Deriving Constrained Dose Selection Rules

From a Safety Requirement to a Dose Rule

A dose can have the highest estimated efficacy and still be an unsuitable recommendation if its toxicity is too likely to exceed an acceptable level. A constrained selection rule makes the tradeoff explicit: first require adequate posterior evidence that a dose meets the toxicity standard, then choose the most efficacious dose among those that pass. The rule is a decision procedure based on a specified model, toxicity threshold, and evidential standard; it does not make the safety requirement disappear when the data are uncertain.

Let \(d_1,\ldots,d_K\) be the prespecified candidate doses. For a patient treated at dose \(d_j\), let \(p_{E,j}\) and \(p_{T,j}\) denote the underlying probabilities of the defined efficacy and toxicity outcomes over their specified assessment windows. Write \(\mathcal D\) for the observed data and \(\theta_T\) for the largest acceptable toxicity probability. The posterior safety probability for dose \(d_j\) is \(q_j=P(p_{T,j}\leq\theta_T\mid\mathcal D)\). A design specifies a minimum value \(\gamma\), between zero and one, that this probability must meet.

The parameter \(\theta_T\) states what toxicity rate is considered acceptable, while \(\gamma\) states how much posterior support is required that a candidate’s true rate is at or below that limit. These are distinct design choices. For example, \(\theta_T=0.30\) and \(\gamma=0.80\) require at least an 80% posterior probability that a dose’s toxicity probability is no greater than 30%. They do not assert that the dose has an 80% chance of being safe in a future patient; the probability concerns the unknown model parameter, conditional on the data and model.

Why the Rule Has This Form

Suppose the decision maker’s reward for selecting dose \(d_j\) is the probability of efficacy \(p_{E,j}\), and the toxicity requirement is a hard constraint rather than a penalty traded against benefit. The posterior expected reward for that decision is \(E(p_{E,j}\mid\mathcal D)\). The rule follows by restricting the available decisions to those that satisfy the posterior safety requirement, then maximizing expected reward over the remaining finite set.

This is a constrained optimization rule, not a single combined efficacy-toxicity score. It uses posterior uncertainty in toxicity through \(q_j\), and posterior uncertainty in efficacy through its expectation. If two doses have equal posterior mean efficacy, both are maximizers unless a tie-breaking convention is added. The rule uses the marginal posterior distributions of efficacy and toxicity; it does not require those parameters to be posterior-independent. However, the model must produce valid posterior distributions for both, and any dependence represented by the model should be retained when calculating them.

For a simple derivation of the posterior mean in a common model, suppose the efficacy outcomes at one dose are Bernoulli with probability \(p_E\), and use a \(\operatorname{Beta}(a,b)\) prior. With \(x\) responses among \(n\) patients, the likelihood is proportional to \(p_E^x(1-p_E)^{n-x}\). Multiplying by the prior density gives a \(\operatorname{Beta}(a+x,b+n-x)\) posterior. The mean of a \(\operatorname{Beta}(u,v)\) distribution is \(u/(u+v)\), so the posterior mean efficacy is \((a+x)/(a+b+n)\). This posterior mean, rather than the observed proportion alone, is the efficacy criterion in the rule.

Posterior Uncertainty Is Part of the Constraint

The posterior safety probability is not interchangeable with comparing the posterior mean toxicity to \(\theta_T\). A posterior mean describes the center of a distribution; \(q_j\) also reflects how much posterior mass lies on either side of the toxicity limit. Two doses with the same posterior mean toxicity can have different safety probabilities if their posterior uncertainty differs. Conversely, a mean at or below the limit does not guarantee that a specified fraction of the posterior distribution is below it.

Posterior probabilities can be evaluated from an analytic posterior distribution or estimated from posterior simulation draws. With \(M\) draws \(p_{T,j}^{(1)},\ldots,p_{T,j}^{(M)}\), the Monte Carlo estimate is \(\hat q_j=M^{-1}\sum_{m=1}^M I(p_{T,j}^{(m)}\leq\theta_T)\), where \(I(\cdot)\) equals 1 when its condition is true and 0 otherwise. A value near \(\gamma\) warrants enough simulation draws to determine whether Monte Carlo error could change eligibility. This numerical uncertainty is additional to the posterior uncertainty about toxicity itself.

Ties, Diagnostics, and Alternatives

If multiple feasible doses attain the same posterior mean efficacy, the mathematical rule returns all of them. A single recommendation requires a tie-breaking policy stated before applying the rule. Possible policies include selecting the lower dose or, among efficacy ties, selecting the dose with the lower posterior mean toxicity. Such a policy is an additional decision convention, not a consequence of the efficacy maximization itself. If estimates are rounded for display, apparent ties may be artifacts; comparisons should use unrounded values and a prespecified numerical tolerance.

Before using the rule, verify that the toxicity endpoint, observation window, threshold, and candidate doses match the intended clinical decision. Check posterior predictive or model-fit diagnostics appropriate to the dose-response model, and assess whether conclusions are sensitive to plausible prior and model choices. For simulation-based posteriors, check that estimates of \(q_j\) are stable and that enough effective draws were used. A dose just above or below \(\gamma\) may change eligibility with small changes in assumptions or data, so report the probability itself rather than only a pass/fail label.

The rule differs from two common alternatives. A plug-in rule compares a point estimate, often a posterior mean toxicity, with \(\theta_T\); it does not directly enforce a posterior probability requirement. A rule that maximizes a weighted efficacy-toxicity score allows a specified penalty for toxicity to trade against efficacy, whereas the constrained rule excludes doses that fail the threshold regardless of their efficacy mean. The constrained rule is transparent, but its result can change sharply around the eligibility boundary. If no dose is feasible, choosing the dose with the highest efficacy anyway would violate the rule; the design must instead specify what happens when \(\mathcal A\) is empty.

The output is conditional on the candidate set, outcome definitions, prior, likelihood, \(\theta_T\), and \(\gamma\). It is not a guarantee that the selected dose’s true toxicity rate is acceptable, nor does it establish that efficacy is clinically worthwhile. In particular, maximizing posterior mean efficacy among feasible doses can favor a dose with only a small efficacy advantage if that advantage is estimated imprecisely; the objective is the posterior mean, not a requirement for a particular efficacy probability. These limitations should be visible when interpreting a recommendation.