Risk-Sensitive Dose Selection Under Asymmetric Harm

Why Average Performance May Not Be Enough

Two candidate doses can have similar average performance but very different consequences in an unfavourable scenario. One dose may cause a modest shortfall from the desired effect for many patients; another may usually perform well but carry a small chance of severe toxicity. When underdosing and toxicity do not have equal consequences, a dose rule based only on an unweighted average can conceal an important clinical trade-off. And even after the harms have been weighted, an average alone does not say whether decision makers wish to avoid unusually poor outcomes.

This tutorial compares three elements of a decision: the clinical weights assigned to different harms, the expected loss across uncertainty, and an additional penalty for losses in the adverse tail. The tail penalty represents a preference for avoiding bad outcomes; it does not replace the clinical judgment needed to define which harms matter and how they should be measured.

The severity measures and their weights need a meaningful interpretation. For example, \(U\) and \(T\) might be points on a prespecified clinical harm scale, and \(c_T>c_U\) would make a unit of toxicity severity count more than a unit of underdosing severity. The numerical scale must be chosen so that this comparison is defensible; the formula does not discover the weights from the dose-response model. Other clinically relevant outcomes can be added if their definitions and weights are justified.

The states and their probabilities may summarize uncertainty relevant to the decision, such as uncertainty in model parameters and predicted outcomes. They must represent a coherent probability distribution for the decision being considered. The calculations below treat that distribution and the loss scale as specified. They do not account automatically for model misspecification, unrepresented outcomes, or uncertainty about the clinical weights.

Expected Loss and Downside-Risk Criteria

A risk-neutral rule selects the candidate dose with the smallest expected loss:

This criterion already accounts for asymmetric harm if the loss function uses unequal weights. “Risk-neutral” here means that, for a fixed loss scale, the choice depends on the average loss rather than an additional preference for or against variability. It does not mean that the clinical consequences are treated as symmetric.

A risk-sensitive rule adds a penalty for a particular part of the loss distribution. One option is a downside penalty based on excess loss above a prespecified reference \(L_0\), such as \(\mathbb{E}[\max\{L(d)-L_0,0\}]\). Another option is a tail criterion: conditional value at risk (CVaR), which averages losses in the worst part of the distribution. For an upper-tail fraction \(1-q\), define the loss quantile and CVaR as follows.

With this convention, a larger \(q\) focuses attention on a narrower, more extreme upper tail. In a discrete distribution, the tail may cut through a probability mass at the quantile. The CVaR calculation must then include only the fraction of that mass needed to make the tail probability exactly \(1-q\); simply averaging every outcome at or above the quantile can give the wrong answer.

The expected loss and CVaR are in the same loss units, so \(\lambda\) is dimensionless. At \(\lambda=0\), the score reduces to expected loss. At larger \(\lambda\), the decision maker gives more weight to severe tail losses. This is a transparent decision rule only if \(q\), \(\lambda\), and the loss scale have been justified and reported.

Worked Dose Comparisons

Justifying the Tail Criterion

CVaR is useful because it makes “bad outcomes matter more” operational without discarding the rest of the distribution. At \(\operatorname{CVaR}_{0.80}\), for example, the rule summarizes the average loss in the worst 20%. A short justification follows from the quantile definition: if losses are ordered from low to high, integrating \(\operatorname{VaR}_u\) from \(q\) to 1 accumulates exactly the losses occupying the upper \(1-q\) probability of the distribution. Dividing by \(1-q\) converts that total tail contribution into an average. If the tail starts partway through an atom, the quantile integral includes only the needed fraction of that atom, as in the preceding example.

The added penalty changes the decision only when doses trade off average and tail loss. If dose A has no greater expected loss and no greater CVaR than dose B, then \(J(A)\leq J(B)\) for every \(\lambda\geq0\); B is not preferred by this score. If A has lower expected loss but higher CVaR, the selection can depend on \(\lambda\). The equality point, when one exists, can be found by solving:

This is a comparison under the specified score, not a universal ranking of doses. A reference-based downside penalty and CVaR answer different questions: the former penalizes loss beyond a chosen threshold, whereas CVaR averages a chosen upper tail. A variance penalty is another possible summary, but it can be sensitive to unusually high as well as low outcomes and does not isolate severe harms in the upper tail of loss. The choice of criterion should follow the decision concern, not convenience alone.

Assumptions, Diagnostics, and Limitations

Before using a risk-sensitive score, check that its ingredients are suitable for the decision:

• Loss definition: severity measures and weights must reflect the clinical consequences being compared. If outcomes overlap, adding their penalties may count the same harm twice.
• Scenario distribution: probabilities must be nonnegative, sum to one, and describe the uncertainty relevant to the decision. Check that important adverse states have not been omitted.
• Tail estimation: a very narrow tail can be estimated poorly when only a small number of simulated states are available. Inspect tail counts and quantile stability rather than trusting a single estimated CVaR.
• Decision sensitivity: report how the selected dose changes over plausible values of \(\lambda\), \(q\), and clinical weights. A recommendation that changes with a small input change should be presented as conditional on those inputs.
• Scale and interpretation: keep the loss units consistent across doses. State the dose set, the chosen tail fraction, and the risk penalty so another analyst can reproduce the comparison.

Risk adjustment does not repair biased predictions or incorrect probabilities. If the model understates toxicity in an adverse state, a precise CVaR calculation will still understate the tail loss. Nor does a CVaR score alone specify an acceptable probability of toxicity; a chance constraint directly limits the probability of an event, while CVaR summarizes the severity of losses in a tail. The earlier material on chance-constrained dose selection addresses that different form of decision rule. Expected loss is simpler and appropriate when averages are the intended criterion; minimax regret, covered in the earlier material on minimax regret for dose choice, instead compares a decision with the best dose in each state. These approaches answer distinct questions and need not select the same dose.

Common Pitfalls

• “The expected-loss winner is safest.” A lower average does not imply a smaller worst-tail loss. State whether the choice is based on an average, a tail measure, or both.
• “Toxicity is already weighted, so risk adjustment is redundant.” Unequal weights encode the relative severity of underdosing and toxicity. A tail penalty separately encodes a preference about the distribution of the resulting loss.
• “CVaR is the average of every outcome at or above the quantile.” In a discrete distribution, that can include too much probability at the quantile. Average exactly the worst \(1-q\) probability, using only a fraction of a tied or boundary mass if needed.
• “A larger \(\lambda\) is automatically more clinically appropriate.” It increases the influence of CVaR by construction; its value requires a defensible decision rationale and should be shown transparently.