Two Objectives, Two Questions
A dose can produce a high average response while leaving many patients below a clinically important threshold. Another dose can have a lower average but place a larger proportion of patients just above that threshold. These are not contradictory findings: the mean and the proportion meeting a target summarize different features of the response distribution. Dose selection can therefore depend on which clinical question is being asked.
Let \(R(d)\) denote the response for a patient from the population of interest at dose \(d\), measured at a specified time and on a specified endpoint scale. Assume, for now, that larger responses are better and that a clinically defined target is \(c\). The population mean summarizes average response; the target-attainment rate summarizes the fraction of the population expected to reach or exceed \(c\). The target-attainment objective is introduced in the earlier material on Target Attainment as a Dose-Finding Objective; here the emphasis is on how its dose ranking can differ from ranking by mean response.
The probability \(q_c(d)\) is a population proportion, not an individual-level guarantee. For example, \(q_c(d)=0.70\) means that the model predicts 70% of the specified population will meet the target at dose \(d\), under the stated treatment and outcome definitions. It does not mean every patient has a 70% chance of success, nor does it describe uncertainty about the estimated rate. Parameter uncertainty can make the rate estimate uncertain in addition to the patient-to-patient variation represented by \(R(d)\).
Why the Rankings Can Differ
The mean uses the magnitude of every response, whereas the target rate classifies each response only as below or at/above \(c\). A very high response can raise the mean substantially even if it occurs in few patients. Conversely, a distribution concentrated just above \(c\) can have a high target rate without having the largest mean. The distinction can be seen directly from indicator variables.
Here \(I\{R(d)\geq c\}\) equals 1 when the response reaches the target and 0 otherwise. The target rate averages these binary indicators; the mean averages the original response values. Neither quantity generally determines the other.
For a nonnegative response, the mean can also be expressed as an integral over all thresholds. Since \(R(d)=\int_0^\infty I\{R(d)>t\}\,dt\), taking expectations gives the identity below. It shows that the mean reflects the entire survival curve across thresholds, while one target rate samples the curve at a particular threshold. Thus, ordering doses at one threshold need not order them by their overall area.
The exchange of expectation and integration is valid for a nonnegative response. If the response may be negative, an appropriate shifted or general tail-integral expression can be used; the central distinction remains that a mean aggregates response magnitudes while \(q_c(d)\) evaluates one threshold event.
Worked Example: Normal Responses Reverse the Dose Ranking
Suppose two hypothetical doses produce normally distributed responses on the same scale, with higher values better. Dose A has \(R(A)\sim N(12,25)\), and dose B has \(R(B)\sim N(11,1)\). The second argument is the variance, so their standard deviations are 5 and 1 response units, respectively. Let the target be \(c=10\).
The means are \(\mu(A)=12\) and \(\mu(B)=11\) response units. The mean-response objective therefore favors A. For a normal response, standardizing the target gives \(q_c(d)=P(R(d)\geq c)=1-\Phi((c-\mu(d))/\sigma(d))\), where \(\Phi\) is the standard normal cumulative distribution function.
Thus A has the higher mean, but B has the higher target-attainment rate: approximately 65.5% for A versus 84.1% for B. The difference is consistent with the specified distributions: A has a larger mean but much greater variability, and many responses fall below 10. The normal-model assumption is being used here to calculate the probabilities; if the model did not adequately describe the response distributions, these rates would not be reliable.
Worked Comparisons with Discrete Responses
Worked Example: A Few Large Responses Raise the Mean
Consider two hypothetical dose-response distributions. At dose A, 90% of patients have response 9 and 10% have response 30. At dose B, 60% have response 11 and 40% have response 10. All values are in response units; the target is \(c=10\).
For A, the mean is \(0.90(9)+0.10(30)=8.10+3.00=11.10\) response units. For B, it is \(0.60(11)+0.40(10)=6.60+4.00=10.60\) response units. Mean response therefore favors A.
For target attainment, count the probability mass at or above 10. At A, only the 30-unit response meets the target, so \(q_{10}(A)=0.10\). At B, both possible responses meet it, so \(q_{10}(B)=0.60+0.40=1.00\). The target-rate objective strongly favors B, even though its mean is lower. In this example the high mean at A is driven by a response of 30 in a small fraction of patients, while most patients have a response below target.
The result also depends on where the target is set. If the target were \(c=12\), A would still have \(q_{12}(A)=0.10\), while B would have \(q_{12}(B)=0\). This does not make one threshold inherently correct: the target must be chosen for its clinical meaning, rather than selected after examining which dose it favors.
Worked Example: Choosing Among Three Candidate Doses
Suppose three hypothetical candidate doses have the following response distributions, with target \(c=10\): at dose L, 20% of patients have response 20 and 80% have response 8; at dose M, 60% have response 11 and 40% have response 9; at dose H, 50% have response 10.5 and 50% have response 9.5. Responses are in common units and the target is met when the response is at least 10.
For L, \(\mu(L)=0.20(20)+0.80(8)=4.00+6.40=10.40\) units and \(q_{10}(L)=0.20\). For M, \(\mu(M)=0.60(11)+0.40(9)=6.60+3.60=10.20\) units and \(q_{10}(M)=0.60\). For H, \(\mu(H)=0.50(10.5)+0.50(9.5)=5.25+4.75=10.00\) units and \(q_{10}(H)=0.50\).
The mean ranking is L (10.40), M (10.20), then H (10.00), so a mean-response objective selects L. The target-rate ranking is M (0.60), H (0.50), then L (0.20), so a target-rate objective selects M. These calculations assume the stated distributions describe the same population at each dose and that responses equal to 10 count as meeting the target. The dose decision changes because the objectives value different features of those distributions.
Assumptions, Diagnostics, and Comparisons
A meaningful comparison requires the same estimand context for both summaries. In particular, the response definition, assessment time, target population and handling of intercurrent events must be aligned across doses. The target \(c\) and the rule for equality at the threshold should be clinically justified and specified in advance. The model must also provide a credible response distribution at each dose: the mean alone is not enough to calculate a target rate unless additional distributional assumptions are supplied.
For a continuous normally distributed response, the target-rate calculation uses the normal cumulative distribution function as in the first example. If normality is not defensible, a suitable alternative is to calculate the probability from another fitted distribution or from the model's predictive distribution. For discrete responses, sum the probabilities of all response categories that meet the target. For an empirical sample, the estimated rate is the number meeting the target divided by the sample size; an empirical rate can be unstable when the sample is small, especially near the tails or at doses with few observations.
Useful checks include comparing observed and model-predicted response distributions, examining whether dispersion and skewness are adequately captured, and checking predicted probabilities near the target against data where available. A model may fit the mean reasonably while missing the spread or tail behavior that determines \(q_c(d)\). Also examine how the ranking changes under clinically plausible target values and model assumptions. Large changes signal that the decision is sensitive to those choices; they are not a reason to choose whichever threshold gives the preferred result.
| Objective | What it summarizes | Potential limitation |
|---|---|---|
| Mean response, \(\mu(d)\) | Average response magnitude across the population | Can be influenced by a small number of extreme responses and may conceal how many patients fall short of a target |
| Target rate, \(q_c(d)\) | Proportion expected to meet one specified threshold | Does not distinguish responses below the target or responses far above it |
| Patient-level utility or another graded outcome summary | Can assign different values to multiple response levels or consequences | Requires additional definitions and assumptions about how outcomes are valued |
A target-rate objective is especially direct when the decision question is explicitly about the chance of reaching a clinically meaningful threshold. A mean-response objective is more direct when average response magnitude is the primary outcome. Neither is universally preferable. A target rate ignores how far a patient falls below or exceeds the threshold; a mean can obscure the proportion experiencing clinically inadequate response. If harms or multiple outcomes matter, either summary alone may be insufficient to represent the full decision.
A common mistake is to assume that the higher mean must imply the higher target rate because both use the same response scale. This is false: the mean weights response magnitudes, while the target rate weights only whether each response crosses \(c\). The correction is to calculate both summaries from the response distributions and interpret each against its own clinical question. Another pitfall is to treat a target rate as a measure of certainty about the estimated rate. A predicted rate of 0.70 describes population outcome variation under the model; uncertainty about the model parameters and the rate estimate must be assessed separately.
Check Your Understanding
Answer each question using the definitions and calculations above. Then compare your reasoning with the solutions.
- A hypothetical dose has \(R(d)\sim N(14,9)\), with responses in units and target \(c=12\). Calculate its mean and target-attainment rate.
- At dose X, 75% of patients have response 6 and 25% have response 22. At dose Y, 80% have response 10 and 20% have response 8. For target \(c=10\), calculate both means and target rates, and state which dose each objective favors.
- Why can the target-rate ranking change when the target threshold changes, even though the response distributions do not change?
- Suppose lower values of an endpoint are better and the clinical target is response no greater than \(c\). How should the target-rate definition be written?
Solutions. (1) The mean is \(\mu(d)=14\) response units and the standard deviation is 3. Therefore \(q_{12}(d)=1-\Phi((12-14)/3)=\Phi(0.6667)\approx0.7475\), or about 74.8%. (2) For X, the mean is \(0.75(6)+0.25(22)=4.50+5.50=10.00\) units, and the target rate is 0.25 because only response 22 reaches 10. For Y, the mean is \(0.80(10)+0.20(8)=8.00+1.60=9.60\) units, and the target rate is 0.80 because response 10 meets the inclusive target. The mean favors X; the target rate favors Y. (3) The target rate is the probability of an event defined by the threshold. Changing the threshold changes which responses count as successes, so it can change the probability and the dose ranking while leaving the means unchanged. (4) For a lower-is-better endpoint, define attainment as \(q_c(d)=P(R(d)\leq c)\), with equality included if the clinical rule says that a response exactly at \(c\) meets the target.