From a TITE-CRM Update to an Assignable Dose
A TITE-CRM model can use patients whose toxicity observation windows are not yet complete. That makes an updated model useful during ongoing accrual, but it does not by itself determine which dose may be assigned next. A dose can appear close to the target toxicity rate and still be prohibited by an overdose-control rule, an escalation limit, or a rule against skipping untried doses. The operational decision therefore has two parts: update the model using the available follow-up, then restrict the model-preferred choice to doses allowed by the trial protocol.
The introductory material on TITE-CRM establishes how partial follow-up contributes to the likelihood. Here, the emphasis is on applying that update together with dose restrictions. The rules must be stated before making a recommendation: an overdose threshold, the set of doses eligible for the next cohort, and any special response to an observed dose-limiting toxicity (DLT).
Update the Posterior Before Applying the Rules
For a patient treated at dose \(d_i\), let \(y_i=1\) if a DLT has been observed during the window, and \(y_i=0\) otherwise. Let \(w_i\) be the fraction of the planned toxicity observation window completed without a DLT, for a patient who has not yet had one. In the usual weighted-likelihood TITE-CRM formulation, the contribution is \(p_{d_i}(c)\) for an observed DLT and \(\{1-p_{d_i}(c)\}^{w_i}\) for a patient event-free through fraction \(w_i\) of the window. Thus, for these data,
This weighted likelihood is an approximation to the likelihood for time-to-event data. Its interpretation relies on the TITE-CRM time-to-event assumptions used to justify the follow-up weights, including the specified observation window and appropriate handling of censoring. In particular, a patient with partial event-free follow-up contributes less evidence against toxicity than a patient who completed the window without a DLT. A patient with an observed DLT contributes the event likelihood, not an event-free follow-up term. Follow-up for a patient must not be counted twice.
With prior probabilities \(\pi(c)\), the posterior weight for each candidate value of \(c\) is proportional to \(\pi(c)L(c)\). Once these weights are normalized, the overdose probability is the sum of posterior probabilities for parameter values that imply toxicity above \(\theta\) at the dose in question. This is a probability over model uncertainty, not the estimated toxicity rate itself.
Worked Example: Updating With Partial Follow-Up at the Current Dose
Suppose a hypothetical trial has three candidate model values with equal prior probabilities. At dose 1, their modelled DLT probabilities are \(0.10\), \(0.20\), and \(0.30\), respectively. Two patients have been treated at dose 1: one completed the observation window without a DLT, and another is event-free after half the window (\(w=0.5\)). Calculate the posterior probabilities.
The likelihood contributions are \(1-p_1(c)\) for the completed patient and \(\{1-p_1(c)\}^{0.5}\) for the partially observed patient. Their product is \(\{1-p_1(c)\}^{1.5}\). Thus, the likelihoods are
Because the priors are equal, normalize these likelihoods directly. Their sum is \(0.8538+0.7155+0.5857=2.1550\). The posterior probabilities are \(0.8538/2.1550=0.3962\), \(0.7155/2.1550=0.3320\), and \(0.5857/2.1550=0.2718\), to four decimal places. The partially followed patient has contributed evidence, but less than another fully observed event-free patient would have contributed.
Turn Posterior Overdose Control Into an Allowed Set
Posterior overdose control is applied dose by dose. For a finite set of candidate model values, calculate \(q_j\) by adding the posterior probabilities associated with \(p_j(c)>\theta\). The comparison is strict: values equal to \(\theta\) are not counted as exceeding the target. Given threshold \(\xi\), retain only doses with \(q_j\leq\xi\). The resulting set is the overdose-safe set under the specified model and threshold; it is not automatically the set of doses that can be assigned next.
Next intersect that set with protocol restrictions. A common escalation restriction permits escalation by at most one dose level above the highest dose already tried. A no-skipping rule excludes an untried dose above the next sequential dose, even if a higher dose passes overdose control. Some protocols also prohibit escalation for the next cohort after a DLT has been observed at the current dose. Such a rule is an additional operational constraint, not a consequence of the TITE-CRM likelihood, and should be applied exactly as specified by the protocol.
Within the remaining admissible doses, a target-probability CRM policy may select the dose whose posterior mean toxicity probability is closest to \(\theta\). Other protocols may use a different model-based selection criterion. In either case, dose selection happens only after restrictions are applied: the dose with the best target fit is not assignable if it is excluded by a safety or escalation rule.
Worked Example: Correcting for Posterior Rounding in an Overdose Check
Consider a hypothetical finite model with three equally likely parameter values before observing data. At dose 2, the DLT probabilities under these values are \(0.19\), \(0.33\), and \(0.50\). One patient has a DLT at dose 2, so the likelihoods are \(0.19\), \(0.33\), and \(0.50\). Multiplying all likelihoods by two gives the equivalent relative likelihoods \(0.38\), \(0.66\), and \(1.00\). Normalize these values to obtain posterior probabilities.
The relative likelihoods sum to \(0.38+0.66+1.00=2.04\). The posterior probabilities are
Suppose the target is \(\theta=0.40\). Only the third candidate value implies a dose-2 toxicity probability above \(0.40\), since \(0.19<0.40\), \(0.33<0.40\), and \(0.50>0.40\). Therefore \(q_2=0.4902\). If the protocol sets \(\xi=0.50\), dose 2 passes posterior overdose control. The decision is close to the threshold, so the unrounded posterior value should be used for comparison; reporting \(0.4902\) is appropriate, whereas \(0.4903\) would be an incorrect rounding of this calculation.
Combine Safety, Escalation, and Target Fit
The following sequence makes the logic explicit. “Admissible” means that a dose survives every restriction that applies to the current decision. The next assignment is then selected from this set using the protocol’s model-based target criterion.
Use the observed DLTs and event-free follow-up fractions to calculate and normalize the TITE-CRM likelihood, including the prior probabilities.
For each dose, calculate \(q_j=P(p_j(c)>\theta\mid\text{data})\) and retain only doses with \(q_j\leq\xi\).
Remove doses disallowed by the escalation limit, no-skipping rule, or any protocol rule triggered by an observed DLT.
Choose among the remaining doses using the prespecified target criterion. If none remains, do not assign a dose outside the admissible set; follow the protocol’s stopping or review rule.
Worked Example: An Escalation Can Pass Safety but Still Be Restricted
Use the posterior from the first worked example. Suppose the toxicity probabilities at doses 2 and 3, under the three model values, are \(0.20,0.35,0.50\) and \(0.30,0.50,0.70\), respectively. Let \(\theta=0.30\) and \(\xi=0.65\). The two patients have only been treated at dose 1, and neither has had a DLT. The protocol allows escalation by one dose level and does not allow skipping an untried dose.
At dose 2, the probabilities above \(0.30\) occur for the second and third model values. Therefore \(q_2\approx0.6038\), which is no greater than \(0.65\), so dose 2 passes overdose control. At dose 3, the probabilities above \(0.30\) also occur for the second and third values; the first value is exactly \(0.30\), so it is not counted. Thus \(q_3\approx0.6038\), and dose 3 also passes the overdose threshold.
The safety screen alone would retain both doses, but the assignment restrictions do not permit jumping from dose 1 to dose 3. Dose 2 is the only upward dose eligible under the one-level escalation rule. To illustrate the target criterion, the posterior mean toxicity at dose 2 is
This is \(0.0313\) above the target \(0.30\), and dose 2 is admissible. The recommendation is therefore dose 2. The calculation does not imply that dose 3 is unsafe; it shows that passing overdose control does not override the separate no-skipping restriction.
Diagnostics, Alternatives, and Limitations
Several checks help detect an incorrect update or recommendation. Confirm that every patient’s follow-up fraction is between 0 and 1, that a DLT patient is treated as an event rather than as event-free follow-up, and that fully observed event-free patients have weight 1. Check that posterior probabilities sum to 1, that each overdose probability is between 0 and 1, and that a dose passes only when its unrounded \(q_j\) is at most \(\xi\). Finally, list the doses removed at each restriction. This makes it possible to see whether the final recommendation is constrained by overdose risk, escalation policy, or both.
The main alternative to posterior overdose control is selecting by target fit without a posterior safety screen, for example choosing the dose with posterior mean toxicity closest to \(\theta\). That can identify a dose that fits the target on average while retaining an unacceptably high posterior chance of exceeding it. Conversely, a stringent overdose threshold can leave no admissible dose even when one dose has the closest target fit. A protocol may also use a different safety framework, such as a feasibility bound; its rule should not be silently substituted for the posterior overdose threshold.
These restrictions do not remove uncertainty in the dose-toxicity model. Posterior overdose probabilities depend on the CRM model, its prior, the observed data, the TITE follow-up weights, and the chosen target and threshold. Sparse data, misspecified dose-response assumptions, or poorly justified handling of event timing can make the calculated probabilities misleading. Restrictions also do not repair an invalid likelihood: the follow-up and censoring assumptions still require clinical and statistical justification.
Worked Example: A DLT Rule Can Override a Model-Preferred Escalation
Suppose a hypothetical posterior update has already been calculated and gives overdose probabilities \(q_1=0.05\), \(q_2=0.18\), and \(q_3=0.42\), with threshold \(\xi=0.45\). The posterior mean toxicity probabilities are \(0.12\), \(0.24\), and \(0.31\), respectively, and the target is \(\theta=0.30\). A DLT has just been observed at dose 2. The protocol prohibits escalation for the next cohort after a DLT at the current dose and permits de-escalation.
All three doses pass the overdose screen because each \(q_j\leq0.45\). By target fit alone, dose 3 would be closest to the target: \(|0.31-0.30|=0.01\), compared with \(|0.24-0.30|=0.06\) for dose 2. However, the protocol rule triggered by the observed DLT excludes escalation above dose 2. Among the remaining doses, dose 2 is closer to target than dose 1, with distances \(0.06\) and \(0.18\), respectively. Under these stated rules, dose 2 is the admissible model-preferred choice. A protocol that instead requires de-escalation after any DLT would exclude dose 2 too and would lead to dose 1. The example shows why the exact DLT rule must be specified rather than inferred from the posterior alone.
Check Your Understanding
Answer each question using the stated rule definitions. Brief solutions follow the questions.
- A patient is event-free through 60% of the observation window. What likelihood contribution does this patient make at dose \(j\)?
- At a dose, posterior mass \(0.22\) is on parameter values with \(p_j(c)>\theta\). If \(\xi=0.20\), does the dose pass overdose control?
- Dose 3 passes overdose control, but patients have been treated only at dose 1 and the protocol allows escalation by one level. Can dose 3 be assigned next?
- Explain why a dose can be closest to the target toxicity probability but not be the recommended assignment.
Solutions. (1) The contribution is \(\{1-p_j(c)\}^{0.60}\), under the stated weighted-likelihood assumptions. (2) No: \(0.22>0.20\), so it fails the overdose rule. (3) No. The one-level escalation limit allows at most dose 2; passing the safety screen does not remove that restriction. (4) The dose may fail overdose control, violate a no-skipping or escalation rule, or be excluded by a protocol response to an observed DLT. Target fit is considered only among admissible doses.