Why a Dose-Selection Question Needs an Estimand
A dose-selection conclusion is meaningful only in relation to a particular population, regimen, treatment duration, outcome, and decision rule. “Choose the effective dose” is not sufficiently precise: effectiveness could mean a mean improvement, a response rate, or a balance of benefit and harm. It could also refer to different patient populations or different times after treatment begins.
An estimand is the precisely defined population quantity that answers the scientific question. It is not the estimate calculated from a particular sample, nor is it by itself a complete decision rule. A dose-selection decision rule states how the estimand, its uncertainty, and any practical constraints will be used to choose a dose.
For example, let \(Y(d,\tau)\) denote a favorable-direction outcome at duration \(\tau\) under regimen \(d\), for an individual in population \(P\). The population mean under dose \(d\) is \(E_P[Y(d,\tau)]\). A causal interpretation of comparisons between doses requires that the regimens and outcomes are well defined and that the data or model identify the relevant population outcomes. In a dose-ranging study, randomization can support comparisons among assigned regimens; model-based projections additionally depend on the model and its assumptions.
Three Distinct Dose-Selection Estimands
The minimum effective dose, the dose maximizing net benefit, and the dose meeting a target response are related but answer different questions. Each requires a defined feasible dose set \(\mathcal D\). That set may consist of tested doses, clinically allowable doses, or doses within a specified range. If a recommendation must be one of the tested doses, the set should say so.
“Effective” here means meeting the stated response threshold; it does not automatically mean statistically significant, safe, or optimal. A different convention—such as defining effectiveness as any positive mean effect—would yield a different estimand and should be stated explicitly. If doses are continuous rather than a finite set, the minimum must actually be attained for the expression to define a minimum; otherwise an infimum may exist without a smallest qualifying dose.
The weights and the utility scales determine how benefits and harms are traded off. They are substantive choices, not quantities supplied automatically by a dose-response model. If harms are treated as unacceptable beyond a limit, the rule can instead exclude doses violating that limit and maximize net benefit over the remaining feasible set.
This target-response estimand concerns a population proportion, not the mean outcome. The individual response threshold \(r^*\) and required proportion \(q\) are separate choices: one defines what counts as a response, and the other defines how common that response must be.
From Estimand to Decision
The estimand is a population quantity; a study produces data and an estimate of that quantity. A decision rule specifies how to act given the estimate and its uncertainty. For instance, one rule might select the lowest dose whose estimated mean improvement exceeds \(\delta\). A more cautious rule might require the lower confidence bound for that improvement to exceed \(\delta\). These rules are not interchangeable: the second requires stronger evidence before selecting a dose. The confidence-bound rule does not change the population estimand; it changes the evidence required for the decision.
The basic minimum-dose rule follows directly from the definition. For each feasible dose, assess whether its population improvement reaches the prespecified threshold. The qualifying doses form the set in braces; selecting the smallest member implements the “minimum” criterion. If the doses are discrete, this is a comparison across those candidates. If the response curve is nonmonotone, it is not valid to assume that every higher dose also qualifies; each candidate must be assessed under the stated rule.
For a net-benefit rule, the corresponding operation is to calculate \(U(d)\) for every feasible dose and select a maximizer. For a target-response rule, assess \(p_R(d)\geq q\) and select the smallest qualifying dose. All three procedures require estimates in practice, but these mathematical definitions refer to population quantities, not noisy sample summaries.
Worked Examples
Worked Example: Finding a Minimum Effective Dose
Hypothetical model-based results. Consider a defined adult population treated for 12 weeks with one specified regimen at doses \(0\), \(10\), \(20\), or \(40\) mg once daily. Let \(Y\) be the improvement in a symptom score at week 12, with larger improvement favorable. Suppose the prespecified meaningful threshold is \(\delta=2.5\) score points, and the estimated placebo-adjusted mean improvements are \(0\), \(1.2\), \(2.8\), and \(3.1\) points at the four doses, respectively. Assume for this example that the estimates adequately represent the population means.
The qualifying condition is \(\Delta(d)\geq2.5\) points. At 10 mg, \(1.2<2.5\), so that dose does not qualify. At 20 mg, \(2.8\geq2.5\), and at 40 mg, \(3.1\geq2.5\). The qualifying doses are therefore 20 and 40 mg, and the smallest is \(d_{\mathrm{MED}}=20\) mg once daily for 12 weeks in this population. This is a threshold-based conclusion; it does not establish that 20 mg is safest or has the greatest net benefit.
Worked Example: Selecting a Dose by Net Benefit
Hypothetical utility calculation. For a specified population and 12-week treatment duration, suppose efficacy benefit, expected harm, and treatment burden have already been converted to a common utility scale. Let the net utility be benefit minus harm minus burden, with each dose’s total deductions shown below. The resulting values are hypothetical population summaries.
| Dose | Benefit | Harm and burden | Net benefit calculation |
|---|---|---|---|
| 0 mg | 0 | 0 | 0 − 0 = 0 |
| 10 mg | 3 | 0.5 | 3 − 0.5 = 2.5 |
| 20 mg | 6 | 2 | 6 − 2 = 4 |
| 40 mg | 7 | 4 | 7 − 4 = 3 |
The largest calculated net benefit is \(4\) utility units at 20 mg. Thus, among these four feasible doses, the net-benefit-maximizing choice is 20 mg. Although 40 mg has greater benefit than 20 mg, its additional harm and burden reduce its net benefit from \(4\) to \(3\) units. This conclusion depends on the stated utility scale, weights, population, and duration; changing those inputs can change the selected dose.
Worked Example: Finding the Lowest Dose Meeting a Response Target
Hypothetical response probabilities. For the same type of specified population and 12-week regimen, define a responder as an individual whose improvement at week 12 is at least 5 score points. Suppose the model-estimated proportions meeting that threshold are 0.30, 0.48, 0.66, and 0.72 at doses 0, 10, 20, and 40 mg. The decision target is that at least \(q=0.65\) of the population meet the individual response threshold.
At 10 mg, \(0.48<0.65\), so the target is not met. At 20 mg, \(0.66\geq0.65\), so it is met. The 40 mg dose also meets the target, but is not the lowest qualifying dose. The target-response choice is therefore 20 mg. This conclusion is about the estimated population proportion achieving at least 5 points of improvement, not about the population mean improvement.
Assumptions, Diagnostics, and Alternatives
Before interpreting any of these results, verify that the underlying predictions refer to the intended population, regimen, and duration. The outcome must be defined consistently across doses, and the comparator and response direction must be clear. If a model supplies the dose-response quantities, its predictions should be credible over the selected dose range; a precise calculation cannot repair a poorly supported extrapolation. The examples above assume that the hypothetical population summaries are available and suitable for the stated decisions. They do not establish that assumption for real data.
Useful checks include plotting estimated outcomes across dose, comparing predictions with observed data at tested doses, and examining uncertainty around the quantities used by the rule. Unexpected reversals in a response curve warrant investigation rather than automatic smoothing or monotonicity assumptions. Also check whether a dose that appears optimal is driven by a small difference that is uncertain, whether one population subgroup has a materially different response, and whether the result changes under plausible alternative thresholds or utility weights.
The principal alternatives reflect different priorities. A minimum-effective-dose rule seeks the lowest dose meeting a specified effect threshold. A target-response rule seeks the lowest dose reaching a specified population response proportion. A net-benefit rule trades benefit against harms and burden and can select a higher or lower dose than either threshold rule. A statistically significant difference from a comparator is another possible decision criterion, but it answers whether the data provide evidence against a null hypothesis under a testing procedure; it does not, by itself, define a clinically meaningful dose or account for harms.
Each approach has limitations. Threshold-based choices can shift when \(\delta\), \(r^*\), or \(q\) changes, so those values should be justified for the intended use. Net-benefit choices depend on how outcomes are placed on a common scale and on the chosen weights. All choices can be sensitive to estimation uncertainty, dose-grid restrictions, and population differences. A recommended dose is therefore conditional on its estimand and decision rule, not a universal property of the drug.
Common Mistake
Wrong reasoning: “The 40 mg dose has the largest mean response, so it must be the best dose.” This ignores whether the extra response is meaningful, whether the target is already met at a lower dose, and whether harm or treatment burden rises with dose. It may also confuse maximizing efficacy with maximizing net benefit.
Correction: State the estimand and rule first. If the objective is minimum effective dose, compare each dose with the prespecified effect threshold and select the smallest qualifying dose. If the objective is net benefit, compare the specified benefit-harm utility across feasible doses. If the objective is target attainment, compare each dose’s response probability with the target proportion. Report the population, regimen, duration, and uncertainty alongside the decision.
Check Your Understanding
For each question, identify the relevant estimand or calculation. Brief solutions follow.
- A study defines meaningful improvement as 3 points. Mean improvements at doses 5, 15, and 30 mg are 2.4, 3.2, and 3.5 points. Which is the minimum effective dose among these candidates?
- In one sentence, explain why a minimum effective dose and a net-benefit-maximizing dose need not be the same.
- A response requires at least 4 points of improvement. The response proportions at two doses are 0.58 and 0.71, and the target is 0.70. Which dose meets the target?
- What must be specified to interpret a dose-selection estimand beyond the numerical dose itself?
Solutions. (1) The 5 mg dose fails because \(2.4<3\); 15 mg qualifies because \(3.2\geq3\), so the minimum effective dose is 15 mg. (2) The minimum effective dose is the lowest dose meeting an effect threshold, whereas the net-benefit dose maximizes a specified balance of benefit, harm, and burden. (3) The dose with response proportion 0.71 meets the target because \(0.71\geq0.70\); the dose at 0.58 does not. (4) Specify the target population, regimen and schedule, treatment duration, outcome summary, comparator or target, feasible dose set, and decision rule.