Deriving Posterior Overdose-Control Rules

From a Toxicity Threshold to an Escalation Rule

A dose can have a toxicity estimate below a target and still be an uncomfortable escalation choice: uncertainty may leave substantial posterior probability that its true toxicity probability exceeds the target. Posterior overdose control makes that uncertainty explicit. It permits a dose only when the posterior probability of exceeding a prespecified toxicity threshold is smaller than a chosen limit.

The earlier material on posterior toxicity probabilities at candidate doses supplies the dose-specific model probability. The decision rule developed here is a further step: it treats that probability as uncertain under the posterior distribution and asks how much posterior mass lies beyond the safety threshold.

Definition: Let $D$ denote the observed trial data, $\theta$ the toxicity-model parameters, and $q_j(\theta)$ the model probability of a dose-limiting toxicity (DLT) at candidate dose $j$. For a prespecified toxicity threshold $\phi$, define the posterior overdose probability at dose $j$ as $$ r_j = P\{q_j(\theta)>\phi\mid D\}. $$ Given a prespecified limit $\xi$, the dose passes the posterior overdose-control rule if $r_j<\xi$. Here $0<\phi<1$ and $0<\xi<1$; $\phi$ is a threshold on the toxicity probability, whereas $\xi$ is a threshold on posterior probability.

The two thresholds answer different questions. For example, $\phi=0.30$ defines a dose as overly toxic when its modeled DLT probability exceeds 30%. A choice of $\xi=0.25$ then permits the dose only if less than 25% of the posterior distribution lies in that overly toxic region. These values are design choices, not quantities learned automatically from the data.

Deriving the Posterior CDF Condition

Let $F_j(t)=P\{q_j(\theta)\leq t\mid D\}$ be the posterior cumulative distribution function (CDF) of the dose-specific toxicity probability. The event $q_j(\theta)>\phi$ is the complement of $q_j(\theta)\leq\phi$. Therefore, without requiring a continuous posterior distribution,

r_j=1-F_j(\phi), \qquad r_j<\xi \ \Longleftrightarrow\ F_j(\phi)>1-\xi.

This is the direct CDF form of the rule. The strict inequality matters: if $r_j=\xi$, the dose does not pass a rule stated as $r_j<\xi$. The identity also accommodates posterior point masses at $\phi$, because $F_j(\phi)$ includes probability at or below $\phi$, while the overdose event is strictly above $\phi$.

A posterior quantile can sometimes provide an alternative description, but it is not generally interchangeable with this event-probability rule. A strict quantile comparison requires additional regularity conditions, such as the posterior CDF being strictly increasing around the relevant quantile and threshold. With gaps in the distribution's support, a quantile can lie below $\phi$ even when the posterior probability above $\phi$ is exactly $\xi$. For computation and decisions, use $r_j=P\{q_j(\theta)>\phi\mid D\}$, or its exact CDF equivalent.

Assumptions and Interpretation

The rule is meaningful only relative to a specified probability model and posterior distribution. Its application assumes that the DLT outcome is defined for a relevant assessment window; the dose-specific probability $q_j(\theta)$ corresponds to that outcome; and the likelihood, dose-toxicity model, and prior adequately represent the information used to form the posterior. The data must also be sufficiently observed or handled by an appropriate model for incomplete follow-up. A precise posterior calculation cannot repair a poorly specified model or an inappropriate DLT definition.

In a binary DLT model, each fully assessed participant contributes an outcome $Y_i$, with $Y_i=1$ for a DLT and $Y_i=0$ otherwise. Conditional on model parameters and dose, the model specifies $P(Y_i=1\mid\theta,\text{dose }j)=q_j(\theta)$. The posterior distribution of $\theta$ given the data induces a distribution for $q_j(\theta)$. The overdose rule uses that induced distribution, not just a single fitted value.

Passing the rule means only that this particular posterior safety condition is met. It does not establish that the dose is safe, nor does it by itself determine which passing dose to select. A study may impose additional constraints, such as dose-order restrictions or clinical considerations. If no dose passes, the rule provides no permission to escalate under this criterion; a separate protocol decision is needed.

Worked Example: Counting Posterior Draws Above the Threshold

Suppose a hypothetical trial uses posterior draws to evaluate a candidate dose. Set $\phi=0.30$ and $\xi=0.10$. From 1,000 effectively independent posterior draws, 73 produce a dose-specific toxicity probability $q_j(\theta)$ greater than 0.30. Estimate the overdose probability and decide whether the dose passes.

The Monte Carlo estimate is the fraction of draws in the overdose event:

\widehat{r}_j=\frac{73}{1000}=0.0730.

Since $0.0730<0.10$, the estimated posterior overdose probability is below the chosen limit, so the dose passes this rule. In context, the posterior assigns an estimated 7.30% probability to the event that this dose's toxicity probability exceeds 30%.

A near-boundary estimate deserves a numerical accuracy check. Under the stated simplifying assumption of 1,000 independent draws, the Monte Carlo standard error is approximately

SE(\widehat{r}_j) =\sqrt{\frac{0.0730(1-0.0730)}{1000}} =\sqrt{0.00006767} \approx 0.00823.

An approximate 95% Monte Carlo interval is $0.0730\pm1.96(0.00823)$, or about 0.0569 to 0.0891. This interval describes simulation error in estimating $r_j$, not uncertainty about the true toxicity probability. Here it remains below 0.10, although the approximation and independent-draw assumption should not be taken for granted with correlated simulation output.

Worked Applications

Worked Example: Applying the Rule to a Posterior Distribution

For a hypothetical candidate dose, suppose the posterior distribution of its toxicity probability is $\operatorname{Beta}(2,8)$. Let $\phi=0.30$ and $\xi=0.20$. Find the posterior overdose probability and determine whether the dose passes.

For a $\operatorname{Beta}(2,8)$ random variable $Q$, its CDF on $0\leq t\leq1$ is $F(t)=1-(1-t)^8(1+8t)$. Thus,

$$ F(0.30)=1-(0.70)^8(1+8(0.30)) =1-(0.05764801)(3.4) =1-0.196003234 =0.803996766. $$

The probability above the threshold is

r_j=1-F(0.30)=0.196003234\approx0.1960.

Because $0.1960<0.20$, the dose passes, but only narrowly. The posterior probability that its toxicity probability exceeds 30% is approximately 19.6%, just below the permitted 20%. The decision should therefore be checked for sensitivity to reasonable modeling choices and numerical precision.

Worked Example: Expressing the Event in a Logistic Model

Suppose the dose-toxicity model is logistic, $q_j(\theta)=\operatorname{logit}^{-1}(\alpha+\beta x_j)$, where $\operatorname{logit}^{-1}(u)=1/(1+e^{-u})$. Assume $\beta>0$, so toxicity probability increases with the ordered dose covariate $x_j$. At a candidate dose with $x_j=1$, use $\phi=0.25$, $\xi=0.25$, and 20 hypothetical posterior draws. The corresponding linear predictors $\eta=\alpha+\beta x_j$ are:

$-2.00,-1.80,-1.60,-1.50,-1.40,-1.30,-1.25,-1.20,-1.15,-1.13,-1.12,-1.11,-1.105,-1.102,-1.101,-1.10,-1.08,-1.00,-0.80,-0.40$.

Because the logistic function is strictly increasing, $q_j(\theta)>0.25$ exactly when $\eta>\operatorname{logit}(0.25)$. The threshold on the linear predictor is

\operatorname{logit}(0.25) =\log\left(\frac{0.25}{1-0.25}\right) =\log(1/3) \approx -1.0986.

Of the 20 listed predictors, the four values $-1.08,-1.00,-0.80,-0.40$ exceed $-1.0986$. Therefore,

\widehat{r}_j=\frac{4}{20}=0.20<0.25.

The candidate passes based on these draws. The calculation uses the monotonicity of the logistic transformation to count exceedances on the linear-predictor scale; it does not replace the posterior event with a comparison of the average predictor or average toxicity probability.

Why the Posterior Mean Is Not the Rule

The posterior mean toxicity probability $E\{q_j(\theta)\mid D\}$ and the posterior overdose probability $P\{q_j(\theta)>\phi\mid D\}$ are different summaries. The first averages toxicity probabilities; the second measures posterior mass above a threshold. A mean below $\phi$ does not guarantee a small overdose probability, and an overdose probability below $\xi$ does not imply the mean must be below $\phi$.

Worked Example: Same Mean, Different Overdose Risk

Compare two hypothetical posterior distributions for a dose-specific toxicity probability, with $\phi=0.30$ and $\xi=0.20$. Under distribution A, $q_j=0.20$ with probability 0.80 and $q_j=0.40$ with probability 0.20. Under distribution B, $q_j=0.30$ with probability 1.

Both posterior means are 0.24 for A and 0.30 for B? For A, the calculation is $0.80(0.20)+0.20(0.40)=0.16+0.08=0.24$; for B, it is $1(0.30)=0.30$. The means are therefore not the same. To make the comparison informative, consider instead distribution A with $q_j=0.10$ at probability 0.50 and $q_j=0.50$ at probability 0.50, and distribution B with $q_j=0.30$ with probability 1. Both means are $0.30$.

For A, $P(q_j>0.30\mid D)=0.50$, so it fails the 0.20 limit. For B, the probability strictly above 0.30 is zero, so it passes. The boundary value $q_j=0.30$ is not in the event $q_j>\phi$. This illustrates why the event and its strict boundary must be stated explicitly.

Diagnostics, Alternatives, and Limitations

Before using a computed rule, check that the posterior distribution represents the intended dose and outcome, that doses are indexed and modeled consistently, and that the posterior computation has converged adequately. For simulation-based estimates, inspect chain mixing and effective sample size. If draws are correlated, the raw number of draws is not the number of independent draws; uncertainty in $\widehat{r}_j$ should reflect effective information. When the estimate is close to $\xi$, increase simulation precision and report the estimate rather than hiding the proximity to the boundary.

Model checks also matter. A model that imposes increasing toxicity with dose should be assessed for whether that structural assumption is appropriate. Prior sensitivity is particularly relevant when toxicity data are sparse: different reasonable priors can move posterior mass across $\phi$, changing $r_j$ and potentially reversing the decision. The chosen $\phi$ and $\xi$ should be justified clinically and in the design; they should not be tuned after seeing results merely to obtain a preferred escalation decision.

Approach	Decision quantity	What it does and does not capture
Posterior overdose control	$P\{q_j(\theta)>\phi\mid D\}$	Directly limits posterior mass above the toxicity threshold.
Posterior mean rule	$E\{q_j(\theta)\mid D\}$	Summarizes average toxicity probability but can conceal substantial tail probability.
Point-estimate rule	A single fitted value of $q_j$	Simple to compute, but does not represent posterior uncertainty about $q_j$.
Posterior interval comparison	A selected posterior bound for $q_j$	Can summarize uncertainty, but the bound and its coverage level must be specified; it is not automatically identical to the event-probability rule.

The rule is a safety constraint, not a complete dose-finding algorithm. It does not define the toxicity threshold or acceptable posterior limit, resolve every decision when several doses pass, or account automatically for benefits and other constraints. Those elements require prespecified design choices. A common mistake is to calculate $E\{q_j(\theta)\mid D\}$, find it below $\phi$, and conclude that overdose control passes. The correction is to calculate the posterior probability of the event $q_j(\theta)>\phi$ and compare that probability with $\xi$.

Key takeaway: Posterior overdose control is the strict event-probability rule $P\{q_j(\theta)>\phi\mid D\}<\xi$, equivalently $F_j(\phi)>1-\xi$. Preserve the event boundary, quantify computational error near the limit, and interpret passing as satisfying one specified safety constraint—not as proof that the dose is safe.

Check Your Understanding

Answer each question using the strict overdose event $q_j(\theta)>\phi$. Full solutions follow.

If $F_j(\phi)=0.84$ and $\xi=0.16$, does the dose pass? Explain the role of equality.
In 2,000 independent posterior draws, 310 have $q_j(\theta)>\phi$. Estimate $r_j$, and apply a limit $\xi=0.17$.
A posterior distribution places probability 0.12 at $q_j=\phi$ and probability 0.18 above $\phi$. What is $r_j$, and does the dose pass when $\xi=0.18$?
Why is a posterior mean below $\phi$ insufficient to establish that the overdose-control rule passes?

Solutions. (1) The CDF identity gives $r_j=1-0.84=0.16$. Since the rule requires $r_j<\xi$, equality does not pass. (2) $\widehat{r}_j=310/2000=0.155$; because $0.155<0.17$, the estimate passes. (3) The atom at $q_j=\phi$ is not part of the strict event, so $r_j=0.18$. Equality with $\xi$ does not pass. (4) A mean averages the posterior toxicity probabilities and does not determine how much posterior mass lies above $\phi$; calculate $P\{q_j(\theta)>\phi\mid D\}$ directly.

Approach	Decision quantity	What it does and does not capture
Posterior overdose control	\(P\{q_j(\theta)>\phi\mid D\}\)	Directly limits posterior mass above the toxicity threshold.
Posterior mean rule	\(E\{q_j(\theta)\mid D\}\)	Summarizes average toxicity probability but can conceal substantial tail probability.
Point-estimate rule	A single fitted value of \(q_j\)	Simple to compute, but does not represent posterior uncertainty about \(q_j\).
Posterior interval comparison	A selected posterior bound for \(q_j\)	Can summarize uncertainty, but the bound and its coverage level must be specified; it is not automatically identical to the event-probability rule.