Computing Posterior Overdose Probabilities

From a Toxicity Curve to an Overdose Probability

A dose can have a posterior mean toxicity below a trial’s upper limit and still carry substantial posterior probability that its true model-implied toxicity exceeds that limit. The relevant calculation is therefore not just the toxicity curve evaluated at an average parameter value. It is the posterior probability of the exceedance event, evaluated across the uncertainty in the BLRM parameters.

Let \(d\) be a candidate dose, \(d_{\mathrm{ref}}\) a reference dose, and \(\phi_U\) the prespecified upper toxicity limit. Write \(p(d;\alpha,\beta)\) for the BLRM toxicity probability at dose \(d\), given parameters \(\alpha\) and \(\beta\). In the parameterization used here,

\operatorname{logit}\{p(d;\alpha,\beta)\} = \operatorname{logit}(p_{\mathrm{ref}}) +\alpha +\beta\log(d/d_{\mathrm{ref}}), \qquad \operatorname{logit}(u)=\log\!\left(\frac{u}{1-u}\right).

Here \(p_{\mathrm{ref}}\) is the reference toxicity probability. The posterior distribution of \((\alpha,\beta)\), conditional on the observed dose-limiting toxicity data and the specified prior, represents uncertainty in the dose-toxicity curve. Under the usual cohort likelihood, outcomes are conditionally independent Bernoulli observations given their dose-specific toxicity probabilities. The calculation also relies on the dose-response model and prior being appropriate for the trial; posterior computation cannot correct a badly specified model.

Definition: The posterior overdose probability at dose \(d\) is \(q(d)=P\{p(d;\alpha,\beta)>\phi_U\mid \text{data}\}\). It is the posterior probability, conditional on the observed data and model, that the dose-specific toxicity probability exceeds the upper limit. It is not the probability that a particular patient will experience a toxicity.

Turning the Event Into a Parameter Boundary

The logistic function is strictly increasing, so comparing \(p(d;\alpha,\beta)\) with \(\phi_U\) is equivalent to comparing their logits. This gives a convenient event to evaluate for every posterior draw.

p(d;\alpha,\beta)>\phi_U \quad\Longleftrightarrow\quad \alpha+\beta\log(d/d_{\mathrm{ref}}) > \operatorname{logit}(\phi_U)-\operatorname{logit}(p_{\mathrm{ref}}).

The right-hand side is the threshold on the linear predictor beyond the reference-dose contribution. The same boundary can be written as a dose-specific intercept threshold for a given slope. When the model constrains \(\beta>0\), increasing \(d\) increases \(p(d;\alpha,\beta)\) for each parameter draw, so a draw that exceeds the limit at a dose also exceeds it at any higher dose. If the fitted model does not constrain the slope to be positive, that monotonicity conclusion does not follow automatically.

This boundary is useful both for checking code and for computation: each draw either falls on the overdose side or it does not. The posterior probability is the posterior mass on that side of the boundary, not a toxicity probability averaged over the boundary.

Worked Example: Counting Exceedances in Posterior Draws

Suppose \(p_{\mathrm{ref}}=0.25\), \(\phi_U=0.30\), and the candidate dose is \(2d_{\mathrm{ref}}\). For illustration, consider eight equally weighted draws from the fitted posterior. All draws have positive slopes.

c=\operatorname{logit}(0.30)-\operatorname{logit}(0.25) =\log(0.30/0.70)-\log(0.25/0.75) =-0.8473-(-1.0986)=0.2513.

At twice the reference dose, \(\log(d/d_{\mathrm{ref}})=\log(2)=0.6931\). A draw is an overdose draw if \(\alpha+0.6931\beta>0.2513\).

Draw	\(\alpha\)	\(\beta\)	\(\alpha+0.6931\beta\)	Exceeds 0.2513?
1	-0.10	0.35	0.1426	No
2	0.05	0.40	0.3272	Yes
3	0.12	0.25	0.2933	Yes
4	-0.02	0.30	0.1879	No
5	0.20	0.20	0.3386	Yes
6	-0.15	0.50	0.1966	No
7	0.08	0.30	0.2879	Yes
8	-0.05	0.25	0.1233	No

Four of the eight draws exceed the boundary. Their empirical posterior overdose probability is \(4/8=0.50\), or 50%. This is an estimate based on only eight draws; it is not a claim that the underlying posterior probability is exactly 50%.

At \(4d_{\mathrm{ref}}\), the log dose ratio is \(1.3863\). For the same eight draws, the smallest linear predictor is \(-0.10+0.35(1.3863)=0.3852\), which is above 0.2513. Thus all eight draws exceed the limit at that dose. That finite-sample result still does not establish that the posterior probability is exactly 1.

Posterior Simulation and Its Error

With \(M\) equally weighted posterior draws \((\alpha^{(m)},\beta^{(m)})\), calculate the overdose indicator for each draw and average the indicators. This directly approximates the posterior probability as an integral over the posterior distribution.

\widehat q(d) = \frac{1}{M}\sum_{m=1}^{M} I\!\left[ \alpha^{(m)}+\beta^{(m)}\log(d/d_{\mathrm{ref}}) > \operatorname{logit}(\phi_U)-\operatorname{logit}(p_{\mathrm{ref}}) \right].

For independent draws from the posterior, each indicator is a Bernoulli random variable with probability \(q(d)\). Consequently, the estimate is unbiased and has Monte Carlo standard error estimated by

\widehat{SE}_{\mathrm{MC}}\{\widehat q(d)\} = \sqrt{\frac{\widehat q(d)\{1-\widehat q(d)\}}{M}}.

This standard error describes simulation error in approximating the posterior probability. It is not the posterior uncertainty in the toxicity probability and is not an interval for a patient’s outcome. For Markov chain Monte Carlo draws, consecutive draws may be correlated, so the independent-draw formula can understate simulation error. Use an effective sample size or a suitable chain-based Monte Carlo error estimate, and check that chains have mixed and converged adequately.

Worked Example: Reporting Monte Carlo Precision

Suppose \(M=1{,}000\) approximately independent posterior draws are evaluated at a candidate dose, and 230 exceed the upper limit. The estimate is

\widehat q(d)=\frac{230}{1000}=0.230.

The estimated Monte Carlo standard error is

\widehat{SE}_{\mathrm{MC}} = \sqrt{\frac{0.230(1-0.230)}{1000}} = \sqrt{0.0001771} \approx 0.0133.

Thus the estimated posterior overdose probability is 23.0%, with Monte Carlo error of about 1.33 percentage points as a standard error. A rough two-standard-error range for simulation error is \(0.230\pm2(0.0133)\), or approximately 20.3% to 25.7%. This rough range describes how much the estimate could vary from finite simulation under the stated independence approximation; it is not a posterior credible interval for \(q(d)\).

If these were correlated chain draws, the calculation would need to reflect their effective sample size rather than treating all 1,000 draws as independent. In either case, increasing the number of effective draws reduces simulation error, but it does not resolve model misspecification.

Numerical Approximation When Draws Are Unavailable

The defining calculation can also be expressed as an integral of an indicator over the joint posterior density. Numerical integration, such as quadrature on a grid or integration of a conditional posterior, can approximate this posterior mass. The integration must cover the joint parameter distribution: separate marginal probabilities for \(\alpha\) and \(\beta\) are not enough because the overdose event depends on their linear combination, and the parameters may be correlated.

A useful special case arises when the joint posterior is approximated by a bivariate normal distribution. A linear combination of jointly normal parameters is normal, so the boundary probability can then be evaluated from a univariate normal tail. This is exact for a genuinely bivariate normal posterior and an approximation otherwise. It should not be mistaken for an exact result merely because the prior was normal; the posterior after a logistic likelihood need not be normal.

Worked Example: A Joint-Normal Posterior Approximation

Suppose \(p_{\mathrm{ref}}=0.25\), \(\phi_U=0.33\), and \(d=2d_{\mathrm{ref}}\). Approximate the joint posterior of \((\alpha,\beta)\) by a bivariate normal distribution with means \(E(\alpha\mid\text{data})=0.05\), \(E(\beta\mid\text{data})=0.45\), variances \(0.0400\) and \(0.0225\), and covariance \(0.0060\).

The overdose boundary is

c=\operatorname{logit}(0.33)-\operatorname{logit}(0.25) \approx -0.7082-(-1.0986)=0.3904.

Let \(L=\alpha+0.6931\beta\). Its approximate posterior mean is

E(L\mid\text{data})=0.05+0.6931(0.45)=0.3619.

Using the covariance as well as the two variances, its approximate posterior variance is

\operatorname{Var}(L\mid\text{data}) = 0.0400+(0.6931)^2(0.0225)+2(0.6931)(0.0060) \approx 0.05913.

Therefore the approximate standard deviation is \(\sqrt{0.05913}\approx0.2432\), and

P(L>0.3904\mid\text{data}) \approx 1-\Phi\!\left(\frac{0.3904-0.3619}{0.2432}\right) = 1-\Phi(0.1172) \approx 0.453.

The approximate posterior overdose probability is 45.3%. Because this uses a normal approximation, it should be checked against posterior simulation or a more direct numerical integration when the posterior is skewed, constrained, or has heavy tails.

Interpretation, Diagnostics, and Alternatives

At each candidate dose, the calculation answers the same specific question: what fraction of the posterior distribution places the dose-specific toxicity probability above \(\phi_U\)? A report should name the dose, state the upper limit, give the estimated probability, and describe whether it came from simulation or approximation. Keep adequate precision, but do not imply more numerical accuracy than the effective simulation size or approximation supports.

The posterior mean toxicity \(E\{p(d;\alpha,\beta)\mid\text{data}\}\) is a different quantity. It averages toxicity probabilities across posterior draws; the overdose probability averages indicators of whether those probabilities cross \(\phi_U\). In general,

P\{p(d;\alpha,\beta)>\phi_U\mid\text{data}\} \ne I\!\left[E\{p(d;\alpha,\beta)\mid\text{data}\}>\phi_U\right].

Nor is \(q(d)\) the probability of a DLT for a future patient at dose \(d\). Under the model, that posterior predictive probability is \(E\{p(d;\alpha,\beta)\mid\text{data}\}\); \(q(d)\) instead concerns whether the underlying dose-specific probability exceeds a prespecified threshold.

Useful checks include verifying that the dose ratio uses the same reference-dose units as the fitted model, that the correct upper limit is used, and that the inequality is evaluated in the correct direction. With a positive-slope model, estimated overdose probabilities across ordered doses should be nondecreasing when computed from the same draws: every draw that exceeds the limit at a lower dose must also exceed it at a higher dose. A violation suggests a coding, indexing, or dose-ordering error. A posterior probability of zero or one from finite simulation should prompt a check of draw count and tail resolution; it does not prove the true posterior mass is exactly zero or one.

Simulation is flexible and accommodates non-normal posterior shapes. Quadrature can be accurate in low dimensions but requires careful coverage of the posterior tails and can become computationally demanding. A normal approximation is quick and interpretable, but may misrepresent boundary probability when the posterior is skewed or strongly constrained. Comparing methods on a few doses, and increasing simulation or integration accuracy where results are near a decision threshold, can reveal numerical instability.

Common Pitfalls

Using the posterior mean curve as the overdose probability. A mean toxicity below the upper limit does not imply a small posterior probability of exceeding it. Evaluate the exceedance indicator for each posterior draw.
Calling \(q(d)\) the chance a patient has a DLT. It is posterior mass on a parameter-defined exceedance event. The posterior predictive DLT probability is a different quantity.
Treating a finite-draw estimate as exact. A value such as \(0/1{,}000\) indicates no exceedance was seen in those draws, not that posterior overdose probability is zero. Improve effective tail sampling or use a validated numerical method.
Ignoring dependence between parameters. In a normal approximation, omitting the covariance term changes the variance of the linear combination and can materially change the estimated tail probability.

Key takeaway: Compute posterior overdose probability by evaluating whether each posterior curve exceeds the upper toxicity limit at the dose of interest, then estimating the posterior mass of those exceedance draws. Report numerical uncertainty and keep this probability distinct from mean toxicity and patient-level DLT risk.

Check Your Understanding

Answer each question, then compare with the solutions below.

In 500 approximately independent posterior draws, 85 exceed the upper limit at a dose. Calculate the estimated overdose probability and its Monte Carlo standard error.
Why can a posterior mean toxicity below the upper limit coexist with a substantial posterior overdose probability?
For a positive-slope BLRM, what ordering should overdose probabilities have across increasing doses when calculated from the same posterior draws?
What does a simulation estimate of zero exceedances in 100 draws establish about the true posterior overdose probability?

Solutions. (1) The estimate is \(85/500=0.170\). Its Monte Carlo standard error is \(\sqrt{0.170(0.830)/500}=\sqrt{0.0002822}\approx0.0168\), or 1.68 percentage points, under the independent-draw approximation. (2) Some posterior draws can imply toxicity above the limit and others below it; averaging toxicity values does not give the fraction of draws above the limit. (3) The estimated probabilities should be nondecreasing with dose, because every positive-slope draw has a nondecreasing toxicity curve. (4) It establishes only that none of those 100 draws exceeded the limit. It does not establish that the posterior probability is zero; the simulation may have missed a low-probability tail region.