactual question.
Defining the Joint Estimand
The four probabilities \(q_{d,00},q_{d,01},q_{d,10},q_{d,11}\) sum to one. They describe, respectively, neither outcome, toxicity without efficacy success, efficacy success without toxicity, and both outcomes. Marginal efficacy and toxicity probabilities follow by summing the relevant cells; for example, \(P(E=1\mid d)=q_{d,10}+q_{d,11}\). In dose finding, the acceptable-dose probability or a utility may depend on these joint cells, so changing the event strategy can change the decision even if a marginal rate changes little.
An intercurrent event is an event after treatment initiation that affects either interpretation or existence of the outcome for the treatment condition of interest. Discontinuation, rescue therapy, and dose modification are common examples. The event is not automatically the same thing as a missing outcome. If a participant discontinues but efficacy and toxicity are still ascertained through the outcome window, those outcomes are observed. If they are not ascertained, the resulting unobserved outcome is also a missing-data problem; choosing an estimand strategy does not recover the missing measurement by itself.
For dose modification, distinguish the assigned starting dose from the dose actually received over time. An estimand may describe assignment to a starting dose under the protocol’s modification rules, or it may ask about sustained treatment at that dose. Those treatment conditions differ. “Dose \(d\)” is not sufficiently precise unless the assigned or received regimen is clear.
Strategies for Handling Events
| Strategy | Question answered | Key interpretation |
|---|---|---|
| Treatment policy | What is the outcome after assignment to dose \(d\), regardless of the event? | Use outcomes whether or not discontinuation, rescue, or modification occurs. |
| Hypothetical | What would the outcome have been if the event had not occurred? | Targets a counterfactual outcome under a defined no-event condition. |
| Composite | What is the outcome when the event is incorporated into the endpoint? | Define a joint outcome category or endpoint rule that includes the event. |
| While-on-treatment | What outcomes occur while participants remain on treatment? | Restrict or summarize outcomes during the specified on-treatment period; this is not automatically a full-window population estimand. |
| Principal stratum | What is the outcome in a subgroup defined by whether participants would experience an event under specified treatments? | Targets a latent subgroup; membership may not be observed for all treatment conditions. |
These strategies are not competing ways to calculate one uniquely correct answer. They target different questions. A treatment-policy strategy is often aligned with assigning a starting dose under a protocol that permits clinical management. A hypothetical strategy may be relevant when the scientific question concerns the effect without rescue, but it requires assumptions about unobserved counterfactual outcomes. A composite strategy can make an event itself count as an unfavorable outcome, provided the rule has clinical meaning. A while-on-treatment summary can be useful for describing observed treatment periods, but it may select participants who tolerate or respond to treatment. Principal-stratum effects can be clinically interesting, yet are generally difficult to identify because the stratum depends on counterfactual event status.
The event rule can differ by event type. For instance, a protocol might use a treatment-policy strategy for dose reduction, a composite strategy for permanent discontinuation due to toxicity, and a hypothetical strategy for rescue medication. Such a combination is legitimate only if each rule is stated clearly and the resulting joint outcome has a coherent clinical interpretation.
Worked Example: Treatment-Policy Joint Probabilities
Suppose, hypothetically, that 40 participants are assigned dose 2. The trial records efficacy and toxicity through week 8, including outcomes after treatment discontinuation or rescue. The observed joint counts are: 3 with both efficacy success and toxicity, 13 with efficacy success and no toxicity, 5 with toxicity but no efficacy success, and 19 with neither. Assume all 40 have evaluable week-8 outcomes.
Under a treatment-policy strategy, include each participant according to the observed outcome regardless of whether an intercurrent event occurred. The estimated joint probabilities are the cell counts divided by 40:
They sum to \(0.075+0.325+0.125+0.475=1.000\). The estimated efficacy probability is \(0.075+0.325=0.400\), and the estimated toxicity probability is \(0.075+0.125=0.200\). The probability of efficacy success without toxicity is \(0.325\). These estimates describe outcomes following assignment to dose 2 within the event and subsequent-care patterns observed in this hypothetical trial; they do not isolate a biological effect of uninterrupted dose 2.
Estimating a Hypothetical No-Rescue Outcome
A hypothetical strategy cannot generally be estimated by simply deleting participants who received rescue. Their outcomes under no rescue are not observed, and the participants who did not receive rescue may differ systematically from those who did. An analysis must specify a model or other identification approach and defend its assumptions.
A simple illustration has one binary event \(R\), where \(R=1\) means rescue occurred before the outcome window ended. Let \(Y^0\) denote the joint outcome that would have occurred under the defined condition of no rescue. For a target population assigned dose \(d\), the law of total probability gives:
The first conditional outcome is observed for participants who did not receive rescue, if their outcomes are ascertained. The second is counterfactual: it must be estimated using assumptions and data, such as a model relating observed outcome and rescue to measured information available before rescue. In a conditional analysis, the terms are standardized over the relevant histories rather than assumed equal across participants.
The basic formula is a short derivation: partition the target population into \(R=0\) and \(R=1\), apply the law of total probability, and use the observed outcome in the first group and the modeled counterfactual outcome in the second. If rescue is nearly inevitable for a type of participant, positivity may fail; data from participants who remain rescue-free cannot reliably determine that participant’s no-rescue outcome.
Worked Example: Standardizing a Hypothetical No-Rescue Estimate
Consider 30 hypothetical participants assigned the same dose; 10 receive rescue and 20 do not. Among the 20 without rescue, the joint outcome counts are \((E=1,T=1):1\), \((E=1,T=0):7\), \((E=0,T=1):2\), and \((E=0,T=0):10\). Suppose a prespecified outcome model, using measured pre-rescue information, predicts the counterfactual no-rescue probabilities among the 10 rescued participants as \(0.10, 0.50, 0.20,\) and \(0.20\) in that same cell order. Assume for this illustration that the model is correctly specified, the necessary event and outcome information is measured, and positivity holds.
The observed no-rescue proportions among 20 participants are \(1/20=0.05\), \(7/20=0.35\), \(2/20=0.10\), and \(10/20=0.50\). Standardize each cell by combining the observed no-rescue group, which is \(20/30\) of the target population, with the model-predicted rescued group, which is \(10/30\):
The cells sum to \(1.000\). The estimated hypothetical efficacy probability is \(0.0667+0.4000=0.4667\), while the estimated toxicity probability is \(0.0667+0.1333=0.2000\). These are model-based counterfactual estimates, not observed rates. If the predictions for rescued participants are wrong, or if important determinants of both rescue and outcome are unmeasured, the arithmetic remains correct but the estimate need not identify the intended no-rescue outcome.
Composite and While-on-Treatment Rules
A composite strategy incorporates an event into the outcome definition. In a joint binary endpoint, one possible rule is to count permanent discontinuation before week 8 as efficacy failure, while retaining toxicity status through the window. This rule must be prespecified and clinically justified. It is not equivalent to treating every discontinuation as toxicity: that would be a different composite.
A while-on-treatment approach instead uses outcome information accrued while treatment continues. For a fixed week-8 binary outcome, this may leave participants who stop early without a full-window outcome. Reporting only the participants who remain on treatment estimates outcomes in that observed subset, not necessarily outcomes for all assigned participants. A time-to-event or longitudinal while-on-treatment estimand needs its own event-time and summary definitions.
Worked Example: Discontinuation Under Two Strategies
In a hypothetical group of 24 participants assigned a dose, 18 complete treatment and have week-8 outcomes: 2 have both efficacy and toxicity, 7 efficacy without toxicity, 3 toxicity without efficacy, and 6 neither. The 6 who discontinue early nevertheless have week-8 outcomes ascertained: 0 have both, 1 has efficacy without toxicity, 1 has toxicity without efficacy, and 4 have neither.
Under treatment policy, the combined counts are \((2,8,4,10)\), in the order \((11,10,01,00)\). Thus estimated efficacy is \((2+8)/24=10/24=0.4167\), toxicity is \((2+4)/24=6/24=0.2500\), and efficacy without toxicity is \(8/24=0.3333\).
Now define a composite strategy that counts discontinuation before week 8 as efficacy failure, while retaining the observed toxicity status. The discontinuers’ one efficacy-without-toxicity outcome is recoded as neither outcome; the other three toxicity and four neither outcomes remain in their toxicity categories. The composite counts become \((2,7,4,11)\). Composite efficacy is \((2+7)/24=9/24=0.3750\); toxicity remains \((2+4)/24=0.2500\); efficacy without toxicity is \(7/24=0.2917\). The difference is produced by the prespecified endpoint rule, not by a new observation of biological efficacy.
If instead the analysis used only the 18 completers, the observed efficacy proportion would be \((2+7)/18=0.5000\), and toxicity would be \((2+3)/18=0.2778\). These figures describe the completer subset. They cannot be interpreted as full-population treatment-policy estimates without additional assumptions.
Choosing and Checking an Event Strategy
State whether the dose means assigned starting dose under protocol rules or a specified received regimen.
Specify what counts as rescue, discontinuation, or dose modification, and when it occurs relative to efficacy and toxicity assessment.
For each event, decide whether outcomes are retained, imagined under a no-event condition, recoded, summarized while on treatment, or restricted to a principal stratum.
Separate events from missing measurements; for hypothetical analyses, examine whether the necessary exchangeability and positivity assumptions are plausible.
Useful diagnostics include tabulating event rates and timing by assigned dose, checking how often outcomes remain observed after each event, and comparing joint cells under prespecified strategies. For a hypothetical analysis, inspect whether histories associated with rescue also occur among participants who remain rescue-free. Sparse or absent overlap signals weak support for the counterfactual prediction. Sensitivity analyses that vary the modeled outcomes after rescue can show whether a dose ranking depends heavily on unverifiable assumptions.
A frequent error is to remove participants who discontinued and call the remaining analysis a hypothetical no-discontinuation estimate. This silently assumes that, after any adjustment, the remaining participants represent those who discontinued under the no-event condition. That assumption is not guaranteed by exclusion. Another error is to label outcomes collected after rescue as “missing”; they may be fully observed and appropriate for a treatment-policy estimand. Conversely, a treatment-policy label does not solve genuine missing outcome data.
Check Your Understanding
Answer each question, then compare your reasoning with the solution immediately below it.
- A participant receives rescue medication and has week-8 efficacy and toxicity measured. Under a treatment-policy strategy, should the participant be excluded? Solution: No. Use the measured joint outcome regardless of rescue. The estimate concerns outcomes following assignment under the care pathway that permits rescue.
- Among 50 assigned participants, 30 have no rescue and 20 receive rescue. The observed no-rescue cell proportion for efficacy without toxicity is \(9/30\). A model predicts this counterfactual cell probability as \(0.30\) among those rescued. What is the standardized hypothetical cell probability? Solution: Weight by the group proportions: \((30/50)(9/30)+(20/50)(0.30)=0.18+0.12=0.30\).
- Why does analyzing only participants who completed treatment not automatically estimate a hypothetical outcome under continuous treatment? Solution: Completion can depend on prognosis, toxicity, or early response. The completers may therefore differ from those who stopped, and exclusion alone does not identify the stopped participants’ counterfactual outcomes.
- A protocol counts discontinuation as efficacy failure but continues to record toxicity. Which strategy is this, and what must be stated? Solution: It is a composite strategy. The protocol must define which discontinuations count, the timing rule, how the event changes the efficacy category, and how toxicity is assigned.
- For a no-rescue estimand, what does a lack of participants who remain rescue-free at histories common among rescued participants imply? Solution: Positivity may fail. The observed data provide little or no support for estimating no-rescue outcomes in those histories, so model extrapolation and sensitivity to assumptions become important.