From Response Targets to Dose Attainment
Locating a minimum effective dose is not always enough to make a dose decision. A candidate may have a promising predicted efficacy response but also a meaningful chance of exceeding a safety limit. Target-attainment analysis makes the decision rule explicit: for each candidate dose, estimate the probability that efficacy reaches its prespecified target and safety remains within its prespecified limit.
The calculation depends on defining what “attainment” means before comparing doses. Specify the efficacy endpoint and its assessment time, the minimum acceptable efficacy value, the safety endpoint and its assessment time, and the direction of the safety limit. The endpoints and thresholds must correspond to the intended clinical decision; changing them can change which dose qualifies.
The probability is meaningful only relative to a specified predictive distribution. That distribution may represent outcomes for future individuals, uncertainty about population-level predictions, or both, depending on the estimand and the model simulation. State which source or sources of uncertainty are included. A probability for a population mean exceeding a target is not the same quantity as the probability that an individual patient’s outcome exceeds it.
In practice, the probability is often estimated by simulation. For each simulated draw, use the same draw to obtain both efficacy and safety outcomes at the candidate dose, then record whether both criteria are met. The paired outcomes matter: the joint event is determined by their joint distribution, not just their separate distributions.
Why the Joint Probability Matters
Write \(A_E(d)=\{Y_E(d)\geq E^*\}\) and \(A_S(d)=\{Y_S(d)<S^*\}\). Then \(A(d)=A_E(d)\cap A_S(d)\), so $$ P(A(d))=P(A_E(d))P(A_S(d)\mid A_E(d)). $$ This identity follows from the definition of conditional probability when \(P(A_E(d))>0\). It shows why multiplying the two marginal probabilities is valid only when the efficacy and safety attainment events are independent. That assumption is often implausible: exposure or individual susceptibility may affect both outcomes.
A useful probability bound follows from the union bound. Since failure to attain the joint target means failing at least one component, $$ P(A(d))=1-P(A_E(d)^c\cup A_S(d)^c) \geq 1-P(A_E(d)^c)-P(A_S(d)^c) =P(A_E(d))+P(A_S(d))-1. $$ Also, the intersection cannot be more probable than either event alone. Thus $$ \max\{0,\;P(A_E(d))+P(A_S(d))-1\} \leq P(A(d))\leq \min\{P(A_E(d)),P(A_S(d))\}. $$ These bounds can be useful as a quick consistency check. They do not replace estimating the joint probability when paired model predictions are available.
If a fitted model supplies a joint predictive density \(f_d(e,s)\), the target probability can also be expressed as an integral over the target region: $$ p_A(d)=\int_{E^*}^{\infty}\int_{-\infty}^{S^*} f_d(e,s)\,ds\,de. $$ Simulation approximates this integral by counting draws in the region. The simulation approach accommodates a range of model distributions, but the simulated draws must preserve the efficacy-safety dependence implied by the model.
Worked Example: Estimating Joint Attainment at One Dose
Suppose a hypothetical model generates \(M=1{,}000\) paired predictive draws at a candidate dose of 20 mg. Efficacy is a reduction in a symptom score, in points, and the target is at least 10 points. The safety outcome is a biomarker increase, in units, and the limit is less than 5 units. Of the 1,000 paired draws, 760 meet the efficacy target, 900 remain below the safety limit, and 700 meet both.
The joint attainment estimate is the number meeting both conditions divided by the number of paired draws: $$ \widehat{p}_A(20)=\frac{700}{1{,}000}=0.700. $$ Thus, under the predictive distribution used in this hypothetical analysis, an estimated 70.0% of outcomes at 20 mg meet the efficacy target while remaining below the safety limit.
The marginal estimates are \(760/1{,}000=0.760\) for efficacy and \(900/1{,}000=0.900\) for safety. Their product is \(0.760\times0.900=0.684\), not the observed joint estimate of 0.700. The product would estimate the joint probability only under independence. For a rough measure of simulation precision, the Monte Carlo standard error of the joint estimate is $$ \sqrt{\frac{0.700(1-0.700)}{1{,}000}}=0.01449. $$ This describes simulation error for the estimated probability, not uncertainty in the clinical model itself.
Turning Probabilities Into an Explicit Dose Rule
A probability estimate alone does not select a dose. A decision rule must specify how much attainment is required and how safety is treated. One transparent rule is to require the estimated joint target-attainment probability to meet a prespecified minimum, \(q\). If an additional safety criterion is wanted, it must also be stated explicitly—for example, requiring the probability of staying below the safety limit to exceed a separate minimum.
The threshold \(q\) is a decision criterion, not a universal statistical constant. It should be chosen for the clinical context and specified before comparing candidate doses. A rule might select the lowest tested dose that meets the joint probability threshold and any additional safety condition. This is a rule for choosing among the candidate doses assessed; it does not, by itself, justify interpolation to an untested dose.
Worked Example: Applying a Prespecified Attainment Criterion
Consider four hypothetical candidate doses. For each dose, 1,000 paired predictive draws were generated. The efficacy target and safety limit are the same at every dose. The prespecified rule is to require joint attainment of at least 0.70 and a safety-limit attainment probability of at least 0.85; among qualifying candidates, choose the lowest tested dose.
| Dose | Meets efficacy target | Below safety limit | Meets both | Joint estimate |
|---|---|---|---|---|
| 10 mg | 620/1,000 | 970/1,000 | 610/1,000 | 0.610 |
| 20 mg | 820/1,000 | 900/1,000 | 760/1,000 | 0.760 |
| 30 mg | 930/1,000 | 780/1,000 | 740/1,000 | 0.740 |
| 40 mg | 960/1,000 | 650/1,000 | 630/1,000 | 0.630 |
For example, the 20 mg joint estimate is \(760/1{,}000=0.760\), and its safety estimate is \(900/1{,}000=0.900\). It meets both requirements. At 10 mg, the safety estimate is \(0.970\), but the joint estimate is only \(0.610\), below the required 0.70. At 30 mg, joint attainment is \(0.740\), but only \(0.780\) of draws remain below the safety limit, short of the required 0.85. The 40 mg dose fails both criteria.
The 20 mg dose is the lowest tested dose that satisfies both parts of the stated rule, so it is selected under that rule. The result is conditional on the thresholds, predictive model, and candidate doses specified; it is not a claim that 20 mg is optimal under every possible objective.
Assumptions, Diagnostics, and Alternatives
A target-attainment estimate is only as relevant as its inputs. Check that the efficacy and safety endpoints, units, assessment times, and threshold directions are correctly implemented. Confirm that model predictions cover the candidate dose and the target population. If predictions are simulated, verify that each simulated efficacy outcome is paired with the corresponding safety outcome, and that the simulation includes the uncertainty required by the chosen estimand.
Simulation diagnostics include checking that the number of draws is adequate for stable estimates, especially when attainment is rare or close to the decision threshold. Repeat the calculation with more draws and compare estimates; large changes suggest Monte Carlo error may affect classification. Inspect marginal probabilities alongside the joint probability and check that the joint estimate lies within the probability bounds derived above. Also inspect the predicted outcome distributions and model fit relevant to each endpoint. A precise simulation cannot correct misspecification or extrapolation beyond informative data.
A common alternative is to compare predicted mean efficacy with the efficacy target and predicted mean safety with the safety limit. This is simpler, but it answers a different question: means may satisfy both limits even when a substantial fraction of predictive outcomes fail one or both. Marginal attainment probabilities are more informative than mean comparisons about each endpoint separately, but still do not determine joint attainment without dependence information. Joint simulation directly addresses the “both conditions” question.
Limitations remain. Results depend on model assumptions, target definitions, the predictive population, and which uncertainty sources are represented. A single probability threshold can conceal how close a dose is to qualifying and how uncertain the estimate is. When estimates are near the decision boundary, report that fact and examine the consequences of plausible changes to assumptions rather than treating a rounded estimate as definitive. More elaborate decision objectives are distinct from this threshold-based attainment rule.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In 500 paired predictive draws, 380 meet the efficacy target, 450 meet the safety limit, and 350 meet both. What is the estimated joint attainment probability?
- Suppose the marginal attainment probabilities are 0.75 for efficacy and 0.80 for safety. Give the lower and upper probability bounds for joint attainment.
- A dose has joint attainment estimate 0.82, but its prespecified minimum is 0.85. Does it qualify under a rule requiring an estimate of at least 0.85?
- Why is it not enough to check that predicted mean efficacy exceeds its target and predicted mean safety is below its limit?
Solutions. (1) The joint estimate is \(350/500=0.700\), or 70.0%. (2) The lower bound is \(\max(0,0.75+0.80-1)=0.55\); the upper bound is \(\min(0.75,0.80)=0.75\). Therefore, the joint probability must lie between 0.55 and 0.75, inclusive. (3) No. Since \(0.82<0.85\), the estimate does not meet the stated criterion. This is a rule-based classification, not proof that the true probability is below 0.85. (4) Means do not reveal how often individual predictive outcomes cross either threshold, nor the probability that both conditions hold for the same outcome. Estimate the joint attainment probability from the relevant joint predictive distribution.