From Two Targets to One Dose Decision
A dose can have an attractive efficacy prediction and still be unsuitable if its toxicity prediction is too high. The decision is more demanding when both predictions are uncertain: the relevant question is not whether the average efficacy and average toxicity look acceptable, but how often both targets are met together. This tutorial works through that calculation using hypothetical joint simulation draws, then compares a probability-constrained choice with a worst-case choice.
The example treats efficacy and toxicity as two dose-specific model predictions, expressed in units chosen for the hypothetical application. Simulation draws represent uncertainty in those predictions under a specified model and analysis. The resulting proportions describe target attainment across those draws; they are not automatically clinical response rates in a new population.
Pairing matters: \(E_s(d)\) and \(T_s(d)\) must come from the same draw \(s\). That pairing preserves the simulated dependence between the two predictions. Combining an efficacy draw from one simulated parameter set with a toxicity draw from another would instead impose a different, usually unjustified, dependence structure.
Hypothetical Joint Predictions
Suppose four candidate doses are under consideration. Efficacy is measured in hypothetical benefit units, and toxicity in hypothetical burden units. The decision targets are efficacy of at least 60 benefit units and toxicity no greater than 20 burden units. Eight paired simulation draws are available at each dose. The values below are illustrative predictions, not results from an actual study.
| Draw | 10 mg: E | 10 mg: T | 20 mg: E | 20 mg: T | 30 mg: E | 30 mg: T | 40 mg: E | 40 mg: T |
|---|---|---|---|---|---|---|---|---|
| 1 | 60 | 8 | 60 | 10 | 62 | 12 | 64 | 17 |
| 2 | 61 | 9 | 62 | 12 | 64 | 14 | 66 | 19 |
| 3 | 62 | 10 | 64 | 14 | 66 | 16 | 68 | 21 |
| 4 | 63 | 11 | 66 | 16 | 68 | 18 | 70 | 23 |
| 5 | 64 | 12 | 68 | 18 | 70 | 20 | 72 | 25 |
| 6 | 65 | 13 | 70 | 20 | 72 | 22 | 74 | 27 |
| 7 | 66 | 14 | 72 | 22 | 74 | 24 | 76 | 29 |
| 8 | 67 | 15 | 58 | 24 | 76 | 26 | 78 | 31 |
Worked Example: Counting Joint Target Attainment
At 20 mg, efficacy must be at least 60 benefit units and toxicity must be at most 20 burden units. Draws 1 through 7 meet the efficacy threshold; draw 8 does not, because its efficacy prediction is 58. Draws 1 through 6 meet the toxicity ceiling; draws 7 and 8 do not. Therefore only draws 1 through 6 meet both targets.
The Monte Carlo estimate is
Thus, 75% of these eight simulated joint predictions meet both targets at 20 mg. This is a simulation-based estimate under the stated model and draw-generation procedure, not a guarantee that 75% of future patients will meet the targets.
For comparison, at 10 mg all eight efficacy predictions are at least 60 and all eight toxicity predictions are at most 20, giving \(8/8=1.00\). At 30 mg, all eight efficacy predictions pass but only the first five toxicity predictions pass, giving \(5/8=0.625\). At 40 mg, efficacy passes on every draw but toxicity passes only on draws 1 and 2, giving \(2/8=0.25\).
Probability-Constrained Choice
A probability-constrained rule sets a required minimum joint-attainment probability before comparing doses. Here, suppose the minimum is \(q=0.75\). Among doses whose estimated joint target-attainment probability is at least \(q\), choose the dose with the highest mean predicted efficacy across draws. This objective and constraint are part of the decision rule: another application could use a different efficacy summary or include additional decision criteria.
Worked Example: Applying the Probability Constraint
The mean efficacy at each dose is calculated by adding its eight efficacy predictions and dividing by 8. For example, at 20 mg:
The same calculation gives a mean efficacy of 63.5 benefit units at 10 mg, 69.0 at 30 mg, and 71.0 at 40 mg. Combining these values with the joint-attainment counts gives:
| Dose | Joint successes | Estimated joint probability | Mean efficacy | Eligible at \(q=0.75\)? |
|---|---|---|---|---|
| 10 mg | 8 of 8 | 1.000 | 63.5 benefit units | Yes |
| 20 mg | 6 of 8 | 0.750 | 65.0 benefit units | Yes |
| 30 mg | 5 of 8 | 0.625 | 69.0 benefit units | No |
| 40 mg | 2 of 8 | 0.250 | 71.0 benefit units | No |
Both 10 mg and 20 mg satisfy the probability constraint. Of those eligible doses, 20 mg has the larger mean efficacy, 65.0 rather than 63.5 benefit units. The probability-constrained choice is therefore 20 mg. The higher mean efficacies at 30 mg and 40 mg do not make those doses eligible: their estimated probabilities of meeting both targets are below 0.75.
Comparison With a Worst-Case Choice
A worst-case rule is more demanding than a probability constraint. For this example, define the simulated scenario set as the eight listed draws. A dose is robustly feasible only if every scenario meets both targets. Among robustly feasible doses, choose the one with the largest minimum efficacy across scenarios. This rule treats the listed draws as a finite set of scenarios to guard against, rather than allowing a specified fraction of them to fail.
Worked Example: Finding the Worst-Case Choice
At 10 mg, the smallest efficacy prediction is 60 benefit units and the largest toxicity prediction is 15 burden units. Thus, even the least favorable of the eight scenarios meets both targets: \(60\ge60\) and \(15\le20\). The dose is robustly feasible, and its worst-case efficacy is 60 benefit units.
At 20 mg, draw 8 has efficacy of 58 benefit units and toxicity of 24 burden units, so it fails both targets. At 30 mg, toxicity reaches 26 burden units, and at 40 mg it reaches 31 burden units; both exceed the ceiling. Neither dose is robustly feasible. Therefore, in this finite scenario set, the worst-case choice is 10 mg.
The two rules answer different questions. The probability-constrained rule chooses 20 mg because six of eight paired predictions pass both targets, exactly meeting the 0.75 requirement, and its mean efficacy exceeds that of 10 mg. The worst-case rule chooses 10 mg because it requires all eight scenarios to pass. Neither answer is universally preferable: the difference reflects the decision-maker’s tolerance for target failure and the meaning assigned to the scenarios.
Why the Pairing and Assumptions Matter
The probability calculation relies on several conditions. First, both endpoint predictions must be generated for every dose and the same simulation draws must be paired across efficacy and toxicity. Second, the efficacy and toxicity definitions, units, thresholds, and candidate doses must be specified before comparing results. Third, the simulations must represent the uncertainty relevant to the decision and be generated from a sufficiently reliable model. Finally, the Monte Carlo proportion is an approximation to the probability implied by that simulation distribution; with only eight draws it is notably coarse.
Worked Example: Why Separate Marginal Probabilities Are Not Enough
At 20 mg, efficacy passes in 7 of 8 draws, while toxicity passes in 6 of 8. Their separate estimated probabilities are therefore \(7/8=0.875\) and \(6/8=0.75\). Multiplying these gives approximately 0.656, but that product would represent joint attainment only under independence of the two pass/fail indicators. The paired draws do not establish that independence.
In the listed data, efficacy fails only in draw 8, which is also a toxicity failure. The six joint successes are therefore the intersection of the two sets of passing draws, not the product of their marginal proportions. The actual paired estimate is \(6/8=0.75\). If the same seven efficacy successes and six toxicity successes were paired differently, the overlap could change even though both marginal proportions stayed the same. The correct calculation counts both conditions on each joint draw.
The choice of probability threshold is also consequential. In the example, reducing the requirement from 0.75 to 0.625 would make 30 mg eligible; among the eligible doses it has the greatest mean efficacy, 69.0 benefit units. Raising the requirement above 0.75 would leave only 10 mg eligible at these simulated proportions. The threshold is a decision requirement, not a number selected after inspecting the results simply to obtain a preferred dose.
Diagnostics, Failure Modes, and Limitations
Before interpreting a recommendation, check that the calculation is working as intended:
- Verify the indicators. For each draw, confirm that efficacy passes when it is at least the target and toxicity passes when it is no greater than the ceiling. Boundary values count as passing under these definitions.
- Check the pairing. Confirm that efficacy and toxicity values labelled with the same draw index come from the same joint simulation. Independently shuffled or separately simulated endpoints can distort their joint probability.
- Inspect the number of draws. With eight draws, probabilities can change only in increments of \(1/8=0.125\). A value of 0.75 here is six successes, not a finely precise probability estimate. More simulation draws can reduce Monte Carlo error, although they cannot repair a misspecified or poorly calibrated model.
- Review the dose pattern. A non-monotone joint-attainment pattern can be plausible when efficacy and toxicity respond differently to dose. Unexpected jumps may also signal a coding, indexing, or simulation problem. Recheck the underlying predictions rather than imposing monotonicity without justification.
- Separate model uncertainty from patient variability. If draws describe uncertainty in model parameters or population predictions, the estimated proportion summarizes uncertainty of that kind. It does not include every source of individual variability unless the simulation explicitly represents it.
A finite-simulation worst case is not automatically a worst case over every plausible parameter value or future outcome. Adding draws can reveal more extreme scenarios and change which dose passes the all-draw requirement. A robust decision therefore needs a defined uncertainty set and a clear account of what its scenarios represent; the minimum of an arbitrary, small simulation sample should not be presented as a general guarantee.
The probability-constrained approach is less conservative in the specific sense that it permits some simulated scenarios to miss one or both targets. It is sensitive to the chosen probability threshold and the predictive distribution. The worst-case approach avoids failures within its stated scenario set, but can be overly restrictive and is highly sensitive to extreme scenarios. Both rules depend on the quality and relevance of the joint predictions; neither removes uncertainty or turns a model-based recommendation into a causal or clinical guarantee.
Check Your Understanding
Use the example’s thresholds and simulated predictions. Solutions are included so the calculations can be checked.
- At 30 mg, how many draws meet both targets, and what is the estimated joint probability?
- Why is 20 mg selected by the probability-constrained rule at \(q=0.75\), rather than 30 mg?
- At 20 mg, what are the separate efficacy-pass and toxicity-pass proportions? Why is their product not the joint estimate from the table?
- If the probability requirement were \(q=0.625\), which dose would the rule select, using mean efficacy as the objective?
- What does the worst-case choice of 10 mg guarantee in this example, and what does it not guarantee?
Solutions. (1) Five draws meet both targets, so the estimate is \(5/8=0.625\). (2) At \(q=0.75\), 30 mg is ineligible because \(0.625<0.75\); 20 mg is eligible with \(0.75\), and has greater mean efficacy than the other eligible dose, 10 mg. (3) Efficacy passes in 7 of 8 draws and toxicity in 6 of 8. Their product assumes independence; the paired joint count is six successes, or \(6/8=0.75\). (4) At \(q=0.625\), 30 mg becomes eligible and has the largest mean efficacy among eligible doses, so the rule selects 30 mg. (5) Within the eight listed scenarios, every prediction at 10 mg meets both thresholds. This does not guarantee target attainment for unrepresented scenarios, future patients, or a different predictive model.