Selecting an Acceptable Dose Interval

From Two Dose Boundaries to an Acceptable Interval

Dose justification often produces two different kinds of boundary. The efficacy analysis identifies a minimum dose that meets a prespecified efficacy criterion; the safety analysis identifies a maximum dose that meets a prespecified safety criterion. The question here is how to combine those boundaries into a range of doses that meets both requirements, and how uncertainty in either boundary changes the conclusion.

The criteria must be stated before interpreting the interval. For example, efficacy might require a specified minimum treatment effect on a clinically interpretable endpoint, while safety might require that a specified risk measure remain below an agreed limit. Those criteria, the dose scale, the target population, and the treatment regimen must be compatible. An interval is only as meaningful as the criteria and models used to define its endpoints.

This definition treats efficacy as requiring doses at or above \(D_E\), and safety as requiring doses at or below \(D_S\). It therefore presumes that these criteria define one lower and one upper boundary on the dose scale. If the dose-response relationship is nonmonotonic, or if a criterion is met in disjoint dose regions, a single interval may not represent the acceptable set. Also, an interval is a range meeting stated criteria, not by itself a recommendation of one uniquely optimal dose.

Point Estimates and the Feasibility Question

A first, descriptive calculation compares the two estimated boundaries. If \(\hat{D}_E\leq\hat{D}_S\), their point estimates define a nonempty estimated interval. If \(\hat{D}_E>\hat{D}_S\), the estimates conflict: the dose needed to meet the efficacy criterion is above the estimated maximum dose meeting the safety criterion. Neither result alone fully describes uncertainty. A narrow estimated interval can be fragile when the boundaries are imprecise, while a modest gap can be persuasive when estimates are precise.

For a discrete set of available regimens, the practically relevant set is the intersection of that set with the interval. For a continuous dose scale, every dose between the boundaries satisfies the two inequalities as defined. In either case, the interval should be reported with units and with the criteria that define its endpoints.

Uncertainty in Whether the Interval Exists

The existence of an acceptable interval depends on the difference between the safety and efficacy boundaries. Define the boundary gap \(G=D_S-D_E\); a feasible interval exists precisely when \(G\geq0\). Its estimate is \(\hat{G}=\hat{D}_S-\hat{D}_E\). If the estimates are approximately jointly normal, then the estimated gap is approximately normal with variance

Writing the standard errors as \(SE(\hat{D}_S)\) and \(SE(\hat{D}_E)\), the estimated standard error of the gap is

This expression follows from the variance rule for a difference: for random variables \(A\) and \(B\), \(\operatorname{Var}(A-B)=\operatorname{Var}(A)+\operatorname{Var}(B)-2\operatorname{Cov}(A,B)\). Thus, uncertainty in both boundaries contributes, and their covariance matters. Shared participants, common model parameters, or jointly fitted dose-response models can make the boundary estimates dependent. Assuming independence without support can misstate uncertainty.

Under an approximate normal sampling model for \(\hat{G}\), a standardized estimate is \(Z=\hat{G}/SE(\hat{G})\). The quantity \(\Phi(Z)\), where \(\Phi\) is the standard normal cumulative distribution function, estimates the repeated-sampling probability that a normally distributed estimate of the gap is positive when the fitted distribution is centered at the observed estimate. It is a useful description of boundary uncertainty under that approximation, but it is not the probability that the true interval exists. That latter probability requires a probability model for the unknown boundaries, such as a Bayesian posterior analysis.

Conservative Bounds for a Decision Range

A different summary uses one-sided confidence bounds. To form a conservative range, use an upper confidence bound for \(D_E\) and a lower confidence bound for \(D_S\). The resulting interval consists of doses above the efficacy boundary bound and below the safety boundary bound. If the lower endpoint exceeds the upper endpoint, this method yields no conservative interval at the chosen confidence levels, even if the point estimates form a nonempty interval.

Separate one-sided confidence statements do not automatically give the same confidence level for their joint coverage. For example, if each bound has 95% marginal coverage, the probability that both cover simultaneously is at least 90% by the union bound; it need not be 95%. The actual joint coverage depends on their dependence. A joint confidence region or a multiplicity adjustment can be used when a specified simultaneous coverage level is required.

Diagnostics, Alternatives, and Limitations

The calculations are only as reliable as the boundary estimates and their uncertainty models. Before presenting an interval, check whether the efficacy and safety criteria are prespecified, expressed on compatible dose and population scales, and supported by observations rather than substantial extrapolation. Inspect dose-response fits near both boundaries, residual or predictive diagnostics, and sensitivity to plausible model forms. If either boundary is poorly identified, a precise-looking interval can be misleading.

• Check the direction of each boundary. Confirm that efficacy is represented by doses at or above the minimum and safety by doses at or below the maximum. A reversed inequality reverses the interpretation.
• Check dependence. Estimate the covariance from a joint fit or resampling procedure when both boundaries use shared data or model parameters. Do not silently set the covariance to zero.
• Check boundary stability. Compare estimates across reasonable model specifications and examine whether the boundary is driven by a few dose groups or by extrapolation beyond tested doses.
• Check the form of the feasible set. A single interval is inadequate if criteria are met in separated dose regions or if the assumed monotone boundary interpretation does not fit the response pattern.
• Check the regimen definition. A daily dose interval does not itself address changes in schedule, treatment duration, or exposure; those features must match the analysis defining the boundaries.

The methods answer related but distinct questions. Comparing point estimates is simple and communicates the estimated range, but does not quantify its uncertainty. A normal approximation for the boundary gap gives a compact summary when the joint sampling distribution is reasonably represented by a normal distribution. A joint bootstrap or simulation can preserve dependence and accommodate nonlinearity, but its results depend on the resampling or simulation model and may be unstable with sparse data. One-sided confidence bounds offer a conservative decision range with transparent marginal coverage, though they can be overly cautious and require care about joint coverage. A posterior probability that \(D_S\geq D_E\) is another possible summary, but it depends on the likelihood and prior distributions and is not interchangeable with a frequentist confidence statement.

No summary repairs misspecification, weak data near a boundary, or criteria that do not represent the intended clinical requirements. An empty conservative interval does not prove that no usable dose exists; it may reflect uncertainty too large to support a conservative range. Conversely, a nonempty point-estimate interval does not establish adequate evidence for every dose inside it. Report the two boundaries, their uncertainty and dependence, the decision criteria, and the method used to combine them.