Updating TITE-CRM With Pending Toxicity Outcomes

Why Pending Outcomes Belong in the Update

In dose escalation, a patient who has not yet completed the dose-limiting toxicity (DLT) assessment window is not equivalent to a patient with a completed DLT-free outcome. But the patient is not uninformative either: remaining free of DLT through part of the window provides evidence about the probability of DLT by the end of the full window. TITE-CRM uses that partial follow-up to update the dose-toxicity model without requiring every patient's assessment window to finish before an update.

The earlier material on deriving TITE-CRM weights from partial follow-up motivates the weight used here. This tutorial focuses on applying those weights consistently to a collection of observed and pending outcomes, then calculating a posterior dose recommendation. The update is Bayesian: each possible model parameter value is weighted by its prior probability and by how well it predicts the observed data.

Definition: Let \(p_j(\theta)\) be the model probability of a DLT by the end of the assessment window at dose \(j\), for model parameter \(\theta\). For patient \(i\), let \(d_i\) be the assigned dose, \(y_i=1\) if a DLT has been observed and \(y_i=0\) if no DLT has yet been observed, and \(w_i\) be the fraction of the assessment window completed without a DLT. The weighted TITE-CRM likelihood contribution is \(p_{d_i}(\theta)^{y_i}\{1-p_{d_i}(\theta)\}^{w_i(1-y_i)}\). Thus an observed DLT contributes \(p_{d_i}(\theta)\); a DLT-free patient with completed follow-up contributes \(1-p_{d_i}(\theta)\); and a pending DLT-free patient contributes \(\{1-p_{d_i}(\theta)\}^{w_i}\).

For a pending patient, \(0\leq w_i<1\); if follow-up is measured as elapsed time since dosing over a common assessment window of length \(T\), a usual choice is \(w_i=t_i/T\), where \(t_i\) is the event-free time observed so far. A completed DLT-free window has \(w_i=1\). Once a DLT is observed, its contribution is \(p_{d_i}(\theta)\), regardless of the fraction of the window elapsed. The contribution does not separately model the timing of that DLT.

Bayesian Update and Its Assumptions

For patients with conditionally independent outcomes given \(\theta\), multiply their likelihood contributions. If \(\pi(\theta)\) is the prior density or probability mass function, the posterior is proportional to that prior times the combined likelihood:

\pi(\theta\mid\text{data})\propto \pi(\theta) \prod_i p_{d_i}(\theta)^{y_i} \{1-p_{d_i}(\theta)\}^{w_i(1-y_i)}.

With a finite set of parameter values \(\theta_k\), prior probabilities \(\pi_k\), and patient-level likelihood \(L_k\), the calculation is particularly direct: form \(u_k=\pi_k L_k\), then divide each \(u_k\) by \(\sum_\ell u_\ell\). These normalized values are posterior probabilities. A posterior mean toxicity probability at dose \(j\) is \(\sum_k P(\theta_k\mid\text{data})p_j(\theta_k)\).

The update relies on several assumptions. The DLT endpoint and assessment window must be defined consistently across patients; a recorded DLT must be classified using that endpoint; and follow-up counted in \(w_i\) must be free of DLT up to the stated time. The chosen dose-toxicity model and prior must adequately represent plausible dose-risk patterns. In addition, the weighted likelihood treats the weight as fixed and uses partial event-free follow-up as fractional information. Its interpretation depends on the weighting approximation, not on a complete model of event times. In particular, the formula does not use the precise time of an observed DLT.

Justification: Taking logs of the likelihood gives a sum of patient-level terms. A pending, event-free patient's term is \(w_i\log\{1-p_{d_i}(\theta)\}\), whereas a completed DLT-free patient's term is \(\log\{1-p_{d_i}(\theta)\}\). The pending patient therefore contributes a fraction \(w_i\) of the full-window no-DLT log-likelihood term. Multiplication by the prior and normalization then gives the ordinary Bayes update over the model parameter values.

Worked Posterior Updates

Worked Example: One Observed DLT and One Pending Patient

Hypothetical data. Suppose a three-state model has prior probabilities \(0.20, 0.50, 0.30\) for states A, B, and C. At dose 1, the respective DLT probabilities are \(0.05, 0.10, 0.20\). At dose 2 they are \(0.08, 0.20, 0.40\), and at dose 3 they are \(0.30, 0.45, 0.65\). One patient at dose 1 has an observed DLT. Another patient at dose 3 has remained DLT-free through 90% of the assessment window, so \(w=0.90\).

The likelihood in each state is the dose 1 DLT probability multiplied by the dose 3 event-free contribution:

L_k=p_1(\theta_k)\{1-p_3(\theta_k)\}^{0.90}.

The unnormalized posterior weights are \(0.20(0.05)(0.70^{0.90})=0.00725418\), \(0.50(0.10)(0.55^{0.90})=0.02919419\), and \(0.30(0.20)(0.35^{0.90})=0.02332451\). Their sum is \(0.05977288\). Dividing each weight by this sum gives posterior probabilities A \(=0.1214\), B \(=0.4884\), and C \(=0.3902\), to four decimal places.

For example, the posterior mean DLT probability at dose 2 is \(0.121362(0.08)+0.488419(0.20)+0.390219(0.40)=0.2635\). At dose 1 it is \(0.1330\), and at dose 3 it is \(0.121362(0.30)+0.488419(0.45)+0.390219(0.65)=0.5098\). If the design's stated recommendation rule is to select the dose whose posterior mean is closest to a target of \(0.25\), dose 2 is selected: its absolute difference is \(0.0135\), compared with \(0.1170\) at dose 1 and \(0.2598\) at dose 3. This is the result of that selection rule, not a guarantee that dose 2 has true risk \(0.25\).

Worked Example: A Pending Outcome Is Not a Completed Outcome

Hypothetical data. Consider two equally likely model states, A and B. At dose 1 the DLT probabilities are \(0.10\) and \(0.20\); at dose 2 they are \(0.25\) and \(0.40\); and at dose 3 they are \(0.50\) and \(0.65\). At dose 2, one patient completed the window without DLT and a second patient remains DLT-free through 25% of the window. Thus the likelihoods are \((1-p_2)^{1.25}\) in the two states.

The likelihoods are \(0.75^{1.25}=0.69795\) in state A and \(0.60^{1.25}=0.52807\) in state B. Multiplying each by its prior probability \(0.50\) gives unnormalized weights \(0.34898\) and \(0.26403\), which sum to \(0.61301\). Normalization gives posterior probabilities \(0.5693\) for A and \(0.4307\) for B.

The posterior means are \(0.5693(0.10)+0.4307(0.20)=0.1431\) at dose 1, \(0.5693(0.25)+0.4307(0.40)=0.3146\) at dose 2, and \(0.5693(0.50)+0.4307(0.65)=0.5646\) at dose 3. With target \(0.30\) and the closest-posterior-mean rule, dose 2 is recommended. Treating the pending patient as if their full window were complete would instead use exponent \(2\), overstating the amount of DLT-free information from that patient.

Worked Example: Applying an Overdose Constraint After Updating

Hypothetical data. Suppose the prior probabilities for states A, B, and C are \(0.25, 0.50, 0.25\). At doses 1, 2, and 3, the DLT probabilities in those states are respectively \((0.05,0.10,0.20)\), \((0.15,0.30,0.45)\), and \((0.30,0.50,0.70)\). The data comprise one completed DLT at dose 2, one completed DLT-free outcome at dose 1, and one patient at dose 3 who is DLT-free through half the window. The likelihood in state \(k\) is \(p_2(\theta_k)\{1-p_1(\theta_k)\}\{1-p_3(\theta_k)\}^{0.5}\).

The likelihoods are \(0.15(0.95)\sqrt{0.70}=0.119224\), \(0.30(0.90)\sqrt{0.50}=0.190919\), and \(0.45(0.80)\sqrt{0.30}=0.197180\). Multiplying by the priors gives \(0.029806\), \(0.095459\), and \(0.049295\), with total \(0.174560\). The posterior state probabilities are approximately \(0.1707, 0.5469, 0.2824\).

Suppose the target is \(0.30\) and the design permits a dose only if its posterior probability of exceeding that target is at most \(0.25\). At dose 1, no state exceeds \(0.30\), so that probability is \(0\). At dose 2, only state C exceeds \(0.30\), giving probability \(0.2824\), which fails the constraint. At dose 3, states B and C exceed \(0.30\), giving \(0.5469+0.2824=0.8293\), which also fails. Dose 1 is therefore the only eligible candidate and is selected under this constrained rule, even though its posterior mean risk is below the target. The constraint and the target-closest rule are distinct decision rules and should not be conflated.

Diagnostics, Alternatives, and Limitations

A reliable update begins with a patient-level audit. Check that every observed DLT contributes \(p_{d_i}(\theta)\), every completed DLT-free patient contributes \(1-p_{d_i}(\theta)\), and every pending DLT-free patient has a weight between zero and one that matches their recorded follow-up. Confirm that the prior-weighted likelihoods are nonnegative and that normalized posterior probabilities sum to one, allowing for rounding. Also inspect posterior dose probabilities across the candidate doses: if the model is intended to be monotone, a violation in the computed summaries may signal a coding or indexing error.

The principal alternative is to wait until all patients have completed the assessment window before updating. That avoids using fractional follow-up, but delays model-based decisions and discards interim information. TITE-CRM trades some of that simplicity for earlier updates under its weighting approximation. A full time-to-event model is another alternative: it explicitly models event timing and censoring rather than representing event-free partial follow-up by a fractional likelihood contribution. Such a model can use timing information more directly, but requires additional modeling assumptions and is not the same update as the weighted TITE-CRM formula.

The weighted update can be sensitive to the follow-up weights, the prior, and the dose-toxicity model. It can be misleading if follow-up duration is recorded incorrectly, if pending status depends on toxicity in a way not represented by the method, or if the assessment window or endpoint differs across patients. It also does not make a pending patient equivalent to a completed non-event, and it does not eliminate the need to apply the trial's escalation, de-escalation, or stopping rules. A posterior recommendation is conditional on the model, prior, data, and decision rule used.

Common Mistake

Wrong reasoning: “The patient has had no DLT so far, so I will enter a completed DLT-free outcome.” This uses the factor \(1-p_{d_i}(\theta)\) even when only part of the window has elapsed.
Correction: Use \(\{1-p_{d_i}(\theta)\}^{w_i}\) for a pending DLT-free patient. The smaller exponent reflects less than a full window of observed event-free follow-up. Conversely, once a DLT is observed, use the DLT contribution \(p_{d_i}(\theta)\); do not multiply it by the partial-follow-up weight.

Another common error is to choose the dose with the smallest posterior mean difference from target and then assume it must satisfy an overdose constraint. A target-matching criterion does not itself impose a safety bound. Evaluate any posterior safety rule separately, as in the third example.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A patient at dose \(j\) is DLT-free at 40% of the assessment window. What is the patient's likelihood contribution under weighted TITE-CRM?
For a completed DLT-free patient at dose \(j\), what exponent is used on \(1-p_j(\theta)\)?
Two equally likely states predict dose 1 risks \(0.10\) and \(0.20\). One completed DLT-free outcome occurs at dose 1. What are the posterior state probabilities?
Why does an observed DLT contribute \(p_j(\theta)\), rather than \(\{p_j(\theta)\}^{w}\), in the weighted TITE-CRM likelihood?

Solutions. (1) The contribution is \(\{1-p_j(\theta)\}^{0.40}\), because the patient has remained DLT-free through 40% of the window. (2) The exponent is \(1\), giving \(1-p_j(\theta)\). (3) The unnormalized weights are \(0.50(0.90)=0.45\) and \(0.50(0.80)=0.40\), with total \(0.85\). The posterior probabilities are \(0.45/0.85=0.5294\) and \(0.40/0.85=0.4706\). (4) The weighting rule uses the probability of an observed DLT by the end of the window, \(p_j(\theta)\); the partial-follow-up exponent applies to a patient who has remained event-free, not to an already observed event.