From Model Output to a Defensible Recommendation
A dose recommendation is not just the dose with the largest estimated efficacy. It is a decision made using defined endpoints, a model for their joint behavior, and a rule that balances benefit against risk. A report should make each part visible: readers need to know what outcomes were measured, what population the estimates describe, how much uncertainty remains, and why the chosen dose meets the stated criteria.
The earlier material on joint dose models provides the modeling context. Here the emphasis is on reporting the model’s decision-relevant results accurately. The numbers in the examples are hypothetical; they illustrate reporting choices rather than results from an actual trial.
Define the Outcomes Before Reporting Estimates
For each endpoint, state what counts as an event, when it is assessed, and which participants are represented. For example, “efficacy” could mean a confirmed response by week 8, whereas “toxicity” could mean at least one dose-limiting toxicity (DLT) within 28 days after the first dose. Those are different outcomes with different windows; vague labels alone do not tell readers what was estimated.
Also report how the analysis handles treatment discontinuation, additional treatment, death, and outcomes that are missing or unevaluable. These choices determine the estimand—the treatment effect or outcome quantity the analysis targets. State whether the analysis population is defined by assignment or treatment received, and explain how the model uses incomplete observations. Do not imply that a missing outcome was observed to be a success or a failure unless that is the stated analysis rule.
For a simple illustration, let \(E=1\) indicate an efficacy response by week 8 and \(T=1\) indicate a DLT within 28 days. At dose \(d\), define \(p_E(d)=P(E=1\mid d)\), \(p_T(d)=P(T=1\mid d)\), and \(p_{E,\mathrm{noT}}(d)=P(E=1,T=0\mid d)\). The population, endpoint windows, and handling of intercurrent events must accompany these symbols in an actual report.
The distinction follows directly by partitioning responders into those with and without toxicity:
so
The independence calculation \(p_E(d)\{1-p_T(d)\}\) is justified only if efficacy and toxicity are independent at that dose under the model. A report should either provide a joint outcome estimate or clearly state the assumption used to derive it from marginal estimates.
State the Model and Decision Rule
A concise report identifies the model used to estimate dose-specific outcomes and the assumptions that matter for interpretation. In this setting, assumptions may include the dose-response form, the model for dependence between endpoints, and any covariate or historical-information structure. State the dose range and the population to which the model’s estimates apply. Explain how model fit and convergence were assessed and disclose material concerns; a polished estimate table cannot compensate for an unstable or poorly fitting model.
A recommendation also requires an explicit decision rule. One transparent rule is to retain doses whose posterior probability of meeting a toxicity limit exceeds a pre-specified threshold, then require sufficient posterior support for an efficacy target. Among doses meeting both conditions, the protocol can specify how to choose—for example, by maximizing expected utility or the probability of joint success. Give the thresholds and the order of operations. A rule chosen after examining the results should be identified as exploratory, not presented as pre-specified.
Worked Example: Applying a Joint Safety-and-Efficacy Rule
Suppose a hypothetical trial compares three doses. The efficacy endpoint is confirmed response by week 8; the risk endpoint is at least one DLT within 28 days. The report targets patients eligible for the trial, uses a joint Bayesian dose model, and gives posterior means and central 90% credible intervals (CrIs). Assume the model has converged, the planned checks found no material lack of fit, and the outcome windows and missing-data rules were applied as specified.
| Dose | Response probability, posterior mean (90% CrI) | DLT probability, posterior mean (90% CrI) | \(P(p_T(d)\leq 0.25\mid\mathrm{data})\) | \(P(p_E(d)\geq 0.35\mid\mathrm{data})\) | \(P(p_E(d)\geq0.35,\ p_T(d)\leq0.25\mid\mathrm{data})\) |
|---|---|---|---|---|---|
| 1 | 0.22 (0.12, 0.34) | 0.08 (0.03, 0.16) | 0.97 | 0.08 | 0.05 |
| 2 | 0.40 (0.27, 0.54) | 0.17 (0.09, 0.27) | 0.86 | 0.72 | 0.68 |
| 3 | 0.52 (0.37, 0.66) | 0.31 (0.21, 0.43) | 0.38 | 0.94 | 0.35 |
For this hypothetical decision, the rule requires at least 0.80 posterior probability that the DLT rate is no greater than 0.25, and at least 0.70 posterior probability that the response rate is at least 0.35. Dose 1 meets the safety requirement (\(0.97\)) but not the efficacy requirement (\(0.08\)). Dose 2 meets both (\(0.86\) and \(0.72\)). Dose 3 meets the efficacy requirement (\(0.94\)) but not the safety requirement (\(0.38\)). The rule therefore recommends dose 2.
The last column reports the posterior probability that both parameter criteria hold together. It is not obtained by multiplying the two preceding probabilities unless those dose-specific posterior events are independent. Here the joint probabilities are supplied by the joint model. State the recommendation as “dose 2 meets the stated criteria,” not as proof that its true rates satisfy the limits.
Report Uncertainty and Sensitivity
Point estimates alone can make a dose appear more certain than the data warrant. Report uncertainty for both endpoints at each dose, and give the uncertainty summaries a precise name: a credible interval summarizes posterior uncertainty under a Bayesian model; a confidence interval has a different repeated-sampling interpretation. Report the interval level and, where decision criteria are posterior probabilities, report those probabilities too. A marginal interval for each endpoint does not by itself describe uncertainty about a joint event.
Sensitivity analysis asks whether reasonable alternatives change the conclusion. Relevant alternatives include a different dependence assumption, dose-response structure, prior specification, missing-outcome assumption, endpoint threshold, safety margin, or benefit-risk weight. Identify each alternative and show which estimates or decision probabilities change. If the recommendation changes, report that directly. Sensitivity analysis is informative when it distinguishes a stable recommendation from one that depends on a particular modeling or value judgment.
Worked Example: Showing Dependence and Utility Sensitivity
Using the posterior mean response and DLT probabilities in the table above, suppose a secondary summary defines expected net benefit at dose \(d\) as \(U(d)=p_E(d)-w p_T(d)\), where one response has benefit weight 1 and each DLT has cost \(w\). This is a simplified, explicitly chosen utility summary, not a substitute for the primary decision rule.
With equal weights (\(w=1\)), the dose 2 calculation is \(U(2)=0.40-1(0.17)=0.23\). The corresponding values are \(U(1)=0.22-0.08=0.14\) and \(U(3)=0.52-0.31=0.21\). Dose 2 has the largest score.
If a DLT is assigned three times the cost of a response (\(w=3\)), the calculations become \(U(1)=0.22-3(0.08)=-0.02\), \(U(2)=0.40-3(0.17)=-0.11\), and \(U(3)=0.52-3(0.31)=-0.41\). Dose 1 now has the largest score. This alternative weighting changes the ranking; it does not establish that dose 1 satisfies the primary efficacy criterion. A report should explain the clinical meaning of the weights and keep this sensitivity result distinct from the primary rule.
Dependence assumptions can also alter the estimated probability of joint success while leaving marginal means unchanged. If a report uses independence as a sensitivity scenario, it should calculate the joint probability under that scenario, compare it with the primary joint model, and label the assumption. For example, at dose 2 the independence plug-in calculation using marginal posterior means is \(0.40(1-0.17)=0.332\). This is an approximation based on those means, not the posterior probability that both rate thresholds are satisfied.
What a Complete Dose-Level Report Looks Like
The table is only one part of the report. A reader should be able to trace the recommendation from the observed endpoints through model summaries to the decision rule. When useful, include the four observed joint-outcome categories—response with DLT, response without DLT, no response with DLT, and neither outcome—by dose. These counts make dependence visible and help distinguish observed data from model-based estimates.
Worked Example: Reporting a Dose With Observed Joint Outcomes
In a separate hypothetical dose group, 40 participants have complete assessment of both endpoints under the stated windows and rules. The observed counts are: 5 with both response and DLT, 11 with response and no DLT, 3 with no response and DLT, and 21 with neither.
The response count is \(5+11=16\), so the crude response proportion is \(16/40=0.400\), or 40.0%. The DLT count is \(5+3=8\), giving \(8/40=0.200\), or 20.0%. The observed proportion with response and no DLT is \(11/40=0.275\), or 27.5%.
The independence product of the two crude marginal complements would be \(0.400(1-0.200)=0.320\), not the observed joint proportion of 0.275. In these data the observed response-with-DLT count is \(5/40=0.125\); accordingly, \(0.400-0.125=0.275\). In a report, label these as crude proportions and keep them separate from model-based estimates, which may borrow information across doses and include uncertainty.
A useful dose-level report therefore includes the denominators and observed outcome counts, the model-based efficacy and toxicity estimates with intervals, any joint outcome estimate, and the decision probabilities relevant to the rule. It should identify which doses are eligible, which criterion excludes a dose, and whether any dose meets all requirements. If none qualifies, say so; do not force a recommendation merely because one dose ranks highest.
Diagnostics, Comparisons, and Limits
Before interpreting the recommendation, check that each endpoint’s denominator and assessment window are clear, that outcome classifications are consistent, and that the estimates come from a converged, adequately checked model. Investigate sparse dose groups, unexpectedly wide intervals, sensitivity to priors or dependence assumptions, and influential observations. If model diagnostics indicate lack of fit or unstable estimation, describe the issue and qualify the recommendation rather than reporting model-based precision without reservation.
Different summaries answer different questions. Marginal estimates answer how often efficacy or toxicity occurs at a dose; a joint outcome probability answers how often a specified combination occurs. A safety-constrained rule makes a toxicity limit explicit before ranking eligible doses. A utility rule makes trade-offs explicit through weights, but its ranking depends on those weights. Reporting more than one summary can be useful, provided the primary decision rule is clearly distinguished from supplementary analyses.
No summary removes the need for judgment about clinical relevance. A threshold may be uncertain, a trial population may not represent all patients, and a model may extrapolate beyond well-supported doses. Posterior probabilities are conditional on the model, prior, and data; they are not guarantees about future patients. A transparent report states these limits and avoids turning a model-based recommendation into a claim of causal certainty or universal optimality.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- At a dose, the response probability is 0.45 and the DLT probability is 0.20. Is the probability of response without DLT necessarily \(0.45(1-0.20)\)? Explain.
- A dose has posterior probability 0.90 of meeting the toxicity limit and 0.60 of meeting the efficacy target. The rule requires at least 0.80 and 0.70, respectively. Does it qualify?
- Why should a report distinguish a model-based posterior mean from a crude proportion observed in a dose group?
- If changing the DLT cost weight changes the dose ranking, what should the report say?
Solutions. (1) No. The product \(0.45(0.80)=0.36\) follows only under independence; without that assumption, the joint probability requires information about the dependence between outcomes. (2) No. It meets the toxicity criterion but fails the efficacy criterion because \(0.60<0.70\). (3) A crude proportion summarizes that group’s observed outcomes, while a model-based posterior mean is conditional on the fitted model and may reflect information sharing, assumptions, and uncertainty. (4) Report the alternative weight and resulting ranking, explain the clinical interpretation of the weight, and identify whether the primary rule still supports its original recommendation.