Estimands For Dose-Level Benefit And Harm

Why Dose-Level Estimands Need More Than a Dose Label

A statement such as “efficacy is 40% at this dose” is incomplete until it specifies the population, what counts as efficacy, when it is assessed, and how events such as treatment discontinuation are handled. The same is true of a toxicity rate. These details define the question a dose-finding trial is intended to answer; they are not choices made after looking at the data.

The introductory material on defining joint efficacy and toxicity outcomes distinguishes the possible outcome pairs for a participant. Here the focus is on the population quantities those outcomes define at each dose. A central distinction is between marginal summaries, such as the probability of efficacy, and the joint distribution, which describes how efficacy and toxicity co-occur in the same population.

Potential Outcomes and Dose-Specific Marginal Estimands

For clarity, suppose the outcomes are binary and are assessed over a prespecified period after dose assignment. For participant \(i\), let \(E_i(d)=1\) indicate that the participant would meet the efficacy definition under dose \(d\), and let \(T_i(d)=1\) indicate that the participant would meet the toxicity definition under dose \(d\). Each is zero if the corresponding event does not occur under the specified outcome definition and period. The population of interest must also be stated, for example, eligible patients represented by the trial’s intended use population.

The potential outcomes \(E_i(d)\) and \(T_i(d)\) are defined for each dose, although a participant in an ordinary parallel-group trial is observed at only the assigned dose. The population marginal estimands at dose \(d\) are:

Here \(p_E(d)\) is the proportion of the target population who would meet the efficacy definition under dose \(d\), and \(p_T(d)\) is the proportion who would meet the toxicity definition. A dose contrast can be defined for either outcome. For example, comparing dose \(d_1\) with dose \(d_0\), the risk differences are:

A positive \(\Delta_E\) means a higher efficacy probability at \(d_1\); a positive \(\Delta_T\) means a higher toxicity probability at \(d_1\). Those are separate statements. Neither contrast alone says whether efficacy and toxicity occur in the same participants, and neither establishes that a dose is preferable overall.

Joint Efficacy-Toxicity Estimands

At each dose, the pair \(\{E_i(d),T_i(d)\}\) has four possible values. Define \(q_{ab}(d)=P\{E_i(d)=a,T_i(d)=b\}\), where \(a,b\) are each zero or one. Thus \(q_{11}(d)\) is the probability of both efficacy and toxicity, and \(q_{10}(d)\) is the probability of efficacy without toxicity. The four probabilities sum to one.

The marginal estimands are sums of cells in this joint distribution:

Consequently, the marginal probabilities do not determine the joint distribution. If the clinical question concerns the probability of an efficacy event without toxicity, \(q_{10}(d)\) is a direct estimand. If it concerns efficacy among participants who experience toxicity, the corresponding conditional probability is \(P\{E_i(d)=1\mid T_i(d)=1\}=q_{11}(d)/p_T(d)\), provided \(p_T(d)>0\). A conditional quantity answers a different question from the unconditional efficacy probability.

Worked Example: Marginal Dose Contrasts and Joint Outcomes

A Short Justification: Marginals Do Not Determine Association

For binary outcomes at a fixed dose, a simple measure of within-person association is the covariance between the two potential outcomes in the target population. Since \(E_i(d)T_i(d)=1\) only for the \((1,1)\) cell, its expectation is \(q_{11}(d)\). Therefore:

This shows directly why the two margins are not enough: the covariance also depends on \(q_{11}(d)\). The joint cell must be known or estimated to describe this association. The allowable range of that cell follows from nonnegative cell probabilities. Since \(q_{11}(d)\) is a probability and \(q_{10}(d)=p_E(d)-q_{11}(d)\), it cannot exceed either marginal. Also \(q_{00}(d)=1-p_E(d)-p_T(d)+q_{11}(d)\) cannot be negative. Combining these restrictions gives:

For instance, if \(p_E(d)=0.50\) and \(p_T(d)=0.30\), then \(q_{11}(d)\) can range from 0 to 0.30. This is a probability constraint, not a model for the association. It permits many different joint distributions with the same two marginal rates.

Worked Example: Same Margins, Different Joint Distributions

Specify the Outcome Strategy Before Estimation

An estimand must make clear what happens in its definition when a participant discontinues treatment, receives rescue therapy, or has an outcome that cannot be assessed. For example, a treatment-policy strategy defines the outcome according to what is observed over the follow-up period regardless of discontinuation or subsequent treatment. A different, explicitly defined strategy could ask what the outcome would have been had discontinuation not occurred. Those strategies target different quantities and require different data and assumptions.

The assessment period also matters. “Toxicity by 28 days” and “toxicity by 90 days” are distinct outcomes; a participant can be counted differently under the two definitions. For a joint estimand, the efficacy and toxicity definitions must be clear about their periods and event handling. If their periods differ, the joint pair is still definable, but it refers to those two specified periods rather than a single shared window.

Identification, Diagnostics, and Common Pitfalls

In a randomized dose-assignment trial, the dose-specific marginal probabilities and joint cell probabilities can be estimated from the corresponding outcomes in participants assigned to that dose, under standard conditions. These include consistent implementation of the dose and outcome definitions, random assignment with a positive probability of receiving each dose in the target population, and no interference between participants. Outcome ascertainment must be adequate, or missing outcomes must be addressed using defensible assumptions and methods. Randomization supports comparison across assigned doses; it does not automatically repair outcome misclassification, missing follow-up, or deviations from the specified treatment strategy.

A basic diagnostic is to verify that the four estimated joint cell proportions are nonnegative and sum to one, and that their row and column sums reproduce the reported efficacy and toxicity margins. Check denominators and follow-up completeness separately for each outcome. Large or dose-differential amounts of missing data can make complete-case joint proportions unrepresentative; a tidy four-cell table does not itself establish validity.

The main alternatives answer different questions. Separate marginal probabilities give direct summaries of benefit and harm but discard co-occurrence information. A joint distribution retains that information but is harder to estimate precisely when some cells are rare. A conditional probability, such as efficacy among those with toxicity, is clinically interpretable for a specified group but conditions on a post-assignment outcome and should not be treated as a causal effect of toxicity. A composite outcome can reduce multiple outcomes to one summary, but its meaning depends on the chosen rule and it can conceal which component drives the result.

• Mistake: “The efficacy rate is 50% and toxicity is 30%, so 15% had both.” This multiplies the margins, which is justified only under independence. Correction: Estimate or specify \(q_{11}(d)\) directly; the margins alone do not determine it.
• Mistake: “At the higher dose, more participants responded, so it benefits each individual.” A population contrast compares probabilities across dose strategies; it does not reveal which particular participants would respond under both doses. Correction: State the population-level estimand and do not infer unobserved individual dose responses from group rates.
• Mistake: Reporting a toxicity percentage without an assessment window or event-handling rule. Correction: Include the population, assigned dose, toxicity definition, follow-up period, and strategy for intercurrent events in the estimand description.

These estimands describe outcomes and contrasts; they do not by themselves rank doses or prescribe a selection rule. Their value is that they make the evidence being summarized explicit, including whether the question concerns separate event rates, their co-occurrence, or a contrast between dose assignments.