Why Compare Each Dose With Its Scenario-Specific Best?
A dose can perform reasonably well across several plausible parameter values without being the dose that maximizes utility under any one of them. Minimax regret makes this trade-off explicit: for each parameter value, it asks how much utility would be lost by choosing a particular dose instead of the best feasible dose for that same value. It then selects the dose whose largest such loss is smallest.
This criterion is useful when a decision maker wants protection against a large avoidable loss but does not want to choose solely for the worst absolute outcome. It complements robust dose selection across plausible parameters: the two rules answer different questions. A robust maximin rule compares the lowest utility of each dose; minimax regret compares each dose with the best achievable utility in each scenario.
For a finite list of plausible scenarios, the supremum is simply the largest regret in that list. The word “best” always refers to the same feasible dose set \(D\) for every scenario. If a dose is excluded by a safety constraint, it cannot be used as the benchmark \(U^*(\theta)\). Utilities must also be comparable across scenarios: regret is a difference on the utility scale, not a difference between unscaled outcomes with unrelated units.
The calculation is conditional on the specified model, utility function, feasible doses, and plausible parameter set. It does not assign probabilities to the scenarios. In particular, a parameter value’s inclusion in \(\Theta\) means it is being considered plausible for this analysis, not that it is equally likely as every other value.
Calculating Regret for Candidate Doses
For each scenario, first find the largest utility among all feasible doses. Subtract each dose’s utility from that scenario-specific maximum to obtain its regret. Finally, take the largest regret for each dose and choose the dose with the smallest maximum. A tie means that more than one dose minimizes the criterion; additional clinical or operational considerations may then distinguish them.
Worked Example: A Discrete Dose Table
Suppose three feasible doses have hypothetical utilities, in utility points, under three plausible parameter scenarios. Higher utility is preferred.
| Scenario | 25 mg | 50 mg | 75 mg | Best utility |
|---|---|---|---|---|
| Low-response parameter | 8 | 7 | 2 | 8 |
| Intermediate parameter | 6 | 10 | 8 | 10 |
| High-response parameter | 3 | 9 | 12 | 12 |
For the low-response scenario, the best utility is 8 points, so the regrets at 25, 50, and 75 mg are \(8-8=0\), \(8-7=1\), and \(8-2=6\) points. For the intermediate scenario, they are \(10-6=4\), \(10-10=0\), and \(10-8=2\) points. For the high-response scenario, they are \(12-3=9\), \(12-9=3\), and \(12-12=0\) points.
The maximum regret for 25 mg is \(\max(0,4,9)=9\) points; for 50 mg it is \(\max(1,0,3)=3\) points; and for 75 mg it is \(\max(6,2,0)=6\) points. Thus 50 mg is the minimax-regret dose, with worst-case regret of 3 utility points. It does not maximize utility in every scenario: it gives up 1 point relative to 25 mg in the low-response scenario and 3 points relative to 75 mg in the high-response scenario. Its appeal is that its largest shortfall from a scenario-specific best is smaller than that of either alternative.
Why the Rule Balances Scenario-Specific Choices
The definition yields a direct justification. For every \(d\) and \(\theta\), \(U^*(\theta)\) is at least \(U(d,\theta)\), so regret cannot be negative. For each fixed scenario, subtracting utility from the scenario best expresses the cost of not knowing which dose would have been optimal if that parameter value were known. Taking the maximum over \(\Theta\) protects against the largest such shortfall; minimizing that maximum chooses the dose with the best worst-case shortfall.
With two scenarios and two candidate actions, a minimax-regret solution often equalizes the two regrets. The reason is that if one regret is strictly larger, a small move toward the action that performs better in that scenario may reduce the larger regret while keeping the other regret below it. This equalization is not guaranteed for discrete choices, irregular utility functions, or more scenarios, so the maximum must be calculated rather than assumed.
Worked Example: Net Utility With Efficacy and Toxicity
Consider three hypothetical dose choices. For illustration, define utility as efficacy benefit minus toxicity cost, with both outcomes already expressed in the same utility-point scale. Under a low-response parameter, benefits are 6, 8, and 9 points and toxicity costs are 1, 3, and 6 points at 20, 40, and 60 mg. Under a high-response parameter, benefits are 4, 9, and 13 points and costs are 1, 3, and 6 points.
The utilities \(U=\text{benefit}-\text{cost}\) are:
In the low-response scenario, the best utility is 5 points, giving regrets \((0,0,2)\). In the high-response scenario, the best is 7 points, giving regrets \((4,1,0)\). Therefore the maximum regrets at 20, 40, and 60 mg are 4, 1, and 2 points, respectively. The minimax-regret choice is 40 mg, whose largest shortfall is 1 point.
For comparison, the minimum utilities of the three doses across the two scenarios are 3, 5, and 3 points. A maximin-utility rule would choose 40 mg in this example too, but that agreement is not automatic: maximin focuses on the dose’s lowest utility, whereas minimax regret focuses on how far it falls below the best utility in each scenario. The calculations also depend on the chosen utility definition; subtracting efficacy and toxicity is meaningful here only because their effects were placed on a common scale.
A Continuous Example and a Short Derivation
A continuous dose set makes the structure of the criterion particularly clear. Suppose \(D=[20,80]\) mg, \(\theta\in[30,70]\) mg represents the dose that would be ideal if known, and utility is \(U(d,\theta)=-0.01(d-\theta)^2\) points. Assume this quadratic utility is an adequate description over the stated dose and parameter ranges. For each \(\theta\), the feasible dose \(d=\theta\) attains utility zero, so \(U^*(\theta)=0\). Hence regret is \(R(d,\theta)=0.01(d-\theta)^2\) points.
For a fixed dose, the greatest squared distance from \(\theta\in[30,70]\) occurs at an endpoint. Therefore:
The smallest possible maximum distance occurs when the distances to the two endpoints are equal. To see why, if \(d<50\), then \(70-d> d-30\); moving \(d\) upward reduces the larger distance until the distances match. If \(d>50\), then \(d-30>70-d\); moving downward reduces the larger distance. At \(d=50\), both distances are 20 mg, so the maximum regret is \(0.01(20)^2=4\) utility points. Thus 50 mg minimizes maximum regret. The result depends on the symmetric interval and quadratic loss; it is not a general rule that the midpoint of a plausible parameter range is optimal.
Worked Example: Checking the Continuous Calculation
Using the preceding hypothetical model, compare 40 mg, 50 mg, and 60 mg. At 40 mg, the largest distance to a parameter endpoint is \(70-40=30\) mg, so maximum regret is \(0.01(30)^2=9\) points. At 50 mg, it is \(0.01(20)^2=4\) points. At 60 mg, the largest distance is \(60-30=30\) mg, giving \(0.01(30)^2=9\) points.
The three calculations confirm that 50 mg has the lowest maximum regret among these doses. The continuous derivation establishes that it is also optimal over every dose in \([20,80]\) mg, not merely among the three checked values. For example, at 45 mg the endpoint distances are 15 and 25 mg, so the maximum regret is \(0.01(25)^2=6.25\) points, which exceeds the 4-point value at 50 mg.
Assumptions, Diagnostics, and Alternatives
A minimax-regret analysis is only as defensible as its inputs. It requires a clearly specified set of feasible doses, a utility \(U(d,\theta)\) that represents the decision objective, and a plausible parameter set \(\Theta\) over which the maximum is taken. If utility is predicted from a dose-response model, the relevant model and uncertainty assumptions must support those predictions. The rule itself does not establish that the model is correct or that every value in \(\Theta\) is clinically plausible.
For a finite scenario table, inspect which scenario produces each dose’s maximum regret. A single scenario may dominate the decision; if it is an implausible boundary value, verify the parameter range and model behavior there. Recompute the choice after reasonable changes to \(\Theta\), the candidate doses, and the utility definition. If small changes switch the selected dose, report that instability rather than presenting the choice as robust. For continuous sets, verify that the maximum has been found over the whole set: checking only a few grid points can miss an interior maximum unless the function’s shape justifies the grid.
| Decision rule | Quantity minimized or maximized | Interpretation |
|---|---|---|
| Minimax regret | Minimizes \(\max_{\theta\in\Theta}\{U^*(\theta)-U(d,\theta)\}\) | Limits the largest utility shortfall from the scenario-specific best |
| Maximin utility | Maximizes \(\min_{\theta\in\Theta}U(d,\theta)\) | Chooses the dose with the best worst-case utility |
| Expected utility | Maximizes an average of \(U(d,\theta)\) under a probability distribution for \(\theta\) | Balances outcomes according to assigned parameter probabilities |
Minimax regret is not a probability-weighted average, and it does not say that the selected dose is most likely to be optimal. It can be conservative because a scenario with very low probability still determines the maximum if it is included in \(\Theta\). Conversely, omitting a clinically important plausible scenario can make the reported worst-case regret falsely reassuring. A regret value is measured in utility units; it is not a probability, a dose difference, or necessarily a clinically important loss without interpretation of the utility scale.
A common mistake is to find the dose with the highest utility in each scenario and then choose the dose that wins in the most scenarios. That ignores the size of each loss: a dose might be nearly best in two scenarios but disastrous in a third. The correction is to subtract utility from the scenario-specific best, take the maximum regret for each dose, and minimize that maximum. Another mistake is to take each dose’s worst utility and call the smallest worst-case value its regret. That is a maximin calculation with the direction reversed, not minimax regret.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For a dose with utilities 4 and 7 under two scenarios whose best utilities are 6 and 9, what are its two regrets and its maximum regret?
- In the first worked example, why is the regret for 50 mg under the high-response scenario 3 points rather than 0?
- Could a scenario with a very small assigned probability determine the minimax-regret dose? Explain.
- Under the quadratic continuous example, what is the regret at 45 mg when \(\theta=70\) mg?
- State one difference between minimax regret and maximin utility.
Solutions. 1. The regrets are \(6-4=2\) and \(9-7=2\) points, so maximum regret is 2 points. 2. The best utility in that scenario is 12 points at 75 mg; 50 mg has utility 9, so its regret is \(12-9=3\) points. 3. Yes. Minimax regret uses the largest regret over the specified set and does not weight scenarios by probability. 4. The dose-parameter difference is \(70-45=25\) mg, so regret is \(0.01(25)^2=6.25\) points. 5. Minimax regret minimizes the greatest shortfall from the scenario-specific best, while maximin utility maximizes the lowest utility of the dose itself.