Why Target Uncertainty Changes Dose Justification
A dose can appear adequate or inadequate depending on which exposure target is used. The target may be uncertain because the exposure-response evidence supports more than one clinically plausible threshold, or because the evidence does not identify a single value precisely. In either case, dose justification should show how conclusions change across alternatives rather than conceal the uncertainty behind one selected number.
A different problem arises when exposure must be high enough for efficacy but low enough to limit safety risk. A dose that improves efficacy attainment may also increase the probability of exceeding a safety limit. The decision is then about meeting both goals—not about reporting the efficacy and safety percentages separately and assuming they imply a particular level of joint success.
This builds on the earlier material on defining thresholds, windows, and joint targets: the event being assessed must be stated precisely. Target uncertainty is not the same as variability in individual exposure. Nor is either one identical to uncertainty in estimated model parameters. A target-scenario analysis asks how the decision changes when the clinically plausible target definition changes; it does not by itself quantify every source of uncertainty.
Prespecify Alternatives and Keep Their Conclusions Separate
When evidence supports multiple plausible target values, define those alternatives before comparing doses. For each scenario, state the exposure metric, threshold or window, relevant time point, and population or clinical context. Keep the scenarios plausible and tied to the evidence; arbitrary values do not become informative merely because they are arranged in a sensitivity table.
Report the dose results separately for each scenario. If a dose meets a decision benchmark under one target but not another, that is the result: the dose justification depends on target choice. Do not average the thresholds or the scenario-specific attainment probabilities unless a defensible probability distribution over target values and an appropriate decision rule have been specified. A simple average would otherwise give unsupported weights to competing clinical interpretations.
Worked Example: Two Plausible Efficacy Thresholds
Suppose a hypothetical exposure analysis considers trough concentration \(C_{\min}\) in mg/L. Two prespecified efficacy scenarios are \(C_{\min}\geq 1.0\) mg/L and \(C_{\min}\geq 1.2\) mg/L. The following hypothetical values are the proportions of the target population meeting each threshold at each dose.
| Dose | At least 1.0 mg/L | At least 1.2 mg/L |
|---|---|---|
| 100 mg | 0.76 | 0.61 |
| 150 mg | 0.88 | 0.78 |
| 200 mg | 0.95 | 0.90 |
For illustration, suppose the decision team has prespecified 0.80 as an operational benchmark for this analysis. Under the 1.0 mg/L scenario, 150 mg exceeds it because \(0.88\geq 0.80\). Under the 1.2 mg/L scenario, 150 mg does not, because \(0.78<0.80\). The 200 mg dose exceeds the benchmark in both scenarios: \(0.95\geq 0.80\) and \(0.90\geq 0.80\).
The conclusion is not that 150 mg is universally adequate or inadequate. Its status depends on which clinically plausible threshold is accepted. The analysis makes that dependence visible; it does not determine which threshold is clinically correct.
Competing Thresholds on the Same Exposure
Consider an efficacy requirement \(X_D\geq L\) and a safety requirement \(X_D\leq U\), where \(X_D\) is the same exposure metric at dose \(D\), and \(L\leq U\). Define \(q_E=P(X_D\geq L)\), \(q_S=P(X_D\leq U)\), and \(q_J=P(L\leq X_D\leq U)\). Here \(q_E\) and \(q_S\) are marginal attainment probabilities, while \(q_J\) is the probability of meeting both requirements.
The union covers every possible value of \(X_D\): a value either is at least \(L\), or is below \(L\) and therefore no greater than \(U\), since \(L\leq U\). Inclusion–exclusion then gives an exact relationship between the marginal and joint probabilities.
This identity applies because the requirements use the same exposure metric and compatible thresholds. If \(L>U\), no value can satisfy both, so \(q_J=0\). If efficacy and safety use different exposure metrics, such as AUC and \(C_{\max}\), the identity generally does not apply: the joint probability depends on how those metrics co-occur in the same individuals.
Worked Example: A Shared Exposure Window
Suppose a hypothetical assessment uses one exposure measure \(X_D\) in ng/mL, with an efficacy threshold of \(L=50\) ng/mL and a safety limit of \(U=100\) ng/mL. The hypothetical marginal probabilities at four doses are shown below. Because \(50\leq100\), the joint probability is calculated as \(q_J=q_E+q_S-1\) at each dose.
| Dose | \(q_E\) | \(q_S\) | Joint calculation | \(q_J\) |
|---|---|---|---|---|
| 100 mg | 0.72 | 0.98 | 0.72 + 0.98 − 1 | 0.70 |
| 150 mg | 0.86 | 0.94 | 0.86 + 0.94 − 1 | 0.80 |
| 200 mg | 0.91 | 0.90 | 0.91 + 0.90 − 1 | 0.81 |
| 250 mg | 0.96 | 0.82 | 0.96 + 0.82 − 1 | 0.78 |
For example, at 200 mg, \(q_J=0.91+0.90-1=0.81\), or 81%. The joint value is not 84%; it must agree with the two marginals under the stated same-exposure events. At 250 mg, efficacy attainment is higher, but safety attainment is lower, and joint attainment is 78%. The result illustrates why neither the efficacy percentage nor the safety percentage alone describes success on both requirements.
If the prespecified operational benchmark were 0.80, the joint results at 150 mg and 200 mg would meet or exceed it, while those at 100 mg and 250 mg would not. This comparison is conditional on the hypothetical inputs and the specified benchmark; it is not a universal dosing rule.
Different Metrics Require Joint Information
When efficacy and safety depend on different exposure metrics, their marginal probabilities do not usually identify the probability of meeting both. For events \(E\) and \(S\), the general bounds are
The upper bound follows because the intersection is a subset of each event. The lower bound follows from \(P(E\cup S)\leq1\) and inclusion–exclusion. These bounds can be informative, but they may be too wide to support a precise dose conclusion. Multiplying the marginal probabilities is justified only if independence of the two events is defensible; it should not be used as a default.
Worked Example: AUC Efficacy and Cmax Safety
Suppose a hypothetical assessment finds that 88% of individuals meet an AUC-based efficacy target and 90% meet a \(C_{\max}\)-based safety limit. Write \(E\) for meeting the efficacy target and \(S\) for meeting the safety limit. The marginals alone bound joint attainment as follows:
Thus the joint probability could be anywhere from 78% to 88% based on these marginals alone. If paired hypothetical results for the same individuals gave \(P(E\cap S)=0.84\), that value would be within the bounds and could be reported as 84%. By contrast, assuming independence would produce \(0.88\times0.90=0.792\), or 79.2%; that is a different result and requires a separate independence assumption. Marginal results by themselves do not establish it.
Diagnostics, Alternatives, and Limitations
A sound analysis makes its target assumptions auditable. Check that each alternative is clinically plausible, that its exposure metric and units are explicit, and that the efficacy and safety events correspond to the same population and dose when a joint result is reported. For same-exposure compatible thresholds, verify the identity \(q_J=q_E+q_S-1\); a discrepancy indicates inconsistent inputs, events, or calculations. Also check that all reported probabilities lie between 0 and 1.
For different metrics, use paired joint outcomes from a coherent analysis rather than reconstructing the joint probability from marginals. If only marginals are available, report the bounds and state the limitation. A joint probability outside those bounds signals an arithmetic or definition error. A probability below zero or above one is likewise impossible.
Three approaches to target uncertainty answer different questions:
- Scenario analysis reports results separately under prespecified plausible targets. It is transparent and does not require assigning probabilities to the alternatives.
- Probabilistic target uncertainty can summarize over alternatives only when a defensible probability distribution over target values and a decision framework are available. The result depends on those choices.
- Robust assessment asks whether the dose conclusion holds across all specified plausible scenarios. This is conservative when scenarios are credible, but it does not show which scenario is more likely.
None of these approaches resolves weak exposure-response evidence or guarantees that a selected target is clinically valid. A scenario analysis can show the consequences of uncertainty, not remove it. Target definitions may also omit other clinically important outcomes, and a dose that satisfies a modeled exposure requirement does not by itself establish a clinical benefit–risk conclusion.
A Common Mistake
A related mistake is to choose whichever target produces the preferred dose after seeing the results. Prespecifying plausible alternatives and reporting the full scenario comparison limits this form of selective interpretation. If the decision changes across scenarios, state that dependence plainly.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- At a dose, the estimated proportions meeting two plausible efficacy thresholds are 0.84 and 0.73. What should a scenario analysis report, and why should it not simply average the two values?
- For one exposure metric, let \(L=40\), \(U=90\), \(q_E=0.92\), and \(q_S=0.87\). Find the joint probability of meeting both thresholds.
- For AUC efficacy and \(C_{\max}\) safety, suppose the marginal probabilities are 0.85 and 0.80. Give the bounds on joint attainment. Is the joint probability necessarily 0.68?
- For a single exposure metric, what is the joint probability of meeting \(X_D\geq 110\) and \(X_D\leq 100\)? Explain.
Solutions. (1) Report 0.84 and 0.73 separately for their respective prespecified thresholds. Their simple average, \( (0.84+0.73)/2=0.785\), assigns equal weight without justification and does not represent either target scenario. (2) Since \(40\leq90\), \(q_J=q_E+q_S-1=0.92+0.87-1=0.79\), or 79%. (3) The bounds are \(\max(0,0.85+0.80-1)=0.65\) and \(\min(0.85,0.80)=0.80\), so \(0.65\leq P(E\cap S)\leq0.80\). The value 0.68 is the product under an independence assumption, not a value determined by the marginals. (4) The requirements are incompatible because \(110>100\); no exposure can meet both, so the joint probability is 0.