Robust Dose Selection Across Plausible Parameters

Why Optimize Worst-Case Performance?

A dose recommendation can depend on uncertain model parameters even when the dose-response model and the decision criterion are fixed. Chance-constrained dose selection addresses this by requiring a specified probability of satisfying a target, under a probability distribution for uncertainty. Robust dose selection answers a different question: which dose has the best performance in the least favorable parameter scenario within a defined plausible set? It is useful when decision-makers want a recommendation that does not rely on assigning precise probabilities within that set.

The result depends on the set being defended as plausible. A set that is too narrow can omit consequential parameter values; a set that is too broad can make the least favorable case implausible or excessively pessimistic. Robustness is therefore not a guarantee that a dose performs well under every imaginable circumstance. It is a guarantee, conditional on the model, utility, dose range, and uncertainty set, about the worst case represented there.

Definition: Let $d$ be a dose from an admissible dose set $\mathcal{D}$, let $\theta$ denote uncertain model parameters, and let $\Theta$ be a specified plausible parameter set. If $U(d,\theta)$ is the utility of dose $d$ at parameters $\theta$, the worst-case utility of $d$ is $$ W(d)=\inf_{\theta\in\Theta} U(d,\theta). $$ A robust dose is $$ d_{\mathrm{rob}} \in \arg\max_{d\in\mathcal{D}} W(d). $$ If the extrema are attained, the infimum and supremum can be written as a minimum and maximum. A robust dose maximizes the least utility over $\Theta$; it does not maximize average utility or minimize regret.

For the rule to be interpretable, specify the utility scale and the uncertainty set before optimizing. In the examples below, efficacy $E(d,\theta)$ is measured in response units and toxicity $q(d,\theta)$ is a probability. They are not directly subtractable. We convert efficacy to utility points using $\kappa_E$, measured in points per response unit, and assign a toxicity penalty using $\lambda$, measured in points per probability unit:

U(d,\theta)=\kappa_E E(d,\theta)-\lambda q(d,\theta).

Thus, both terms on the right are in utility points. The coefficients encode a decision-maker's value trade-off; they are not statistical parameters estimated by the dose-response model. Other utility functions are possible, but their units and decision meaning must also be clear. Here, $U(d,\theta)$ is a conditional expected utility at fixed $\theta$, not the utility of a particular patient's realized outcome.

Assumptions and the Worst Case

The rule is well-defined only after the decision problem is specified. The admissible doses $\mathcal{D}$ must reflect what can actually be administered. The parameter set $\Theta$ must be explicit, nonempty, and relevant to the population and setting in which the recommendation will be used. The utility must put all outcomes on a common scale. Finally, the minimum over parameters and maximum over doses must exist, or a numerical procedure must approximate them closely enough for the decision.

A common case is a rectangular set, such as $\Theta=[a_{\min},a_{\max}]\times[b_{\min},b_{\max}]$. It allows any combination of parameter values inside the ranges, including combinations that may be poorly supported jointly. If utility decreases as $a$ decreases and increases as $b$ decreases, then the worst case on this rectangle is $a=a_{\min}, b=b_{\max}$. This conclusion follows from monotonicity, not simply from the fact that these values are endpoints. If utility is non-monotone, or the plausible set is constrained or correlated, the worst case may be elsewhere on its boundary or in its interior.

Worked Example: Comparing Worst-Case Utility at Candidate Doses

Consider a hypothetical efficacy-toxicity model with dose $d$ in dose units, response $E$ in response units, and toxicity probability $q$: $E(d,a)=a d/(2+d)$ and $q(d,b)=bd$. Suppose $d$ is restricted to 1, 2, or 4; $a\in[8,10]$ response units and $b\in[0.01,0.02]$ per dose unit. Let $\kappa_E=10$ utility points per response unit and $\lambda=500$ utility points per probability unit. Thus, $U=10E-500q$, in utility points. Assume the ranges define a rectangular plausible set and that these model forms are the stated decision model over the candidate doses.

For each dose, efficacy increases with $a$, while toxicity probability increases with $b$. Therefore, its worst-case utility occurs at $a=8$ and $b=0.02$. At $d=1$, $E=8(1)/(2+1)=2.667$ response units and $q=0.02(1)=0.020$. Hence $U=10(2.667)-500(0.020)=26.67-10.00=16.67$ points.

At $d=2$, $E=8(2)/(2+2)=4.000$ response units and $q=0.02(2)=0.040$. The worst-case utility is $U=10(4.000)-500(0.040)=40.00-20.00=20.00$ points.

At $d=4$, $E=8(4)/(2+4)=5.333$ response units and $q=0.02(4)=0.080$. The worst-case utility is $U=10(5.333)-500(0.080)=53.33-40.00=13.33$ points.

The worst-case utilities are 16.67, 20.00, and 13.33 points at doses 1, 2, and 4, respectively. The robust choice among these candidates is therefore $d=2$. This conclusion is conditional on the specified model, utility trade-off, and rectangular parameter set; it is not a statement that dose 2 is best for every plausible decision criterion.

Deriving a Continuous-Dose Robust Choice

When doses vary continuously, the same maximin definition applies, but $W(d)$ must be optimized over the full admissible range. A useful justification for reducing the parameter optimization is to prove which parameter values minimize utility. In the model above, suppose $d\in[0,5]$, $a\in[8,10]$, and $b\in[0.01,0.02]$. For every fixed $d\geq0$, the efficacy term increases with $a$ and the toxicity penalty increases with $b$. Hence the worst case is $a=8,b=0.02$, and

W(d)=10\left(\frac{8d}{2+d}\right)-500(0.02d) =\frac{80d}{2+d}-10d.

For an interior maximum, differentiate and set the derivative equal to zero:

W'(d)=\frac{160}{(2+d)^2}-10=0 \quad\Longrightarrow\quad (2+d)^2=16 \quad\Longrightarrow\quad d=2,

where the other algebraic root, $d=-6$, is outside the admissible range. To verify that this stationary point is the global maximum, the second derivative is $W''(d)=-320/(2+d)^3<0$ for every $d\in[0,5]$. Thus $W$ is strictly concave on the dose interval, and its stationary point is its unique maximum. The boundaries confirm the result: $W(0)=0$, $W(2)=20$, and $W(5)=80(5)/7-50=7.143$ points. This derivation is valid because monotonicity identifies the least favorable parameter values for every dose, and concavity rules out a better dose elsewhere in the interval.

Worked Example: How a Parameter-Set Boundary Changes the Choice

Return to the three candidate doses and utility from the first example. Now expand the upper toxicity-slope boundary from $b_{\max}=0.02$ to $b_{\max}=0.03$, keeping $a_{\min}=8$ and the same rectangular-set and monotonicity assumptions. The worst case is now $a=8,b=0.03$.

At $d=1$, the worst-case efficacy remains $2.667$ response units and $q=0.03$. Thus $U=10(2.667)-500(0.03)=26.67-15.00=11.67$ points.

At $d=2$, efficacy remains $4.000$ response units and $q=0.06$. Thus $U=10(4.000)-500(0.06)=40.00-30.00=10.00$ points.

At $d=4$, efficacy remains $5.333$ response units and $q=0.12$. Thus $U=10(5.333)-500(0.12)=53.33-60.00=-6.667$ points. The robust choice changes from dose 2 to dose 1. The change is driven by the added high-toxicity boundary, which penalizes larger doses more strongly. It should prompt scrutiny of whether $b=0.03$ is genuinely plausible for this decision, not an automatic conclusion that the expanded set is preferable.

Set Shape, Diagnostics, and Alternatives

Rectangular parameter ranges are convenient but can be conservative if they combine values that cannot plausibly occur together. To see why the geometry matters, retain the same model, utility, and candidate doses, but suppose the plausible set consists only of two hypothetical parameter scenarios: $(a,b)=(8,0.01)$ and $(10,0.02)$. The worst-case utilities at doses 1, 2, and 4 are, respectively, 21.67, 30.00, and 26.67 points. For example, at dose 4 the two utilities are $10(8\cdot4/6)-500(0.01\cdot4)=33.33$ points and $10(10\cdot4/6)-500(0.02\cdot4)=26.67$ points; the smaller is 26.67. The robust dose remains 2, but its worst-case utility differs from the rectangular-set result of 13.33 points at dose 4. The rectangle includes the crossed combination $(8,0.02)$, which is not in this two-scenario set. A set that represents parameter dependence can therefore avoid treating an unsupported combination as the worst case.

Before accepting a numerical robust dose, check the following:

Check units and direction. Confirm that efficacy and toxicity have been converted to a common utility scale, that probabilities are valid over the dose range, and that the assumed monotonic directions hold.
Inspect the worst-case parameter values. Verify that they are in the stated set and scientifically plausible. If an optimizer repeatedly returns a boundary, determine whether that is expected from monotonicity or a sign of an overly restrictive set or an implementation error.
Check the dose optimization. For a continuous dose range, evaluate endpoints and interior stationary points, and use a sufficiently fine numerical search or a suitable optimizer. A coarse grid can miss an interior optimum; a single local search can miss other optima when $W(d)$ is not concave.
Vary the set deliberately. Report how the worst-case utility and selected dose change under defensible boundary changes. A recommendation that changes sharply deserves particular attention to those boundaries.

A chance-constrained rule instead imposes a probability requirement, such as a minimum probability of acceptable efficacy and safety under a specified uncertainty distribution. It answers a probability-based question and depends on that distribution. Robust selection uses the least utility over a set and does not assign probabilities to its parameter values. Expected-utility selection averages utility using a specified distribution; a high average need not imply acceptable worst-case performance. Minimax regret is another decision rule, but its objective is regret relative to scenario-specific optimal decisions rather than the worst attainable utility. These approaches are alternatives, not interchangeable interpretations of one calculation.

Robust selection has important limitations. If $\Theta$ is misspecified, the guarantee applies to the wrong set; if the model is misspecified, optimizing within it cannot repair that defect. A very broad set may force a dose that is unduly conservative, while a narrow set can create false reassurance. The rule also treats all included scenarios as relevant to the worst case, even if some are much less plausible than others. The set, utility trade-off, and model therefore require clinical and scientific justification, and the resulting dose should be presented with the assumptions that define it.

Key takeaway: A robust dose maximizes the minimum utility over a declared plausible parameter set. The recommendation is only as defensible as the utility scale, the model, and the set boundaries used to define that worst case.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

For the original model and utility, if $b_{\max}$ increases to 0.03, what are the three worst-case utilities and which candidate dose is selected?
If efficacy is 3.2 response units and toxicity probability is 0.04, calculate utility when $\kappa_E=10$ points per response unit and $\lambda=500$ points per probability unit.
In one sentence, distinguish robust dose selection from chance-constrained dose selection.
For $W(d)=80d/(2+d)-10d$ on $[0,5]$, state the stationary dose and justify why it is the global maximum.

Solutions. 1. With $a=8,b=0.03$, the utilities at doses 1, 2, and 4 are $10(8/3)-500(0.03)=11.67$ points, $10(4)-500(0.06)=10.00$ points, and $10(16/3)-500(0.12)=-6.667$ points. Dose 1 has the largest worst-case utility. 2. $U=10(3.2)-500(0.04)=32-20=12$ utility points. 3. Robust selection maximizes utility's minimum over a specified parameter set, whereas chance-constrained selection requires a probability condition under a specified uncertainty distribution. 4. Differentiating gives $W'(d)=160/(2+d)^2-10$, which is zero at $d=2$ within the interval. Since $W''(d)=-320/(2+d)^3<0$ throughout $[0,5]$, $W$ is strictly concave there, so the stationary point is the unique global maximum.

Robust Dose Selection Across Plausible Parameters

What You'll Learn