CRM Dose Selection With Posterior Overdose Control

Why Add a Safety Constraint to CRM?

A continual reassessment method (CRM) uses accumulated toxicity data and a dose–toxicity model to recommend a dose whose estimated toxicity is near a prespecified target. With limited data, however, a dose can have an estimated toxicity close to target while still having substantial posterior probability of exceeding that target. A safety constraint makes this uncertainty part of the selection rule: a dose is eligible only if its posterior overdose probability is no greater than a prespecified ceiling.

This rule has two distinct parts. First, the CRM identifies how close each dose’s estimated toxicity is to the target. Second, posterior overdose control screens out doses whose probability of being above the target is too large. The selected dose is the closest-to-target dose among those that pass the screen; it is not necessarily the unconstrained CRM recommendation.

Model and Decision Quantities

Suppose there are $J$ ordered dose levels, indexed by $j=1,\ldots,J$, and a binary dose-limiting toxicity outcome measured over a prespecified, complete assessment window. Let $Y_i=1$ indicate that participant $i$ experiences the outcome in that window and $Y_i=0$ otherwise. Let $p_j$ denote the probability of toxicity at dose $j$, and let $p_T$ be the target toxicity probability. The accumulated, fully observed trial data are denoted by $D$.

For a standard one-parameter CRM specification, a fixed increasing skeleton $0<d_1<\cdots<d_J<1$ is transformed by a parameter $a$, for example $p_j(a)=d_j^{\exp(a)}$. A prior distribution is assigned to $a$, and the observed outcomes update that prior. The particular skeleton, transformation, and prior must be fixed for the analysis. Other CRM parameterizations are possible; the decision rule below applies to the resulting posterior dose-toxicity probabilities.

Assumptions for this rule: The toxicity outcome and its assessment window are defined consistently for all participants; the outcomes used in the update are observed over that full window; the chosen CRM model and prior provide a posterior distribution for toxicity at every dose; and the dose ordering is clinically meaningful. The safety ceiling and the target $p_T$ are specified before applying the rule. The rule does not correct a misspecified model or incomplete outcome follow-up.

For each dose, define its posterior mean toxicity probability $m_j=E(p_j\mid D)$, and define its posterior overdose probability $q_j=P(p_j>p_T\mid D)$. The first quantity summarizes the estimated toxicity level; the second is the posterior probability that the dose exceeds the target. Let $\xi$ be the prespecified maximum acceptable posterior overdose probability.

Definition: The unconstrained CRM recommendation is any dose minimizing $|m_j-p_T|$. The safety-eligible set is $F_\xi=\{j:q_j\leq \xi\}$. The CRM dose selection with posterior overdose control is any dose $$ j^*\in\underset{j\in F_\xi}{\operatorname{argmin}}\;|m_j-p_T|. $$ Thus the rule minimizes distance from the target posterior mean toxicity, subject to satisfying the posterior overdose-probability ceiling.

The posterior overdose probability is calculated from the posterior distribution, not by comparing the posterior mean with the target. In a continuous-parameter model, it can be written as $q_j=\int I\{p_j(a)>p_T\}\,\pi(a\mid D)\,da$, where $\pi(a\mid D)$ is the posterior density and $I\{\cdot\}$ is one when the condition holds and zero otherwise. In practice, posterior draws can estimate both $m_j$ and $q_j$.

Why the Constrained Rule Has This Form

The CRM target criterion and the safety criterion answer different questions. The distance $|m_j-p_T|$ ranks dose estimates by closeness to target; the condition $q_j\leq\xi$ determines which doses are admissible. Combining them as a constrained minimization preserves the target-seeking objective while enforcing the safety screen.

Compute posterior summaries.
For every dose, obtain $m_j$ and $q_j$ from the posterior distribution under the fitted CRM.

Apply the safety screen.
Include dose $j$ in $F_\xi$ exactly when $q_j\leq\xi$. Equality passes because the rule uses a non-strict ceiling.

Rank eligible doses by target distance.
For each eligible dose, calculate $|m_j-p_T|$, and select a dose with the smallest value.

To justify the last step, consider the constrained decision problem of minimizing $L_j=|m_j-p_T|$ subject to $q_j\leq\xi$. The constraint is equivalent to restricting the search to $F_\xi$; therefore the minimizer is exactly $\operatorname{argmin}_{j\in F_\xi}L_j$. If an unconstrained CRM minimizer is eligible, it remains a constrained minimizer. If it is ineligible, it cannot be selected, and the rule chooses the eligible dose with the smallest remaining target distance. This is a direct consequence of constrained minimization, not a change to the CRM posterior.

Under a monotone CRM model in which each posterior draw assigns nondecreasing toxicity probabilities across ordered doses, $q_j$ is nondecreasing in dose as well: if $j<k$, then $p_j\leq p_k$ for every draw, so $I\{p_j>p_T\}\leq I\{p_k>p_T\}$, and taking posterior expectations gives $q_j\leq q_k$. Consequently, eligible doses form a lower segment of the dose grid, possibly empty. The selection rule still uses the explicit eligibility check and target-distance ranking.

Worked Decisions

Worked Example: The CRM Recommendation Passes the Safety Screen

Consider a hypothetical trial with target $p_T=0.25$ and ceiling $\xi=0.25$. Assume the posterior summaries below have been computed from the completed binary toxicity data using the prespecified CRM. Posterior means are probabilities, and $q_j$ is the posterior probability that toxicity exceeds 0.25.

Dose	$m_j$	$q_j$	$\|m_j-0.25\|$	Eligible?
1	0.08	0.02	0.17	Yes
2	0.16	0.08	0.09	Yes
3	0.24	0.22	0.01	Yes
4	0.34	0.58	0.09	No

For dose 3, the target distance is $|0.24-0.25|=0.01$, and its overdose probability $0.22$ is at most $0.25$. Dose 3 is eligible and has the smallest target distance among eligible doses. The selected dose is therefore dose 3. Dose 4 is not eligible because $0.58>0.25$, even though its posterior mean is also relatively close to the target.

Worked Example: The Unconstrained CRM Recommendation Is Excluded

In a second hypothetical trial, let $p_T=0.25$ and $\xi=0.20$. The posterior summaries are:

Dose	$m_j$	$q_j$	$\|m_j-0.25\|$	Eligible?
1	0.06	0.01	0.19	Yes
2	0.13	0.04	0.12	Yes
3	0.23	0.18	0.02	Yes
4	0.31	0.41	0.06	No
5	0.40	0.79	0.15	No

Without the safety constraint, dose 3 is the CRM recommendation because $0.02$ is the smallest target distance. Since $q_3=0.18\leq0.20$, dose 3 remains eligible and is selected. To see the effect of the constraint on a less favorable posterior, suppose instead that the computed overdose probability at dose 3 were $0.24$, with all other entries unchanged. Dose 3 would then fail the ceiling, and dose 2 would be selected: it is eligible and has the smallest distance among the remaining doses, $|0.13-0.25|=0.12$. This comparison shows that the constraint acts on the posterior overdose probability, not on the posterior mean alone.

Worked Example: Equal Target Distances Require a Tie Rule

Suppose a hypothetical analysis has $p_T=0.25$, $\xi=0.30$, and these posterior summaries:

Dose	$m_j$	$q_j$	$\|m_j-0.25\|$	Eligible?
1	0.12	0.01	0.13	Yes
2	0.21	0.12	0.04	Yes
3	0.29	0.27	0.04	Yes
4	0.32	0.39	0.07	No

Doses 2 and 3 both pass the safety ceiling. Their target distances are $|0.21-0.25|=0.04$ and $|0.29-0.25|=0.04$, so both minimize the constrained objective. The rule alone does not distinguish them. If the protocol prespecifies choosing the lower dose in a tie, dose 2 is selected; another tie convention could give a different decision. A tie rule should be explicit rather than inferred from rounded posterior summaries.

Checking the Analysis and Understanding Its Limits

Before acting on the selected dose, verify that the posterior summaries correspond to the same model, data cut, target, and assessment window. The rule is only as reliable as those inputs. In particular, it assumes toxicity outcomes are sufficiently observed to be classified over the defined window. Counting a participant without a completed window as having no toxicity can understate toxicity and distort both $m_j$ and $q_j$.

Several checks can reveal problems in the calculation or its interpretation:

Check dose ordering. Under a monotone CRM, posterior mean toxicity and posterior overdose probability should be nondecreasing across ordered doses. A reversal may indicate a coding, indexing, or posterior-summary error, or that the fitted model does not impose the assumed monotonicity.
Check the event in the safety probability. The screen concerns $P(p_j>p_T\mid D)$, not $P(p_j>m_j\mid D)$, not the probability of observing a toxicity in the next participant, and not a probability that the hypothesis $p_j\leq p_T$ is true.
Check the boundary and precision. A dose with $q_j=\xi$ passes the stated rule. Use unrounded posterior probabilities for eligibility; rounding can turn a value just above the ceiling into an apparent pass.
Check model dependence. A sparse set of outcomes can leave posterior summaries sensitive to the skeleton or prior. Posterior calculations should be examined for computational stability and interpreted in light of the prespecified CRM model.

The rule differs from unconstrained target-probability CRM selection only by the safety screen: unconstrained selection minimizes target distance over all doses, whereas this rule minimizes it over eligible doses. It also differs from EWOC, which uses a posterior constraint involving whether the maximum tolerated dose lies below a candidate dose. Here the screened event is directly whether the candidate dose’s toxicity probability exceeds $p_T$. These probabilities describe different posterior events and should not be treated as interchangeable.

A posterior overdose ceiling is not a guarantee that the selected dose is safe, nor does it establish a frequentist error rate for the entire adaptive trial. It is a model-based decision criterion conditional on the data and prior. It can be sensitive to the ceiling, the CRM specification, and the available dose grid. If no dose satisfies the ceiling, the constrained minimization has no eligible candidate; a separate protocol rule must govern that situation. The selection rule also does not by itself define escalation restrictions, stopping decisions, or how to choose the ceiling.

Common Mistake

Common mistake: “The posterior mean toxicity at dose 4 is below the target, so dose 4 must pass safety control.” A mean below target does not imply that the posterior probability of exceeding target is below $\xi$. The correction is to calculate $q_4=P(p_4>p_T\mid D)$ from the full posterior distribution, check $q_4\leq\xi$, and only then compare the dose’s target distance with those of other eligible doses.

Check Your Understanding

Use the rule as stated above. Give a dose decision where possible, and explain the reason for each conclusion.

Let $p_T=0.20$, $\xi=0.25$. Doses 1–3 have $(m_j,q_j)$ values $(0.08,0.02)$, $(0.18,0.20)$, and $(0.24,0.26)$. Which dose is selected?
A dose has posterior mean toxicity $m_j=0.19$, target $p_T=0.20$, and overdose probability $q_j=0.31$. Does it pass a ceiling of $\xi=0.25$? Explain why its mean does not settle the question.
Two eligible doses have posterior mean toxicities 0.17 and 0.23 when $p_T=0.20$. Which doses minimize the target-distance criterion?
State the safety event used by this rule, and distinguish it from a probability about whether the null hypothesis is true.

Solutions. (1) Dose 2 is selected. Its overdose probability $0.20\leq0.25$, and its distance is $|0.18-0.20|=0.02$. Dose 3 fails because $0.26>0.25$; dose 1 is eligible but has distance $0.12$. (2) No. Since $0.31>0.25$, the dose fails the safety screen. A mean of 0.19 does not determine how much posterior probability lies above 0.20. (3) Both minimize: $|0.17-0.20|=0.03$ and $|0.23-0.20|=0.03$. A prespecified tie rule is needed to select one. (4) The event is that the dose-specific toxicity probability exceeds the target, $p_j>p_T$, conditional on the observed data. The resulting posterior probability is not the probability that $H_0$ is true; it is a probability about a model parameter under the posterior distribution.