Bayesian Utility Maximization For Dose Selection

From Joint Outcomes to a Dose Decision

Dose selection requires more than estimating response and toxicity separately. A decision maker must also state how the possible clinical outcomes compare in value. A dose that produces more efficacy may also produce more toxicity; whether that tradeoff is worthwhile depends on the clinical consequences assigned to each joint outcome. Bayesian utility maximization makes this tradeoff explicit and combines it with uncertainty about dose-specific outcomes.

The decision considered here is which dose to select from a prespecified set after observing data. It is not, by itself, a complete escalation algorithm, a safety stopping rule, or a guarantee that a dose is acceptable. Those operational rules can be added as constraints, but the utility calculation and the constraints should be kept conceptually distinct.

Definition: Let $d$ denote a candidate dose, $Y_d$ the joint clinical outcome for a patient treated at dose $d$, and $u(d,Y_d)$ the utility assigned to that dose-outcome pair. Given observed data $\mathcal D$ and a posterior distribution for model parameters $\theta$, the posterior expected utility of dose $d$ is $$ EU(d\mid\mathcal D) =E\{u(d,Y_d)\mid\mathcal D\} =\int \left[\sum_y u(d,y)P(Y_d=y\mid\theta)\right]\pi(\theta\mid\mathcal D)\,d\theta. $$ A utility-maximizing choice among the available doses is any dose $d^\star$ satisfying $d^\star\in\arg\max_d EU(d\mid\mathcal D)$. If a separate eligibility or safety rule excludes doses, the maximization is over the remaining eligible doses.

The inner sum calculates expected utility conditional on the model parameters. The outer integral averages that quantity over posterior uncertainty about those parameters. Equivalently, one can first average each joint-outcome probability over the posterior to obtain the posterior predictive probability $P(Y_d=y\mid\mathcal D)$, then weight each outcome's utility by that probability.

For this calculation to be meaningful, the outcome categories must be defined consistently across doses, the utility values must reflect a stated clinical perspective, and the fitted joint model must adequately represent the relevant outcomes. In particular, if utility depends on the joint occurrence of efficacy and toxicity, marginal probabilities alone do not determine expected utility. Dependence matters because it changes the probabilities of the joint categories.

Deriving the Calculation for Binary Efficacy and Toxicity

Suppose efficacy $E$ and toxicity $T$ are each coded as 1 when present and 0 when absent. Assign utility $u(d,e,t)=e-\lambda t$, where one efficacy event has utility 1 and a toxicity event has penalty $\lambda\geq 0$. This additive scale is a simplifying choice; it is not a universal clinical standard. For each dose, the four possible outcome utilities are determined by substituting the relevant values of $e$ and $t$.

\begin{aligned} EU(d\mid\mathcal D) &=\sum_{e=0}^1\sum_{t=0}^1(e-\lambda t)P(E=e,T=t\mid d,\mathcal D)\\ &=P(E=1\mid d,\mathcal D)-\lambda P(T=1\mid d,\mathcal D). \end{aligned}

To see the second equality, expand the sum. The terms containing $e$ add to the probability of efficacy, because $e=1$ in the two efficacy categories and $e=0$ in the other two. The terms containing $-\lambda t$ add to $-\lambda$ times the probability of toxicity. Thus this particular additive utility depends only on the two posterior predictive marginal probabilities. A utility that assigns different values to, for example, efficacy with toxicity and efficacy without toxicity generally requires the full joint outcome distribution.

If posterior simulation is used, the same result can be computed by averaging across posterior draws. For draw $m$, calculate the expected utility at each dose using that draw's joint probabilities; then average those values over the draws. This is a Monte Carlo approximation to the posterior integral, not a substitute for defining the utility or checking the model.

Worked Example: Comparing Two Doses Across a Toxicity Penalty

Hypothetical data and model output. Suppose the posterior predictive joint outcome probabilities for two candidate doses are as follows. They are assumed to have been estimated from a joint dose model and each row sums to 1.

Dose	Efficacy, no toxicity	Efficacy and toxicity	Neither	Toxicity, no efficacy
A	0.35	0.15	0.35	0.15
B	0.45	0.25	0.15	0.15

Use $u=e-\lambda t$. For dose A, the posterior predictive efficacy probability is $0.35+0.15=0.50$, and the toxicity probability is $0.15+0.15=0.30$. For dose B, these probabilities are $0.45+0.25=0.70$ and $0.25+0.15=0.40$, respectively. Therefore

\begin{aligned} EU(A\mid\mathcal D)&=0.50-0.30\lambda,\\ EU(B\mid\mathcal D)&=0.70-0.40\lambda,\\ EU(B\mid\mathcal D)-EU(A\mid\mathcal D) &=(0.70-0.40\lambda)-(0.50-0.30\lambda)\\ &=0.20-0.10\lambda. \end{aligned}

At $\lambda=1$, $EU(A\mid\mathcal D)=0.50-0.30=0.20$, whereas $EU(B\mid\mathcal D)=0.70-0.40=0.30$. The difference is $0.30-0.20=0.10$ utility units in favor of B. Equivalently, substituting $\lambda=1$ into the difference gives $0.20-0.10=0.10$. In general, B is preferred when $\lambda<2$, the doses tie when $\lambda=2$, and A is preferred when $\lambda>2$. The recommendation therefore depends on the relative penalty assigned to toxicity, rather than on efficacy alone.

Utility Maximization Is Not an Acceptability Threshold

A threshold-based rule asks whether a dose satisfies a specified condition, often expressed through a posterior probability such as the probability that its underlying efficacy rate exceeds a minimum while its toxicity rate remains below a maximum. A rule may then call a dose acceptable only if that posterior probability reaches a chosen cutoff. Such a rule answers a classification question: does the dose meet the stated conditions with enough posterior support?

Utility maximization answers a different question: which candidate has the greatest posterior average of the consequences that have been assigned value? It can rank a dose with appreciable toxicity above a safer dose when its expected benefit compensates for that harm on the chosen scale. Conversely, a threshold rule can exclude a dose even when its expected utility is highest. The two approaches need not agree, and neither should be described as automatically implementing the other.

Worked Example: A Threshold Rule and a Utility Rule Choose Differently

Hypothetical posterior summaries. Consider two doses, L and H. Suppose an acceptability rule defines acceptable doses as those with posterior probability at least 0.80 of meeting both a minimum underlying efficacy rate and a maximum underlying toxicity rate. The joint posterior probability of meeting both conditions is stipulated to be 0.88 for L and 0.68 for H. Thus the threshold rule accepts L and does not accept H.

Now use the joint posterior predictive outcomes below and the utility assignments $u(1,0)=1$, $u(1,1)=0.20$, $u(0,0)=0$, and $u(0,1)=-0.50$, where the first argument is efficacy and the second is toxicity.

Dose	Efficacy, no toxicity	Efficacy and toxicity	Neither	Toxicity, no efficacy
L	0.50	0.05	0.40	0.05
H	0.48	0.24	0.22	0.06

For L, the expected utility is $0.50(1)+0.05(0.20)+0.40(0)+0.05(-0.50)=0.50+0.01-0.025=0.485$. For H, it is $0.48(1)+0.24(0.20)+0.22(0)+0.06(-0.50)=0.48+0.048-0.03=0.498$. On this utility scale, H has the larger expected utility by $0.498-0.485=0.013$ utility units, even though it fails the stipulated acceptability cutoff. The results are not contradictory: the methods use different decision criteria. A protocol could instead use acceptability as a hard eligibility constraint and maximize utility only among accepted doses; under that combined rule, L would be selected.

Posterior Averaging, Dependence, and Sensitivity

The posterior expectation incorporates uncertainty about model parameters rather than treating one estimated parameter value as known. For a finite set of posterior draws $\theta^{(1)},\ldots,\theta^{(M)}$, a direct approximation is

\widehat{EU}(d\mid\mathcal D) =\frac{1}{M}\sum_{m=1}^M\sum_y u(d,y)P(Y_d=y\mid\theta^{(m)}).

When utility is linear in the outcome probabilities, averaging probabilities first or averaging draw-specific expected utilities gives the same result. For nonlinear decisions, such as applying a threshold to an underlying dose-specific rate, the order matters: the posterior probability of meeting the threshold must be calculated from the posterior distribution, not by applying the threshold to a posterior mean.

Worked Example: Averaging Utility Over Posterior Draws

Hypothetical posterior output. Suppose two posterior draws receive weights 0.25 and 0.75. Under the utility $u=e-t$, the draw-specific posterior predictive marginal probabilities are:

Dose	Draw weight	Efficacy probability	Toxicity probability
C	0.25	0.80	0.40
C	0.75	0.40	0.10
D	0.25	0.65	0.15
D	0.75	0.55	0.20

For C, the draw-specific utilities are $0.80-0.40=0.40$ and $0.40-0.10=0.30$. Its posterior expected utility is $0.25(0.40)+0.75(0.30)=0.10+0.225=0.325$. For D, they are $0.65-0.15=0.50$ and $0.55-0.20=0.35$, giving $0.25(0.50)+0.75(0.35)=0.125+0.2625=0.3875$. D is preferred by $0.3875-0.325=0.0625$ utility units. The posterior average, rather than the most favorable draw, determines this comparison.

Several checks help reveal when a utility analysis is going wrong:

Check the outcome definition. Confirm that efficacy and toxicity windows, grading, and any composite categories mean the same thing at every dose.
Check joint probabilities. Each dose's mutually exclusive joint categories should sum to 1, and marginal probabilities derived from them should agree with reported summaries.
Check the utility scale. State whose preferences the values represent and whether the scale is interpretable across outcomes and doses. Arbitrary values can produce precise calculations but an unjustified decision.
Check posterior computation. Compare direct summation over outcome categories with simulation-based estimates when feasible, and investigate unstable estimates or poor mixing in posterior draws.
Check sensitivity. Recalculate rankings over plausible utility values, model assumptions, and dependence structures. A small utility difference may not support a robust recommendation.

How the Approaches Differ, and What Utility Cannot Do

Approach	Decision quantity	What it emphasizes	Main caution
Posterior expected utility	Posterior mean of assigned outcome utility	Tradeoffs across outcomes on an explicit scale	Results depend on utility elicitation and model adequacy
Posterior acceptability threshold	Posterior probability of meeting stated conditions	Meeting specified efficacy and safety criteria	A cutoff can make a borderline dose switch from accepted to rejected
Selection by a single estimated endpoint	One fitted efficacy or toxicity summary	Simplicity and a narrow target	Can ignore the other endpoint, posterior uncertainty, or their joint distribution

A utility maximum is a ranking under the specified model, candidate set, and utility scale. It does not establish that the selected dose is safe, clinically worthwhile, or suitable for every patient. A utility function can incorporate treatment burden, severe toxicity, or a no-treatment option, but these additions must be defined rather than assumed. Safety constraints may appropriately rule out a high-utility dose, and a trial protocol should state how such constraints interact with the ranking.

A common mistake is to treat “largest expected utility” as synonymous with “acceptable.” For example, one might argue that a dose should be selected because its expected benefit exceeds its expected harm, without checking whether the utility scale is justified or whether a separate safety requirement has been met. The correction is to report both the utility comparison and any acceptance or safety criteria, then apply the prespecified decision rule. Another mistake is to combine efficacy and toxicity marginal probabilities when the utility distinguishes joint outcomes. In that case, use the joint posterior predictive probabilities; two doses with the same marginals can still have different probabilities of the outcome combinations that carry the greatest reward or penalty.

Key takeaway: Bayesian utility maximization selects the dose with the greatest posterior expected value under an explicitly defined utility scale. It is a tradeoff rule, not an automatic safety filter; constraints and posterior acceptability thresholds must be stated separately if they are part of the decision.

Check Your Understanding

Try each question, then compare your reasoning with the solutions.

For $u=e-2t$, a dose has posterior predictive efficacy probability 0.60 and toxicity probability 0.20. What is its posterior expected utility?
Why can posterior expected utility require joint outcome probabilities even when efficacy and toxicity marginal probabilities are known?
A dose has the largest expected utility but fails a prespecified safety eligibility rule. What is the appropriate dose-selection procedure?
Two doses have expected utilities 0.41 and 0.40. What additional analysis should inform whether the ranking is useful?

Solutions. (1) $EU=0.60-2(0.20)=0.20$ utility units. (2) If utilities assign different values to different efficacy-toxicity combinations, the expected value weights each joint category separately; the marginals do not determine those category probabilities without further assumptions about dependence. (3) Exclude the ineligible dose and maximize expected utility over the remaining eligible candidates, as specified by the protocol; if none remain, follow the prespecified no-selection or stopping rule. (4) Assess sensitivity to plausible utility values and model assumptions, and examine posterior uncertainty in the utility difference. A small point difference alone does not show that the ranking is robust.