Why a Selected Dose Needs an Uncertainty Summary
Dose selection often produces a concise recommendation: for example, advance 60 mg once daily. That number can be useful, but by itself it hides how stable the decision is. Nearby doses might have almost the same expected benefit, the estimated best dose might vary substantially across plausible data sets, or the recommended dose might only sometimes meet an efficacy or safety criterion. A complete report therefore describes not only which dose was selected, but also how uncertain the selection is and what that uncertainty means for the decision.
Uncertainty about a selected dose is not the same as interindividual variability. Interindividual variability describes differences in responses or exposures among patients at a given dose. Decision uncertainty describes how the recommended dose or its predicted performance could change because the evidence and model parameters are uncertain. Both can matter, but they answer different questions and should not be combined into a single unexplained range.
The quantity being summarized must be named. An interval for the dose maximizing estimated utility is not an interval for the range of doses that meet a target, and neither is an interval for patient responses. Likewise, a probability that a dose meets a criterion needs a precise event definition: for example, “the probability that the population mean benefit exceeds 10 points” differs from “the probability that an individual patient improves by at least 10 points.”
Intervals for a Selected Dose
A confidence interval can summarize repeated-sampling uncertainty in a parameter such as the dose maximizing a specified utility. Its interpretation relies on the method’s assumptions, including an adequate model, appropriate sampling behavior, and a valid calculation of uncertainty. A 95% confidence procedure has 95% coverage in repeated applications under those assumptions. After observing the data, it is not correct to say that the fixed parameter has a 95% probability of lying in the particular interval.
A Bayesian credible interval instead summarizes posterior uncertainty conditional on the data, model, and prior. A 95% posterior credible interval contains 95% of the posterior probability for the specified quantity. That interpretation is conditional on the modeling and prior assumptions; it is not an assumption-free statement about long-run coverage.
For dose selection, the interval should match the quantity and selection procedure. A confidence interval for the response at a dose does not automatically quantify uncertainty in the dose that maximizes utility. If the target is \(d^*\), uncertainty should be computed for \(d^*\), for example by applying a valid uncertainty method to the model’s optimizer. A normal approximation based on the delta method can be useful when the optimizer is a smooth function of well-estimated model parameters and the optimum is not near a boundary. Bootstrap or posterior draws can be more informative when the optimizer is irregular, bounded, or concentrated at several candidate doses; their validity still depends on how the resampling or probability model represents the data-generating process.
Worked Example: An Interval for the Utility-Maximizing Dose
In a hypothetical dose-ranging analysis, a smooth utility model gives an estimated optimum of 60 mg. The estimated standard error of this optimizer, calculated from the fitted model’s parameter covariance using the delta method, is 12 mg. Assume the optimizer is sufficiently far from the allowable-dose boundaries, the model is adequate, and the sampling distribution is approximately normal.
Under these assumptions, an approximate 95% confidence interval is the estimate plus or minus 1.96 standard errors:
The margin of error is \(1.96\times12=23.52\) mg, so the interval is \((60-23.52,\ 60+23.52)=(36.48,\ 83.52)\) mg, or about 36.5–83.5 mg. The interval indicates substantial uncertainty about the location of the optimum: it does not establish that every dose in the interval is equally effective, or that each dose in it meets a clinical target. Its frequentist interpretation is that the interval procedure would cover the fixed optimum in about 95% of repeated comparable studies if the assumptions hold.
The symmetry and normal approximation deserve checking. If the optimum were close to a dose boundary, or if estimated utility were nearly flat across a broad range, a symmetric interval could be misleading. In that setting, report a suitable bootstrap or posterior summary and show how its bounds relate to the allowable dose range.
Probabilities of Meeting a Decision Criterion
A probability can answer a more direct decision question than an interval: how likely is a candidate dose to meet a prespecified criterion? The event must be defined in terms of the intended estimand. For example, it might be that the population mean treatment benefit exceeds a clinically meaningful threshold. Under a Bayesian analysis, the probability is calculated from the posterior distribution of that estimand. Under a frequentist analysis, uncertainty can instead be expressed through a confidence bound or a procedure designed to control a specified error rate; these are not interchangeable interpretations.
When a decision requires both efficacy and safety conditions, calculate the probability of their joint occurrence rather than multiplying two marginal probabilities unless independence has been justified. Efficacy and safety outcomes may be related, and joint posterior draws preserve that dependence. Report the threshold values, the population or patient-level quantity to which they apply, and whether the probability is conditional on a particular model.
Worked Example: Probability of Meeting an Efficacy Target
Suppose, hypothetically, the posterior distribution for population mean benefit at 60 mg is \(N(16,6^2)\) benefit points. The prespecified target is a mean benefit of at least 10 points. The posterior probability of meeting the target is the probability that this normal random variable is at least 10:
Thus, conditional on this model, prior, and data, the posterior probability that the population mean benefit at 60 mg reaches 10 points is about 84.1%. This is not the probability that 84.1% of patients respond, nor does it show that the dose is safe. If the decision also requires a safety limit, that event must be evaluated as well.
For probabilities estimated by simulation, Monte Carlo error should be small enough that it does not obscure the decision. If \(M\) posterior draws are used and \(K\) meet the criterion, the estimated probability is \(K/M\). Its approximate Monte Carlo standard error is \(\sqrt{\hat{p}(1-\hat{p})/M}\), where \(\hat{p}=K/M\). This measures simulation noise, not uncertainty in the underlying clinical quantity; the posterior distribution already represents the latter under the model.
Worked Example: Probability of Meeting Joint Efficacy and Safety Criteria
In a hypothetical Bayesian analysis at 60 mg, 1,000 paired posterior draws are used. A draw meets the efficacy criterion if the population mean benefit is at least 10 points, and meets the safety criterion if the population risk of a specified adverse event is no more than 8%. Of the 1,000 draws, 720 meet efficacy, 910 meet safety, and 680 meet both.
The estimated posterior probabilities are \(720/1000=0.720\) for efficacy, \(910/1000=0.910\) for safety, and \(680/1000=0.680\) for meeting both criteria. The joint probability is calculated directly from the paired draws. Multiplying the marginal probabilities would give \(0.720\times0.910=0.6552\), which differs from 0.680 and would only be justified if the two criterion events were independent under the posterior.
For the estimated joint probability, the approximate Monte Carlo standard error is \( \sqrt{0.680(1-0.680)/1000}=\sqrt{0.0002176}\approx0.01475\). Thus, simulation noise is about 1.5 percentage points in standard-error units. The result says that about 68.0% of posterior draws satisfy both stated population-level criteria, conditional on the analysis. It does not mean that 68% of treated patients will benefit safely.
Ranking Candidate Doses
When several doses have similar utility, reporting only the dose with the largest estimated utility can make a small numerical difference look decisive. Ranking probabilities retain information across uncertainty draws. For each draw, compute utility at every candidate dose and identify the dose with the highest utility. The proportion of draws in which a dose ranks first estimates its probability of being best under the posterior analysis. Similar summaries can report the probability of being among the top two or of outperforming a clinically relevant comparator.
A rank probability is not the probability that a dose is clinically worthwhile. A dose can rank first in nearly every draw while all doses have low absolute utility; conversely, several doses can meet the target even though only one ranks first in each draw. Therefore, rankings should accompany absolute performance summaries, such as probabilities of meeting criteria, rather than replace them.
Worked Example: Reporting First-Rank Probabilities
For a hypothetical analysis, utilities at 40, 60, and 80 mg are calculated in each of 1,000 posterior draws. The dose with the largest utility in each draw is assigned rank one. The counts are 180, 520, and 300 draws, respectively.
The first-rank probabilities are \(180/1000=18.0\%\) for 40 mg, \(520/1000=52.0\%\) for 60 mg, and \(300/1000=30.0\%\) for 80 mg. They sum to 100%, because each draw assigns the first rank to one dose. The 60 mg dose ranks first most often, but does so in only 52.0% of draws. Describing it as an unambiguous optimum would overstate the evidence. A clear report would state that 60 mg had the highest first-rank probability, while 80 mg ranked first in 30.0% of draws, and would also present the doses’ probabilities of meeting the clinical criteria.
How to Build a Decision-Focused Summary
A useful communication sequence keeps the decision target visible and prevents distinct uncertainty summaries from being conflated.
Specify whether the target is the optimal dose, utility at a dose, or an event such as meeting both an efficacy threshold and a safety limit. Give the criteria and units.
Use an interval for uncertainty in a continuous quantity, a probability for a defined event, and ranking probabilities for relative ordering across candidates. Identify whether the summary is frequentist or Bayesian.
For intervals, check the approximation or resampling method. For simulated probabilities and rankings, confirm adequate draws and report Monte Carlo error when relevant. Ensure joint criteria are evaluated jointly when dependence matters.
Explain what the evidence favors, where plausible alternatives remain, and whether the leading dose meets absolute clinical criteria. Keep patient-level variability separate from uncertainty in the population decision.
Diagnostics, Alternatives, and Limitations
Several signs should prompt caution. An estimated optimum at the lowest or highest tested dose may indicate that the data do not locate an interior optimum; a symmetric interval can then extend beyond feasible doses or hide boundary behavior. A very broad interval may reflect a flat utility curve: in that case, the data may support a range of practically similar doses rather than a precise optimum. If small changes in model fits or resampling produce abrupt changes in the selected dose, the apparent point recommendation is unstable. Check whether simulations converge, whether draws cover the dose domain, and whether all candidate doses are evaluated consistently.
The main summaries answer different questions. A confidence interval emphasizes repeated-sampling coverage under its assumptions; a Bayesian credible interval expresses posterior uncertainty conditional on the model and prior. A probability of meeting a criterion is directly decision-focused but depends on a precisely stated event. A ranking probability compares candidates but does not measure how much better the leading dose is. These approaches are complementary, not competing substitutes for one another.
All summaries inherit limitations from the evidence and model. They cannot compensate for an inappropriate dose-response model, weak information at relevant doses, or a criterion that does not reflect the clinical decision. Posterior probabilities depend on the prior and likelihood; confidence procedures can have poor coverage when their approximations fail. Monte Carlo precision does not eliminate model uncertainty. Finally, a population-level probability does not describe every patient’s outcome: patient-level prediction requires a quantity that includes interindividual variability.
Check Your Understanding
Answer each question using the definitions and distinctions in this tutorial.
- A 95% confidence interval for the utility-maximizing dose is 36.5–83.5 mg. What does the 95% refer to?
- In 500 posterior draws, a dose meets an efficacy target in 350 draws and a safety criterion in 400 draws. What additional count is needed to calculate the probability it meets both?
- One dose ranks first in 70% of draws. Does this establish that it meets the clinical target in 70% of patients?
- Name one situation in which a symmetric interval for the optimum may be misleading.
Solutions. (1) Under the confidence procedure’s assumptions, intervals constructed in repeated comparable studies would contain the fixed optimum about 95% of the time; it is not a 95% probability statement about this particular fixed optimum. (2) Count how many of the same draws meet both criteria, then divide that count by 500; the marginal counts alone do not determine the joint probability. (3) No. The first-rank probability describes how often the dose has the highest utility among candidates in the posterior draws, not the proportion of patients meeting a target. (4) For example, if the optimum is near a dose boundary or utility is nearly flat, a symmetric interval may not describe the uncertainty well.