From a Dose-Toxicity Model to Dose-Specific Risk
A fitted dose-toxicity model does not generally produce one known toxicity probability at each dose. It produces a posterior distribution over model parameters, and each possible parameter value implies its own probability of toxicity at every candidate dose. Summarizing that distribution is important: a single fitted curve can hide substantial uncertainty, especially when few participants have been treated or when a candidate dose lies near the edge of the observed range.
Suppose \(Y_{ij}\) is the binary dose-limiting toxicity (DLT) outcome for participant \(i\) at candidate dose \(d_j\), assessed over a specified toxicity window. Let \(p_j(\theta)\) be the model-implied probability of a DLT at dose \(d_j\), where \(\theta\) denotes the model parameters. For example, in a logistic dose-toxicity model, \(p_j(\theta)=\operatorname{logit}^{-1}(\alpha+\beta x_j)\), with \(\operatorname{logit}^{-1}(u)=1/(1+\exp(-u))\), and \(x_j\) a dose covariate. Centering \(x_j\) changes the parameter interpretation and numerical stability, not the meaning of \(p_j(\theta)\).
The calculations below assume a correctly specified binary outcome likelihood, a stated prior, and a posterior distribution that accounts for the observed outcomes. In the simple logistic example, outcomes are conditionally independent given the parameters and assigned doses, and higher doses are constrained to have nondecreasing DLT risk by taking \(\beta>0\). Participants are assumed to have complete assessment for the defined toxicity window; delayed or incomplete follow-up requires a likelihood that represents that observation process. Posterior summaries are only as trustworthy as these modeling and data assumptions.
Calculating Posterior Means and Credible Intervals
In practice, posterior summaries are commonly calculated from draws \(\theta^{(1)},\ldots,\theta^{(M)}\) that represent the fitted posterior distribution. At each draw, calculate the risk at each candidate dose, \(p_j^{(m)}=p_j(\theta^{(m)})\). With equally weighted draws, the posterior mean risk is estimated by the arithmetic average of those values. The endpoints of an empirical credible interval are estimated by the corresponding quantiles of the same dose-specific values. If draws have unequal weights, use the normalized posterior weights for the mean and weighted quantiles for the interval.
The “hat” here indicates a Monte Carlo estimate of a posterior summary, not a frequentist estimate of an unknown population proportion. The interval describes uncertainty in the model-implied toxicity probability at the specified dose, conditional on the model, prior, and observed data. It is not an interval for the toxicity outcome of one future participant: that outcome is binary, whereas the interval summarizes uncertainty about its underlying probability.
Worked Example: Summarizing Risk at Three Candidate Doses
Consider five equally weighted draws from a hypothetical fitted posterior for a logistic model. The dose covariates are \(x=-1,0,1\), corresponding to three ordered candidate doses. Each table entry is the model-implied DLT risk calculated from that draw. For this small illustrative set, define the central 60% empirical interval as the second through fourth ordered values. This interval convention makes the quantile calculation explicit; in an actual analysis, the interval level and quantile algorithm should be stated.
| Posterior draw | Risk at \(x=-1\) | Risk at \(x=0\) | Risk at \(x=1\) |
|---|---|---|---|
| 1 | 0.0451 | 0.0832 | 0.1481 |
| 2 | 0.0522 | 0.1192 | 0.2497 |
| 3 | 0.0832 | 0.1545 | 0.2689 |
| 4 | 0.0911 | 0.2059 | 0.4013 |
| 5 | 0.0392 | 0.0998 | 0.2315 |
At \(x=0\), the posterior mean risk is the average of the five draw-specific risks:
Thus, the posterior mean DLT risk at this candidate dose is about 13.3%. Sorting the five values gives \(0.0832, 0.0998, 0.1192, 0.1545, 0.2059\). The central 60% interval, using the stated second-to-fourth order-statistic convention, is \([0.0998,0.1545]\), or about 10.0% to 15.5%. This is posterior uncertainty about the risk, not a statement that 60% of future participants will have risks in that range.
The same arithmetic gives posterior mean risks of \(0.0622\) at \(x=-1\) and \(0.2599\) at \(x=1\). The summaries retain the ordering in this example: risk is higher at the higher candidate doses for every draw. If the model enforces monotonicity, this should hold by construction; it does not prove that the model describes the data adequately.
Why Risk Uncertainty Is Not Outcome Uncertainty
Let \(Y\) denote the DLT outcome for one future participant treated at dose \(d_j\). Conditional on the model parameters, \(Y\) is Bernoulli with probability \(p_j(\theta)\). Averaging over the posterior gives the predictive probability of a DLT:
So the posterior mean risk is also the posterior predictive probability of toxicity for one participant under the model. But a credible interval for \(p_j(\theta)\) is not a predictive interval for \(Y\): \(Y\) can only be 0 or 1. The distinction is also visible in the variance decomposition. Conditional Bernoulli variation and posterior parameter uncertainty are different components:
where \(m_j=E[p_j(\theta)\mid\text{data}]\). The first term is outcome variation conditional on risk; the second is uncertainty about risk. Their sum is the predictive variance of the binary outcome.
A related but separate question concerns dose choice. If a decision rule selects a dose separately at each posterior draw, the resulting distribution over selected doses describes uncertainty in the recommendation. It is not the credible interval for risk at any one dose. That distribution depends on the decision rule as well as on the posterior.
Worked Example: Posterior Uncertainty in the Selected Dose
Use the five posterior draws in the table above. For illustration, define a decision rule that selects the dose whose draw-specific toxicity risk is closest to a target risk of 0.20. This is a hypothetical rule used only to demonstrate recommendation uncertainty; it is not a safety-control rule.
For draw 1, the risks are \(0.0451, 0.0832, 0.1481\). The distances from 0.20 are \(0.1549, 0.1168, 0.0519\), so the rule selects \(x=1\). For draw 2, the risks are \(0.0522, 0.1192, 0.2497\), with distances \(0.1478, 0.0808, 0.0497\); it selects \(x=1\). For draw 3, the risks are \(0.0832, 0.1545, 0.2689\), with distances \(0.1168, 0.0455, 0.0689\); it selects \(x=0\). For draw 4, the risks are \(0.0911, 0.2059, 0.4013\), with distances \(0.1089, 0.0059, 0.2013\); it selects \(x=0\). For draw 5, the risks are \(0.0392, 0.0998, 0.2315\), with distances \(0.1608, 0.1002, 0.0315\); it selects \(x=1\).
The dose-selection proportions are therefore \(2/5=0.40\) for \(x=0\), \(3/5=0.60\) for \(x=1\), and \(0/5=0\) for \(x=-1\). With a larger posterior simulation, the analogous proportions estimate posterior probabilities that this specified rule selects each dose. They are not posterior probabilities that a dose is “correct,” nor are they risk intervals. A different decision rule could produce a different distribution over selected doses from the very same posterior.
Posterior Risk Exceedance and Practical Diagnostics
Posterior draws also allow direct estimation of probabilities about the risk itself. For example, the posterior probability that the risk at dose \(d_j\) exceeds a specified value \(c\) is estimated by the fraction of draws for which \(p_j^{(m)}>c\). This answers a different question from the posterior mean. It can be useful for describing the uncertainty around a risk boundary, but the calculation alone does not prescribe what action should follow.
Worked Example: Probability That Risk Exceeds a Value
At \(x=1\), the five draw-specific risks are \(0.1481,0.2497,0.2689,0.4013,0.2315\). For the illustrative threshold \(c=0.25\), only \(0.2689\) and \(0.4013\) exceed the threshold. The empirical posterior probability is therefore:
This estimate is based on only five draws, so it is imprecise as a Monte Carlo approximation. In a real analysis, one would use many more well-mixed posterior draws and check whether the estimated probability is stable. The result means that 40% of these illustrative posterior draws imply risk above 25%; it does not mean that the probability the threshold is “true” is 40%, or that 40% of participants will experience a DLT.
Several checks help identify unreliable summaries. First, inspect the draw-by-draw dose-risk curves. Unexpected reversals across ordered doses can indicate a model that does not enforce monotonicity, a coding error, or a model that permits nonmonotone risk. Second, inspect whether posterior draws explore the relevant parameter space adequately. Poor mixing, low effective sample size, or chains that have not converged make means, quantiles, and exceedance probabilities unreliable; increasing the number of iterations does not repair a model or sampler that has not converged. Third, compare prior and posterior implications. If the posterior risk distribution barely changes after observing informative outcomes, check the data coding, likelihood, and prior influence. If the data are sparse, a broad interval may be appropriate rather than evidence of a computational failure.
For a candidate dose with no observations or one far beyond the observed dose range, posterior risk may be driven mainly by the model form and prior. Such extrapolation is particularly sensitive to assumptions about the dose-response shape. Report the candidate dose and units, the assessment window for DLT, the posterior mean risk, the interval level and method, and relevant diagnostics. A numerical interval without this context is hard to interpret.
Common Pitfall and Alternatives
A common mistake is to compute toxicity risk only at the posterior mean parameter value and call that the posterior mean risk. In general, these are not equal because the dose-response function is nonlinear:
The correction is to calculate \(p_j(\theta)\) at each posterior draw and then average those risks. A plug-in calculation can be a compact descriptive summary, but it does not represent posterior averaging and provides no credible interval.
Another alternative is to report only a point estimate, such as a maximum-likelihood fitted risk. That may be useful in settings focused on frequentist estimation, but it does not communicate posterior uncertainty or posterior probabilities of risk events. Conversely, a posterior risk interval alone does not communicate how often a decision rule would choose each dose. Report the summary that answers the scientific question, and do not treat these distinct quantities as interchangeable.
Check Your Understanding
Answer each question before reading the solutions.
- At a candidate dose, four equally weighted posterior draws imply risks \(0.10, 0.20, 0.30, 0.40\). Calculate the posterior mean risk.
- For those same draws, estimate the posterior probability that risk exceeds 0.25. Interpret the result precisely.
- Explain why a 90% credible interval for \(p_j(\theta)\) is not a 90% interval for a future participant’s binary DLT outcome.
- Two doses have different posterior mean risks, but a decision rule selects each dose in half of the posterior draws. What does that 50%–50% split describe, and what additional information is needed to interpret it?
Solutions. 1. The mean is \((0.10+0.20+0.30+0.40)/4=1.00/4=0.25\), or 25%. 2. Two of the four draws exceed 0.25, so the estimated posterior probability is \(2/4=0.50\). It describes the fraction of these posterior draws with risk above 25%, not the fraction of participants expected to have a DLT. 3. The credible interval summarizes posterior uncertainty about the probability \(p_j(\theta)\). A future outcome is binary, with predictive probability of toxicity equal to the posterior mean risk; it is not a continuous risk parameter drawn from that interval. 4. The split describes the posterior frequency with which the stated decision rule selects each dose. Interpretation requires knowing the rule, including its objective and any constraints; the split is neither a risk interval nor a probability that either dose is objectively correct.