Estimating Posterior Probabilities of Dose Targets

From Dose Targets to Posterior Probabilities

A dose-response model can describe the expected efficacy response and the safety risk at each dose, while accounting for uncertainty in the model parameters. For dose selection, it is often useful to ask a direct question: given the observed data and the model, how probable is it that a candidate dose meets the clinical targets? Posterior draws allow this question to be answered using the full joint uncertainty in the fitted model, rather than only its most likely parameter values.

The event of interest must be defined before its probability is calculated. For example, suppose efficacy is measured as improvement from baseline, with larger values better, and safety is measured as the probability of a specified adverse event, with smaller values better. A target might require expected efficacy improvement of at least 8 points and an adverse-event probability no greater than 10%. The thresholds, endpoint scale, population, and dose regimen should be tied to the clinical objective, not chosen after inspecting which dose looks best.

Definition: Let \(E(d,\theta)\) be the model-based efficacy quantity at dose \(d\), and \(R(d,\theta)\) the model-based safety risk, where \(\theta\) denotes the dose-response model parameters. For efficacy threshold \(E_{\min}\) and maximum acceptable safety risk \(R_{\max}\), the target event is \(A(d)=\{E(d,\theta)\ge E_{\min},\ R(d,\theta)\le R_{\max}\}\). The posterior target probability is \(P(A(d)\mid y)\), where \(y\) denotes the observed data. It is the posterior probability, conditional on the model, prior, and data, that the model-based quantities at dose \(d\) satisfy both criteria.

This definition concerns uncertainty about model-based dose-response quantities. For example, \(R(d,\theta)\) may be the model-implied population probability of an adverse event. The target event then asks whether that underlying risk is below the specified limit; it does not ask whether a particular future patient will avoid the event. A probability about future patient outcomes is a posterior predictive probability and requires accounting for outcome variability as well as parameter uncertainty.

Estimating the Probability From Posterior Draws

Suppose a fitted model provides \(S\) draws \(\theta^{(1)},\ldots,\theta^{(S)}\) from the posterior distribution given the data. For each draw, calculate the model-implied efficacy and safety quantities at the candidate dose, then record whether both targets are met. The estimated probability is the fraction of draws for which the event holds.

I_s(d)= \begin{cases} 1, & E(d,\theta^{(s)})\ge E_{\min}\ \text{and}\ R(d,\theta^{(s)})\le R_{\max},\\ 0, & \text{otherwise}, \end{cases} \qquad \widehat{P}(A(d)\mid y)=\frac{1}{S}\sum_{s=1}^{S} I_s(d).

The indicator \(I_s(d)\) is one when draw \(s\) meets both requirements and zero otherwise. If posterior draws accurately represent the posterior distribution, their average indicator estimates the posterior probability of the event. This is a Monte Carlo estimate: with a finite number of draws, it has simulation error even when the underlying model and posterior computation are appropriate.

Justification: For a draw from the posterior, \(I_s(d)\) is a Bernoulli random variable with success probability \(q=P(A(d)\mid y)\). For \(S\) independent posterior draws, the sample average has expectation \(q\) and variance \(q(1-q)/S\). Replacing the unknown \(q\) by its estimate gives the approximate Monte Carlo standard error \(\sqrt{\widehat{P}(1-\widehat{P})/S}\).

That standard-error formula assumes independent draws. Markov chain Monte Carlo draws are generally autocorrelated, so the number of stored draws can overstate the amount of independent information. In that setting, assess convergence and mixing, and estimate Monte Carlo error using the effective sample size of the event-indicator sequence \(I_s(d)\), or an equivalent method that accounts for its autocorrelation. A large draw count alone does not establish that the estimate is reliable.

Worked Example: Estimating an Efficacy-Target Probability

Suppose the prespecified criterion is expected efficacy improvement of at least 8 points at dose 30 mg. For each of 1,000 independent posterior draws, the model-implied mean improvement is calculated. In 742 draws it is at least 8 points.

The estimated posterior probability is

\widehat{P}\{E(30,\theta)\ge 8\mid y\} =\frac{742}{1000}=0.742.

Thus, under the fitted model and prior, the data support a 0.742 posterior probability that the population mean improvement at 30 mg meets or exceeds 8 points. The approximate Monte Carlo standard error is

\sqrt{\frac{0.742(1-0.742)}{1000}} =\sqrt{0.0001914} \approx 0.0138.

This standard error describes simulation precision of the estimated posterior probability, not uncertainty in the efficacy parameter itself. For example, an approximate 95% Monte Carlo error range for the probability estimate is \(0.742\pm1.96(0.0138)\), or about 0.715 to 0.769. It is not a 95% posterior credible interval for the dose-response effect.

Joint Efficacy and Safety Criteria

When a dose must satisfy efficacy and safety conditions simultaneously, calculate the joint event in each draw. Do not multiply separately estimated marginal probabilities unless independence of the efficacy and safety quantities under the posterior has been justified. In dose-response models, these quantities may share parameters or be associated through the model, so independence should not be presumed.

Worked Example: Estimating a Joint Target Probability

A candidate dose of 40 mg is assessed against two prespecified criteria: mean efficacy improvement of at least 8 points and adverse-event risk no greater than 0.10. Across 2,000 posterior draws, efficacy meets its target in 1,320 draws, safety meets its target in 1,640 draws, and both meet their targets in 1,210 draws.

The separate probabilities are estimated as

\widehat{P}(\text{efficacy target}\mid y)=\frac{1320}{2000}=0.660, \qquad \widehat{P}(\text{safety target}\mid y)=\frac{1640}{2000}=0.820.

The joint probability is calculated from the draws in which both criteria hold:

\widehat{P}(\text{both targets}\mid y)=\frac{1210}{2000}=0.605.

Multiplying the marginal estimates would give \(0.660(0.820)=0.541\), which is not the joint estimate from these draws. It would be justified only under posterior independence of the two target events. The draw-by-draw calculation preserves their dependence and directly answers the decision question.

The estimated probability should be reported with its event definition, dose, units, data context, and relevant simulation precision. A probability of 0.605 does not itself dictate a decision: the decision rule must specify what probability level is adequate and how uncertainty, clinical consequences, and other evidence enter the choice.

Comparing Candidate Doses and Checking Precision

The same posterior draws can be evaluated at each candidate dose. This gives a probability profile across doses and shows whether a dose has a strong chance of meeting the complete target, not merely whether its posterior mean response appears favorable. The table below summarizes hypothetical results from 2,000 independent posterior draws per dose. The joint count is the number of draws meeting both the efficacy and safety criteria.

Dose	Efficacy target probability	Safety target probability	Joint target probability
10 mg	690/2,000 = 0.345	1,900/2,000 = 0.950	650/2,000 = 0.325
20 mg	1,440/2,000 = 0.720	1,700/2,000 = 0.850	1,280/2,000 = 0.640
40 mg	1,700/2,000 = 0.850	1,160/2,000 = 0.580	1,040/2,000 = 0.520

Suppose the decision criterion, set in advance, requires a joint target probability of at least 0.60. On these hypothetical results, 20 mg meets that probability rule, while 10 mg and 40 mg do not. The 40 mg dose has the highest efficacy-target probability but a lower probability of meeting the safety target; selecting on efficacy alone would miss that trade-off. This conclusion is conditional on the defined criteria and the assumed decision rule, not a universal claim that 20 mg is the best dose.

Worked Example: Assessing Monte Carlo Precision for a Dose Choice

For 20 mg in the table, 1,280 of 2,000 independent draws meet both targets, so the estimated joint probability is \(0.640\). Its approximate Monte Carlo standard error is

\sqrt{\frac{0.640(1-0.640)}{2000}} =\sqrt{0.0001152} \approx 0.0107.

The estimate is therefore about 0.640 with Monte Carlo standard error 0.0107. This indicates relatively small simulation error for the probability estimate, but it does not account for uncertainty from model choice, prior specification, or misspecification. If the posterior draws are autocorrelated, the independent-draw calculation may understate Monte Carlo error; the indicator sequence must be checked using an effective sample size or another autocorrelation-aware estimate.

Specify the target event.
State the efficacy and safety quantities, thresholds, population, dose regimen, and direction of benefit or harm before calculating probabilities.

Evaluate every draw at every candidate dose.
Use the same draw’s model parameters to calculate the efficacy and safety quantities, preserving their joint dependence.

Count target-meeting draws.
Divide the number satisfying the required event by the number of usable posterior draws.

Assess computation and interpret in context.
Check posterior convergence and indicator Monte Carlo error, then compare estimates with the prespecified decision rule.

Assumptions, Diagnostics, and Alternatives

A posterior target probability is meaningful only relative to the model and analysis that generated it. The likelihood must represent the observed endpoints and study design adequately; the dose-response structure and prior must be defensible for the question; and the posterior computation must explore the relevant distribution. The endpoint definitions and thresholds must match the clinical objective. If these conditions are doubtful, a precise-looking probability can still be misleading.

Useful checks include examining trace plots and convergence summaries for parameters and for \(I_s(d)\), assessing effective sample sizes, and verifying that the dose-specific quantities and units were calculated correctly. Inspecting the posterior distributions of efficacy and safety quantities can reveal whether a target probability is driven by a small number of extreme draws or by a threshold lying near a dense part of the distribution. Recalculate probabilities under plausible prior or model assumptions when those choices could materially affect the result. These checks do not prove that the model is correct; they help identify computational or inferential fragility.

A probability-based summary complements, rather than replaces, other dose-uncertainty summaries. A posterior mean or median dose-response curve gives a central estimate but can obscure the chance of meeting a threshold. Profile-likelihood intervals, covered in the earlier material on profile likelihood, quantify uncertainty using a likelihood-based framework; they do not directly give a posterior probability for the joint threshold event. Each approach answers a different question, and results should not be described as interchangeable. A simple decision based on whether point estimates cross thresholds is easier to compute, but ignores uncertainty around those estimates.

Common failure modes include calculating efficacy and safety probabilities separately and multiplying them without an independence argument; treating every autocorrelated draw as independent when reporting Monte Carlo error; and interpreting \(P(A(d)\mid y)\) as the probability that a particular patient will benefit or avoid harm. A further limitation is that posterior probabilities depend on the specified model and prior. They also do not account automatically for uncertainty in the target thresholds, unmeasured outcomes, or consequences of incorrect decisions. Those issues need explicit treatment if they matter to the decision.

Key takeaway: Define the clinical target as an event, evaluate it draw by draw using the joint posterior draws, and estimate its probability by the fraction of draws meeting the event. Report the event and simulation precision clearly, and interpret the result conditional on the model, prior, and data.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In 500 independent posterior draws, 365 meet a prespecified efficacy target. What is the estimated posterior target probability?
For one dose, the efficacy target is met in 0.70 of draws and the safety target in 0.80. Can the joint probability be determined by multiplying these estimates alone? Explain.
A safety criterion requires model-implied adverse-event risk no greater than 0.05. In 800 posterior draws, the risk is no greater than 0.05 in 600. Calculate the estimated safety-target probability.
Why is the posterior probability that mean efficacy exceeds a threshold not the same as the probability that a future patient will respond?

Solutions. (1) \(365/500=0.730\). (2) No. The product is 0.56 only if the two target events are independent under the posterior; estimate the joint probability by counting draws that meet both criteria. (3) \(600/800=0.750\). (4) The first probability concerns uncertainty about a model-based population mean. A future-patient response probability additionally involves outcome variability and is a posterior predictive quantity.