Estimands for Exposure-Response Analyses

From an Exposure-Response Question to a Target Quantity

An exposure-response analysis can answer different questions even when it uses the same patients and data. For example, a comparison of outcomes among patients with higher versus lower exposure is not necessarily the same question as the comparison that would result from assigning patients to regimens intended to produce higher versus lower exposure. The population, exposure conditions, outcome, follow-up period, and handling of events after treatment begins all affect what the comparison means.

An estimand states the target quantity an analysis is intended to describe. Defining it first makes the clinical question explicit; it does not, by itself, specify how to estimate the quantity or establish that the available data can identify it. In exposure-response work, this distinction is especially important because exposure is often observed after treatment starts rather than randomly assigned.

These components should describe one coherent clinical question. “Higher exposure is associated with better response” is incomplete unless it identifies higher relative to what, in whom, for which response, over what period, and under what rules for events such as rescue therapy.

Define the Components Precisely

Target population. State the patients to whom the question is intended to apply: for example, adults with a specified disease severity who meet stated eligibility criteria. The population is not automatically the set of patients with complete exposure and outcome data. If the target is a subgroup, describe it in clinical terms rather than defining it only by a feature that may arise after treatment begins.

Exposure contrast. State the two conditions being compared. They might be specified exposure ranges, two exposure levels, or exposure conditions associated with different regimens. The contrast must be clear enough to distinguish the groups or conditions. Defining the contrast does not settle how exposure should be summarized; that is a separate modeling decision. Nor does an observed contrast imply that exposure was assigned or that the comparison is causal.

Outcome and time horizon. Define what is measured and the time at which, or period over which, it is assessed. “Efficacy” could mean change from baseline on a clinical scale at week 12, while “safety” could mean the occurrence of a specified adverse event by week 24. A time-to-event outcome also needs a defined follow-up horizon or other summary. These details determine which observations contribute to the target quantity.

Intercurrent-event strategy. Decide how events such as discontinuation, rescue therapy, or treatment switching relate to the outcome question. A treatment-policy strategy uses outcomes regardless of the event. A hypothetical strategy asks what the outcome would have been under a specified scenario in which the event did not occur. A while-on-treatment strategy uses outcomes only up to a specified treatment event or period. A composite strategy incorporates the event into the outcome definition, such as classifying rescue use as treatment failure. These strategies answer different questions; none is a universal default.

An event that prevents an outcome from being defined needs particular care. For example, death before an efficacy assessment cannot simply be treated as an unobserved week-12 score without stating what that score is meant to represent. Depending on the clinical question, death might be incorporated into a composite, addressed under a hypothetical strategy, or handled with another explicitly defined outcome.

Population-level summary. Specify how outcomes are compared, such as a difference in mean change, a difference in event risks, or a ratio of event risks. For an outcome \(Y\), a mean contrast between high and low exposure conditions can be written as:

Here \(E_T\) denotes an expectation in the stated target population, with the stated outcome and event-handling strategy. This expression can describe a standardized association. A causal contrast instead concerns outcomes under specified exposure conditions that could be assigned or otherwise defined as interventions. These are not interchangeable targets: defining a causal estimand does not show that the observed exposure groups provide a valid estimate of it. The later material on causal assumptions examines that issue.

Why the Target-Population Definition Matters

The calculation above illustrates a general standardization identity. Suppose the target population has severity groups \(k\), with fixed proportions \(w_k\) that sum to 1. The target-population mean for exposure condition \(e\) is the weighted sum of the group-specific means:

The contrast is consequently:

The equality follows by applying the same target-population weights to both exposure conditions and collecting the terms for each group. If the weights differ between conditions, the contrast combines within-group differences with differences in population composition. It may then answer a different question from the one intended. Standardization clarifies the target summary; it does not remove the need to justify the data or assumptions used to estimate it.

Intercurrent Events Can Change the Question

An event strategy should be chosen for clinical reasons and stated before interpreting a result. Consider rescue medication in an efficacy study. Under a treatment-policy strategy, week-12 response is assessed regardless of rescue use. Under a hypothetical strategy, the target is response that would have occurred if rescue had not been used. Under a composite strategy, rescue might count as failure even if the patient later meets the response criterion. Each produces a different outcome definition or target.

Diagnostics, Alternatives, and Limitations

Before analysis, check whether the estimand can be read as a complete sentence with an unambiguous population, exposure contrast, outcome, horizon, event strategy, and summary. Check that the outcome can be defined for the relevant patients and that the data collection period matches the stated horizon. Also check whether the exposure conditions and outcome assessments are represented in the data for the target population. If a target exposure condition is absent or nearly absent for an important subgroup, a numerical estimate may rely heavily on extrapolation rather than direct information.

Several warning signs suggest the question or its implementation is going wrong:

• The exposure groups are described only as “high” and “low,” without a definition of the contrast.
• The outcome is reported without a measurement time or follow-up period.
• Patients who discontinued or used rescue therapy are excluded without explaining how that exclusion relates to the target question.
• The analysis describes an association, but the conclusion uses causal language such as “increasing exposure improves response.”
• The stated target population differs from the analyzed population, but no target-population summary or justification is given.

The main alternatives are not competing statistical procedures; they are alternative estimands. A treatment-policy question may be more relevant to outcomes under real treatment use, while a hypothetical question may address what would happen absent rescue or discontinuation. A composite may be useful when an event itself represents an unfavorable outcome. A while-on-treatment summary may be relevant to outcomes during exposure, but it does not describe the full follow-up experience. Choose among them by asking which clinical decision the result is intended to inform.

An estimand is also not an estimator. The estimand defines the target; an estimator is the procedure used to calculate an estimate from data. A precisely worded target can still be difficult or impossible to estimate credibly if data are sparse, follow-up is incomplete, or the exposure conditions are poorly represented. Likewise, a plausible estimate cannot repair an unclear target. Exposure-response analyses commonly support important development decisions, so the question should be fixed before interpreting fitted model results.