What a Dose Recommendation Needs to Communicate
A dose recommendation is not just a number. A reader needs to know which rule selected it, what outcomes the model predicts at that dose, how uncertain the selection is, and which assumptions matter. Without that information, the same stated dose could mean “best predicted average benefit,” “the only dose meeting safety constraints,” or “the dose selected most often across model fits.” Those are different decision claims.
This tutorial focuses on reporting the result of an uncertainty-aware dose decision, rather than deriving a pharmacometric model or choosing among all possible decision methods. A report should identify the analysis population and endpoint definitions, state the decision rule, give outcome probabilities in context, quantify uncertainty in the selected dose, and show whether reasonable alternatives change the recommendation. The summary should be concise, but it must not conceal a failed constraint or a fragile result.
Define the Rule Before Reporting Its Result
Let \(d\) denote a candidate dose and let the model estimate, for the intended population, the probability of achieving a prespecified efficacy target and the probability of a prespecified toxicity event. Write these predictive probabilities as \(p_E(d)\) and \(p_T(d)\), respectively. A rule might require \(p_E(d)\geq 0.80\) and \(p_T(d)\leq 0.10\), then select the eligible dose with the greatest expected utility. The endpoints, time horizon, population, and source of uncertainty must be defined in the analysis; a report should not leave “efficacy probability” or “toxicity probability” ambiguous.
Here \(\mathcal{D}\) is the candidate dose set and \(U(d)\) is the stated utility criterion. If no dose is eligible, the rule has not produced a qualifying recommendation; it is not appropriate to silently choose the “least bad” dose and describe it as meeting the criteria. If several doses tie under the criterion, the report should state the tie-breaking rule, such as choosing the lower dose.
Predicted probabilities and uncertainty about the selected dose answer different questions. For example, \(p_E(75)=0.86\) means the model predicts an 86% probability of the defined efficacy event at 75 mg for the specified population, under the stated model and assumptions. It does not mean there is an 86% probability that 75 mg is the correct dose. Uncertainty about the selection is summarized by repeatedly applying the decision rule across uncertainty draws.
Worked Example: State the Rule and Its Predictions
Worked Example: State the Rule and Its Predictions
Hypothetical data. Suppose a model is evaluated for a target population at three candidate doses. Efficacy is the probability of reaching a prespecified clinical response by week 12; toxicity is the probability of a prespecified dose-limiting event by week 12. The analysis uses posterior predictive probabilities, and expected utility is reported on a prespecified relative scale.
| Dose (mg) | Predicted efficacy probability | Predicted toxicity probability | Expected utility |
|---|---|---|---|
| 50 | 0.81 | 0.03 | 0.54 |
| 75 | 0.86 | 0.07 | 0.62 |
| 100 | 0.91 | 0.16 | 0.59 |
The rule requires efficacy probability at least 0.80 and toxicity probability at most 0.10, then maximizes expected utility among eligible doses. At 50 mg, both constraints hold: \(0.81\geq0.80\) and \(0.03\leq0.10\). At 75 mg, both hold: \(0.86\geq0.80\) and \(0.07\leq0.10\). At 100 mg, the toxicity constraint fails because \(0.16>0.10\), even though its efficacy probability is highest.
The eligible doses are therefore 50 and 75 mg. Their utilities are 0.54 and 0.62, so the rule selects \(d^*=75\) mg. A defensible report gives the rejected 100 mg dose’s failed safety constraint as well as the selected dose; otherwise, the trade-off can be obscured.
In context, the model predicts an 86% probability of the defined week-12 response and a 7% probability of the defined toxicity event at 75 mg in the target population. These are separate marginal predictions. They do not provide the probability that both outcomes occur, nor do they imply independence between efficacy and toxicity.
Quantify Uncertainty in the Selected Dose
To summarize uncertainty in the decision, draw uncertain model inputs from the analysis’ uncertainty distribution, calculate the relevant predictions for each draw, and reapply the same decision rule. Denote the dose selected on draw \(b\) by \(d^{*(b)}\). The empirical distribution of these selected doses describes how the decision changes across the represented uncertainty. Its quantiles can be reported as an uncertainty interval, alongside selection frequencies where useful.
The superscript \(b\) indicates that predictions, eligibility, and utility are evaluated for uncertainty draw \(b\). If an uncertainty draw has no eligible dose, the analysis must specify how that event is handled and report how often it occurs. It should not be hidden by excluding those draws without explanation.
A short justification for this procedure follows from the definition of the target quantity: the uncertain object is the output of the decision rule, not merely one model parameter. Applying the rule to each draw transforms uncertainty in model inputs into uncertainty in the selected dose. Taking quantiles of those outputs therefore summarizes the selected-dose distribution. In contrast, taking an interval for a parameter such as an efficacy slope does not directly give an interval for the dose chosen by a nonlinear rule with thresholds.
Worked Example: Report an Uncertainty Interval for the Selected Dose
Hypothetical data. Suppose 1,000 posterior draws are available. On every draw, the same efficacy and toxicity thresholds and utility rule from the preceding example are applied. The resulting selection counts are 180 draws selecting 50 mg, 620 selecting 75 mg, and 200 selecting 100 mg. In this uncertainty analysis, 100 mg can be selected on a draw when that draw’s toxicity prediction satisfies the stated constraint.
The selection frequencies are calculated by dividing each count by 1,000:
Thus, 75 mg is selected most often, in 62.0% of the draws. Ordering the selected doses gives 180 values at 50 mg, then 620 at 75 mg, then 200 at 100 mg. The 5th percentile is 50 mg because the cumulative proportion at 50 mg is 18.0%; the 95th percentile is 100 mg because the cumulative proportion reaches only 80.0% at 75 mg. The reported 90% uncertainty interval is therefore 50–100 mg.
This interval describes variation in the dose selected under the specified rule across posterior draws. It is not a 90% probability statement about the unknown dose being between 50 and 100 mg, and it does not assert that every dose in the interval satisfies the constraints. Because the outcome is discrete, the interval also need not contain every possible selected dose with equal frequency. Reporting the 62.0% selection frequency for 75 mg helps make that limitation visible.
Worked Example: Show How a Changed Threshold Alters the Decision
Worked Example: Show How a Changed Threshold Alters the Decision
Hypothetical data. Return to the predicted probabilities and utilities in the first example. Consider two prespecified sensitivity analyses that change one efficacy threshold at a time while retaining the toxicity ceiling of 0.10 and the utility criterion.
First, raise the minimum efficacy probability from 0.80 to 0.85. At 50 mg, \(0.81<0.85\), so it is no longer eligible. At 75 mg, \(0.86\geq0.85\) and \(0.07\leq0.10\), so it remains eligible. The 100 mg dose still fails toxicity. The selected dose remains 75 mg.
Second, retain the efficacy threshold of 0.80 but lower the maximum toxicity probability from 0.10 to 0.05. At 50 mg, \(0.81\geq0.80\) and \(0.03\leq0.05\), so it is eligible. At 75 mg, \(0.07>0.05\), so it is not eligible; 100 mg also fails. The selected dose changes to 50 mg. If both the higher efficacy threshold of 0.85 and the lower toxicity ceiling of 0.05 are applied together, none of the three doses meets both constraints: 50 mg fails efficacy, while 75 and 100 mg fail toxicity. That result should be reported as no qualifying dose under that combined rule.
A concise interpretation is that the recommendation is unchanged under the stricter efficacy threshold alone, but moves to 50 mg under the stricter toxicity ceiling; under both changes jointly, no candidate dose qualifies. This conveys more than saying generally that “results were robust,” because it identifies which assumption changes the decision and in what direction.
How to Write a Clinically Interpretable Summary
A useful report can be short while preserving the decision’s key information. One practical sequence is: define the population and time horizon; state the endpoints and rule; name the selected dose and its predicted probabilities; give the selected-dose uncertainty summary; then describe material sensitivity results and limitations. State dose units and probability units clearly. If a model-averaged prediction is reported, identify it as such; do not make a single-model probability sound like it incorporates structural uncertainty.
Name the target population, candidate dose range, outcome definitions, and time horizon.
Give eligibility thresholds, the optimization criterion, and any tie-breaking or no-eligible-dose rule.
Give the selected dose, predicted outcome probabilities at that dose, uncertainty interval, and selection frequency or other useful distribution summary.
Summarize changes that alter the decision, relevant model or population assumptions, and important diagnostics or limitations.
For the hypothetical examples, a compact statement could read: “For the specified target population and week-12 endpoints, the rule selected 75 mg: maximize expected utility subject to predicted response probability at least 0.80 and toxicity probability at most 0.10. Predicted probabilities at 75 mg were 0.86 for response and 0.07 for toxicity. Across 1,000 posterior draws, 75 mg was selected in 62.0% of draws; the 5th–95th percentile interval for selected dose was 50–100 mg. The recommendation remained 75 mg when the response threshold was raised to 0.85, changed to 50 mg when the toxicity ceiling was lowered to 0.05, and no dose qualified when both changes were applied.”
That summary is interpretable only if its assumptions are available. At minimum, state the model and uncertainty framework used, how the target population was represented, how missing or unavailable outcomes were handled if relevant, and whether uncertainty intervals include parameter uncertainty, model uncertainty, or both. A report may refer to earlier material on model averaging or structural model uncertainty, but it must say what was actually included in this recommendation.
Diagnostics, Alternatives, and Limitations
Before finalizing a summary, check that the rule has been applied consistently to every uncertainty draw; that the reported probabilities correspond to the stated population, endpoints, and time horizon; and that the units and dose candidates are correct. Inspect whether some draws have no eligible dose, whether selection frequencies are concentrated or diffuse, and whether small changes to clinically plausible thresholds reverse the recommendation. These checks help distinguish a stable decision from a single point estimate that hides substantial variation.
| Reporting approach | What it communicates | Important caveat |
|---|---|---|
| One selected dose with predicted outcome probabilities | The rule’s central recommendation and modeled outcomes at that dose | Can conceal instability in dose selection |
| Selected-dose uncertainty interval | The range of selected values across the specified uncertainty draws | Does not guarantee that all interval doses meet the rule |
| Selection frequencies across candidate doses | How often the rule selects each candidate under uncertainty | Depends on the uncertainty distribution and the rule |
| Sensitivity analysis under alternative rules | Whether clinically meaningful assumptions change the decision | Does not establish which alternative is clinically correct |
These summaries are complementary, not interchangeable. A narrow interval can coexist with a low selection frequency for the reported point dose if the distribution is concentrated on neighboring doses; a wide interval can coexist with one dose selected most often. Sensitivity analysis answers a different question again: it shows how conclusions change under specified alternatives, not the probability that one alternative is true.
A further limitation is that model-based probabilities are conditional on the model, data, endpoint definitions, and target population assumptions. An uncertainty interval that varies parameter estimates but holds model structure fixed does not represent structural model uncertainty. Nor does it automatically account for uncertainty in clinical targets, population composition, or implementation. Name the uncertainty sources included and avoid implying that the reported interval covers sources that were not analyzed.
Common Reporting Pitfalls
- Reporting only the dose. “Recommend 75 mg” is incomplete without the rule, endpoint probabilities, and uncertainty summary. Add those elements so the reader can see why the dose was selected.
- Calling a selection interval a safety range. The uncertainty interval describes variation in selected doses; it does not establish that every dose inside it is safe or effective. Report predicted outcomes and eligibility separately.
- Omitting failed constraints. A dose with high predicted efficacy may still be ineligible because it exceeds a toxicity ceiling. State which constraint rules out such candidates.
- Using “robust” without a comparison. A recommendation is not demonstrated to be robust merely because one alternative was examined. Name the changed assumption, the resulting dose, and whether any dose remains eligible.
- Overstating a predictive probability. A model-predicted 86% response probability is not an 86% probability that the recommendation is correct. Keep outcome prediction and decision uncertainty distinct.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A rule requires efficacy probability at least 0.80 and toxicity probability at most 0.10. At a dose, the predictions are 0.88 and 0.12. Is the dose eligible? Which constraint determines the answer?
- Across 500 uncertainty draws, a dose is selected 350 times. Calculate its selection frequency and interpret it correctly.
- What does a 5th–95th percentile interval for selected dose summarize, and what does it not guarantee?
- A stricter toxicity ceiling changes the recommended dose. What information should a sensitivity summary include?
Solutions. (1) No. Although \(0.88\geq0.80\), the toxicity probability fails the ceiling because \(0.12>0.10\). (2) The frequency is \(350/500=0.700\), or 70.0%. Under the specified uncertainty draws and decision rule, that dose was selected on 70.0% of draws; this is not the probability that the dose is correct. (3) It summarizes the central 90% of selected-dose outcomes across the represented uncertainty draws, using the stated quantile convention. It does not guarantee that every dose in the interval satisfies efficacy or toxicity constraints. (4) State the original and changed ceiling, the selected dose under each rule, the predictions or eligibility changes that explain the shift, and whether any qualifying dose remains under the alternative.