Bayesian Decision Rules for Dose Selection

From posterior evidence to a dose decision

A posterior distribution describes uncertainty about each dose’s effects after observing the data. A dose-selection rule turns that uncertainty into an action: for example, retain doses that have a sufficiently high posterior probability of meeting an efficacy target and a sufficiently high probability of staying below a toxicity limit. Alternatively, choose the dose with the greatest posterior expected utility, where utility gives explicit value to benefit and harm.

These rules answer different questions. A probability threshold asks whether a dose is sufficiently likely to satisfy a specified condition. An expected-utility rule asks which available action has the greatest average value under a stated utility scale. Neither rule is automatic: the target, thresholds and relative value of benefits and harms are decision inputs that should be specified and justified.

For a simple binary-outcome model, each participant at dose \(j\) has an efficacy outcome and a toxicity outcome, each coded 1 if the event occurs and 0 otherwise. The marginal data at a dose can be modelled as binomial counts conditional on \(\theta_{E,j}\) and \(\theta_{T,j}\). A common conjugate specification gives each rate a beta prior; with a \(\mathrm{Beta}(a,b)\) prior and \(y\) events among \(n\) observations, the posterior is \(\mathrm{Beta}(a+y,b+n-y)\). This calculation presumes the binomial sampling model is appropriate and the prior is specified before using the data for the decision.

A rule may require two marginal conditions, such as \(P(\theta_{E,j}\geq\theta_E^*\mid D)\geq\gamma_E\) and \(P(\theta_{T,j}\leq\theta_T^*\mid D)\geq\gamma_T\). Satisfying both is a pair of marginal statements; it does not, by itself, specify the posterior probability that both conditions hold simultaneously. That joint probability requires a joint posterior model. The examples below use marginal criteria or additive utility and do not infer a joint event from marginal probabilities.

Calculating a posterior probability criterion

The posterior probability is evaluated against a prespecified threshold, not against a frequentist significance level. For integer beta parameters, a useful identity gives an exact calculation: if \(X\sim\mathrm{Beta}(a,b)\), then \(P(X>q)=P\{Z\leq a-1\}\), where \(Z\sim\mathrm{Binomial}(a+b-1,q)\). This identity follows from the relationship between the beta cumulative distribution function and binomial tail probabilities.

Screening doses with efficacy and toxicity criteria

A practical probability-based rule can first exclude doses that are insufficiently likely to be effective or sufficiently likely to be too toxic. The decision thresholds should reflect the intended use of the dose and the consequences of errors. For example, a rule might retain dose \(j\) only when both \(P(\theta_{E,j}\geq\theta_E^*\mid D)\geq\gamma_E\) and \(P(\theta_{T,j}\leq\theta_T^*\mid D)\geq\gamma_T\). A further rule is needed if more than one dose remains; one might choose the lowest qualifying dose or use expected utility to rank the qualifying doses.

Choosing by posterior expected utility

A probability screen treats the chosen cutoffs as decisive. Expected utility offers another way to express trade-offs. Suppose a participant’s utility at dose \(j\) is \(u_j\), specified in utility points, and it is additive in efficacy and toxicity:

where \(B>0\) is the utility value assigned to efficacy and \(H>0\) is the disutility assigned to toxicity. If dose burden or treatment cost matters, it can be included as another prespecified utility term. For a finite set of available doses and a common utility scale, select a dose \(j^*\) maximizing \(EU_j\). This rule balances the outcomes through the specified values \(B\) and \(H\), rather than through separate pass/fail probability cutoffs.

The justification is direct. If the action is to choose one dose and the consequence of action \(j\) is utility \(u_j\), its posterior expected utility is \(E(u_j\mid D)\). The posterior expected utility of any randomized choice among doses is a weighted average of their \(EU_j\) values, and cannot exceed the largest of them. Choosing a dose with maximum \(EU_j\) therefore maximizes posterior expected utility among the available actions. This derivation relies on comparing utilities on the same scale and on the stated model for the posterior expectations.

Assumptions, checks and alternatives

Before applying either rule, check that the dose-specific outcome model and prior are appropriate for the available data. In a beta-binomial analysis, confirm that the counts refer to the intended population and follow-up period, that event definitions are consistent across doses, and that the binomial sampling assumption is reasonable. Inspect how much the prior influences the posterior, particularly when dose-specific sample sizes are small. Also check whether the decision thresholds and utility values were set independently of the observed outcomes; choosing them after seeing the results can make the decision difficult to interpret.

A posterior probability rule is transparent when there are clinically meaningful targets and a clear acceptable level of uncertainty. An expected-utility rule can compare several outcomes on one scale and can distinguish doses that all pass a screen. The alternatives are not interchangeable: a dose can pass efficacy and safety thresholds but have lower expected utility than another qualifying dose, or have the highest expected utility while failing a non-negotiable safety constraint. In that situation, a coherent protocol can apply a safety screen first and maximize expected utility among the remaining doses.

• Common mistake: treating two marginal probabilities as a joint guarantee. Passing both separate 0.90 criteria does not establish that the probability of meeting both conditions is 0.90 or greater. The correction is to state the criteria as separate marginal requirements or calculate the joint probability under a suitable joint posterior.
• Common mistake: calling the highest observed efficacy rate the optimal dose. The observed rate ignores posterior uncertainty and toxicity. The correction is to apply the prespecified posterior decision rule to all relevant outcomes and candidate doses.
• Common mistake: treating utility points as objective facts. A numerical utility scale encodes value judgments. The correction is to justify its components and examine whether plausible alternatives would change the choice.

These rules do not, by themselves, establish a dose’s causal effect, guarantee patient benefit, or quantify frequentist error rates. Their conclusions are conditional on the model, prior and decision inputs. They also do not resolve issues such as uncertain control response, complex dependence or computational error; those require additional modelling and assessment.