One Candidate Set, Several Legitimate Answers
A dose comparison can be numerically correct and still produce different recommendations depending on the question being asked. The dose with the largest average response need not be the dose most likely to reach a clinically relevant target. Neither need be the dose with the greatest benefit after accounting for harm. These are not competing estimates of one universal “best dose”; they are answers to different decision objectives.
This tutorial compares those objectives across a fixed set of candidate doses. The calculations use hypothetical population predictions, treated as known for illustration. They do not account for uncertainty in the predictions, and the resulting rankings should not be interpreted as evidence that one dose is best for every patient or clinical setting.
The target \(T\), the definition of harm, and the utility function must be specified before interpreting the rankings. In particular, a target-attainment probability for benefit alone does not automatically incorporate safety. If harm is a separate concern, it must enter through a joint utility or through a separately defined safety constraint.
What Each Objective Rewards
The mean-response objective summarizes the distribution of benefit at each dose with one number. It answers, “Which candidate has the greatest expected response?” This can be appropriate when average response is the decision-relevant goal and the response scale is meaningful for averaging. It does not show how responses are distributed around the average.
Target attainment instead asks how often the response meets or exceeds a specified threshold. It depends on the entire response distribution, not just its mean. A dose can have a smaller mean but a greater probability of crossing the target if its distribution places more probability just above that threshold. Conversely, a large mean can be driven by high responses in some individuals while many remain below target.
A joint utility objective assigns value to benefit and harm on a common decision scale. For example, a utility function might reward higher response and penalize a serious adverse event. The resulting expectation averages that combined value over the outcomes at each dose. Its ranking depends not only on the outcome predictions but also on the utility function, which represents the consequences and preferences relevant to the decision.
Worked Example: Selecting the Dose With the Greatest Mean Response
Suppose four hypothetical candidate doses have the following population response distributions. Within each row, the listed response values and probabilities describe the response for a randomly selected patient at that dose. Response is measured in points on the same scale at every dose.
| Dose | Response distribution |
|---|---|
| 10 mg | 5 points with probability 0.80; 9 points with probability 0.20 |
| 20 mg | 6 points with probability 0.40; 9 points with probability 0.60 |
| 30 mg | 8 points with probability 0.50; 10 points with probability 0.50 |
| 40 mg | 7 points with probability 0.20; 10 points with probability 0.80 |
For a discrete response distribution, the mean is the sum of each possible response multiplied by its probability. At 10 mg:
The same calculation gives:
Among these four candidates, 40 mg has the largest mean response, 9.40 points, so it is selected by the mean-response objective. This conclusion says nothing by itself about how many patients reach a particular target or about the harms at 40 mg.
Target Attainment Uses the Whole Distribution
For a prespecified target \(T=8\) response points, target attainment at dose \(d_j\) is the sum of the probabilities of all response values at least 8. This is a tail probability. The calculation includes a response exactly equal to 8 because the target is defined using \(B\geq T\).
Worked Example: Selecting the Dose Most Likely to Reach Target
Use the same hypothetical response distributions and set the clinically relevant target at \(T=8\) points. At 10 mg, only the response of 9 reaches the target, so:
At 20 mg, only the response of 9 qualifies, giving \(P(B\geq8\mid20\text{ mg})=0.60\). At 30 mg, both possible responses, 8 and 10, qualify, so \(P(B\geq8\mid30\text{ mg})=0.50+0.50=1.00\). At 40 mg, the response of 10 qualifies but the response of 7 does not, giving \(P(B\geq8\mid40\text{ mg})=0.80\).
Thus the target-attainment probabilities are 0.20, 0.60, 1.00, and 0.80 at 10, 20, 30, and 40 mg, respectively. The target-attainment objective selects 30 mg. This is a different answer from the mean-response objective, which selected 40 mg.
The reason is visible in the distributions. At 40 mg, the 20% probability of a 7-point response lies below target, even though the high responses raise the mean to 9.40. At 30 mg, every response is at or above 8. The target choice is conditional on the threshold being clinically justified: changing \(T\) can change the ranking.
Joint Benefit-Risk Utility
For a joint objective, the benefit and harm outcomes must be placed on a common utility scale. For a clear numerical comparison, suppose hypothetical stakeholders have specified the following utility for a patient with benefit \(B\) points and a serious adverse event indicator \(H\), where \(H=1\) if the event occurs and \(H=0\) otherwise:
Under this illustrative preference, one serious adverse event carries a utility penalty of 25 response-point equivalents. This penalty is a preference input, not a biological constant. The additive form also assumes that the event penalty is the same at every benefit level. If consequences depend on the combination of benefit and harm, an interaction or a more detailed utility function would be needed.
The expected utility follows directly from linearity of expectation:
The last equality uses the fact that \(H\) is an indicator: its expectation equals the probability that the event occurs. No independence assumption between benefit and harm is required for this additive utility calculation. However, if the utility assigned to a patient depends on both outcomes jointly in a non-additive way, marginal means and harm probabilities may not be enough; the joint outcome distribution is then needed.
Worked Example: Selecting the Dose With the Greatest Expected Utility
Suppose the hypothetical probabilities of a serious adverse event are 0.03, 0.08, 0.18, and 0.30 at doses 10, 20, 30, and 40 mg. Combine these with the mean responses already calculated and the stated utility \(U(B,H)=B-25H\). For 10 mg:
The corresponding calculations for the other candidates are:
The expected utilities are measured in response-point equivalents under the specified utility scale. Among these candidates, 20 mg has the largest expected utility, 5.80, and is selected by this joint benefit-risk objective. The result is not a claim that 20 mg has the greatest mean benefit: its mean is 7.80 points, below the means at 30 and 40 mg. It is the best of these candidates according to the particular stated trade-off between benefit and harm.
Comparing the Decisions and Their Assumptions
| Objective | Quantity compared | Selected dose in the example | Question answered |
|---|---|---|---|
| Mean response | \(E(B\mid d_j)\) | 40 mg | Which candidate has the greatest average benefit? |
| Target attainment | \(P(B\geq8\mid d_j)\) | 30 mg | Which candidate most often reaches the chosen benefit target? |
| Joint expected utility | \(E\{B-25H\mid d_j\}\) | 20 mg | Which candidate has the greatest average value under the stated benefit-harm trade-off? |
The comparison makes the role of the objective explicit. The mean-response calculation treats the average benefit as decisive. Target attainment emphasizes the proportion meeting a threshold, irrespective of how far above target a response lies. The utility calculation incorporates harm, but its ranking depends on how harm is valued relative to benefit. These objectives need not agree, and none should be described as universally correct without reference to the clinical decision.
The calculations also rely on conditions that should be checked before applying them. Predictions must refer to the same patient population, outcome definition, follow-up time, and dosing context across candidates. The target must be specified on the same response scale used in the distributions. Harm probabilities must refer to a consistent event definition and time horizon. Finally, utility weights should reflect an explicit preference or decision rule, rather than being chosen solely because they produce a desired dose.
Derivation and Checks Before Choosing
The separation between a mean and a target probability can be stated formally. The mean is a probability-weighted sum of all response values, whereas target attainment sums the probabilities of only those values in the target region. For the 30 mg and 40 mg candidates, the arithmetic is:
Both inequalities follow from the given distributions, so there is no contradiction in the rankings. The first compares weighted response values; the second compares probability mass above a threshold. A mean alone cannot recover the target-attainment probability without additional distributional information.
Before reporting a recommendation, check the arithmetic and the inputs: probabilities within each outcome distribution should sum to 1; each target event should include exactly the outcomes satisfying the stated inequality; and expected utility should use the stated utility for each possible outcome. Then confirm that the ranking answers the intended question. A sensitivity check can show whether a utility-based choice changes under plausible alternative preference weights or harm probabilities, but it does not replace eliciting a defensible utility scale.
These fixed-candidate calculations are also limited by the candidate set and predictions. The selected dose is only the best among doses evaluated; a dose between candidates could perform differently. If predictions are uncertain, a ranking based on point predictions may be unstable. This comparison holds predicted quantities fixed to isolate how objectives differ; uncertainty analysis is a separate layer of evidence, not something captured by these numerical rankings.
Common Mistake
Wrong reasoning: “The mean response is greatest at 40 mg, so 40 mg must be the best dose; a lower target-attainment probability or greater harm risk cannot change that.” This treats one objective as if it automatically answered every decision question.
Correction: State the objective first. If the goal is greatest average response, 40 mg wins in this example. If it is greatest probability of reaching 8 points, 30 mg wins. If the stated utility places a 25-point-equivalent penalty on a serious event, 20 mg wins. A recommendation is interpretable only alongside the objective and its assumptions.
Check Your Understanding
Answer each question using the definitions and calculations above. Solutions are included after the questions.
- If the target in the hypothetical example were \(T=10\) points instead of 8, what would the target-attainment probabilities be at 10, 20, 30, and 40 mg?
- Why can the 40 mg dose have a larger mean than 30 mg but a lower target-attainment probability when \(T=8\)?
- If the serious-event penalty were 10 rather than 25 response-point equivalents, calculate expected utility at 20 and 40 mg. Which of those two candidates would have the larger expected utility?
- Does the additive utility calculation require benefit and harm to be independent? Explain briefly.
Solutions. (1) Only a response of at least 10 reaches this target. The probabilities are 0 at 10 mg, 0 at 20 mg, 0.50 at 30 mg, and 0.80 at 40 mg. (2) The 40 mg distribution has a 20% chance of a 7-point response, below 8, but its 80% chance of a 10-point response raises its mean to 9.40. Every 30 mg response is at least 8, so its target probability is 1.00 even though its mean is 9.00. (3) At 20 mg, expected utility is \(7.80-10(0.08)=7.00\). At 40 mg, it is \(9.40-10(0.30)=6.40\). Among these two, 20 mg has the larger expected utility. (4) No. For additive utility \(B-cH\), linearity of expectation gives \(E(B-cH)=E(B)-cE(H)\) without an independence assumption. A non-additive utility that depends on the combination of benefit and harm may require their joint distribution.