Why Partial Follow-Up Matters
In a dose-finding trial, a patient may have received a dose but not yet completed the prespecified toxicity observation window. Waiting for every patient to finish that window can slow accrual and dose decisions. Treating each patient who has not yet had a recorded dose-limiting toxicity (DLT) as fully event-free, however, overstates the evidence: a DLT could still occur during the remaining follow-up.
The time-to-event continual reassessment method (TITE-CRM) addresses this gap by allowing a patient with incomplete follow-up to contribute information before the observation window ends. The key distinction is between a patient with no DLT by the end of the window and a patient who has had no DLT so far. The latter has a pending outcome, not a confirmed non-event for the full window.
The Weighted Contribution
Suppose the CRM dose-toxicity model assigns probability \(p_j(\theta)\) of a DLT within the full window at dose level \(j\), given model parameter \(\theta\). A common CRM specification uses a prespecified increasing skeleton \(q_1,\ldots,q_J\), with \(0<q_1<\cdots<q_J<1\), and sets \(p_j(\theta)=q_j^{\exp(\theta)}\). The precise dose model can vary; the TITE weighting concerns how the observed follow-up contributes to updating that model.
In the standard weighted-likelihood version of TITE-CRM, a patient with a DLT contributes \(p_j(\theta)\). A patient with no DLT yet contributes \(\{1-p_j(\theta)\}^{w_i}\). Thus, partial follow-up contributes less evidence for a low full-window toxicity probability than completed follow-up does. The contribution is a fractional likelihood contribution, not a declaration that the pending patient had no DLT over the entire window.
Here \(j_i\) is patient \(i\)'s dose level, \(y_i=1\) if a DLT has occurred within the window and \(y_i=0\) if none has yet been observed. For a patient with an event, \(t_i\) is taken up to the event time, but the event contribution is \(p_{j_i}(\theta)\), not a follow-up-weighted version of that term. For a patient with no event yet, \(w_i\) measures the fraction of the window observed. With a prior distribution \(\pi(\theta)\), the corresponding weighted-likelihood update is proportional to \(\pi(\theta)L(\theta)\).
This approach requires a prespecified window and a consistent definition of DLT. It also assumes that patients’ outcomes are conditionally independent given the dose-toxicity model, and that the dose model is suitable for the trial. Interpreting elapsed follow-up as a useful fraction of information additionally relies on a reasonable relationship between follow-up time and the opportunity to observe toxicity. In particular, follow-up should be recorded accurately, and withdrawal or missed assessments should not silently be counted as event-free time.
Worked Contributions and Updates
Worked Example: A Pending Patient at One Dose
Consider a hypothetical patient treated at a dose where the model gives a full-window DLT probability of \(p_j(\theta)=0.25\). The observation window is 28 days. At the current review, the patient has had no DLT and has been followed for 11.2 days. Calculate the contribution to the weighted likelihood.
The follow-up fraction is \(w_i=11.2/28=0.40\). Because no DLT has yet been observed, the contribution is:
The patient contributes 0.8913 at this candidate model value. For comparison, a patient who completed all 28 days without a DLT would contribute \(0.75^1=0.75\), which is stronger evidence in favor of a lower toxicity probability. The pending patient has not been classified as event-free for 28 days; the smaller information contribution reflects the 40% of the window observed.
Worked Example: Comparing Two Candidate Toxicity Models
Suppose a hypothetical trial has two patients. Patient A experienced a DLT at dose 1. Patient B has had no DLT after completing half of the window at dose 2, so \(w_B=0.50\). Compare two candidate CRM parameter values: under model 1, the probabilities at doses 1 and 2 are 0.30 and 0.20; under model 2, they are 0.60 and 0.50. For illustration, assign equal prior probabilities to the two candidate models.
Patient A's event contributes the dose 1 toxicity probability. Patient B's pending non-event contributes the square root of one minus the dose 2 toxicity probability. The weighted likelihoods are therefore:
With equal prior probabilities, the normalized weights are proportional to these likelihoods. The weight for model 2 is:
The weighted update assigns about 61.26% of its probability to model 2, compared with 50% before these observations. This is an illustrative calculation using two candidate models, not a claim that all TITE-CRM analyses use a two-point prior. It shows how the event and the partial non-event jointly affect the update.
Worked Example: Follow-Up Fraction Controls the Non-Event Contribution
At a hypothetical dose, let the candidate full-window DLT probability be 0.40. Two patients have had no DLT: one has completed 25% of the window and the other has completed 75%. Find each patient's contribution and their product.
For the patient with \(w_i=0.25\):
For the patient with \(w_i=0.75\):
Their combined contribution is \(0.8801\times0.6817\approx0.6000\). Equivalently, because the patients share the same candidate dose probability, it is \(0.60^{0.25+0.75}=0.60\). A completed non-event at this dose would also contribute 0.60. The equality illustrates how this weighted likelihood aggregates fractional follow-up in this particular calculation; it does not make two patients with incomplete follow-up equivalent to two fully observed patients.
Justification and Practical Use
The weighting has a simple log-likelihood interpretation. A completed non-event at dose \(j\) contributes \(\log\{1-p_j(\theta)\}\). If only a fraction \(w_i\) of the window has been observed without an event, the weighted version uses that same log contribution multiplied by \(w_i\). Exponentiating returns the fractional contribution:
Thus, if \(w_i=0\), the pending non-event contributes 1 to the likelihood and supplies no evidence; if \(w_i=1\), it contributes the ordinary full-window non-event likelihood. For intermediate values, it contributes between these two limits. This is a justification for the weighting rule as a fractional-information approximation, not a derivation of an exact event-time likelihood.
At each analysis, the available information can be represented with three distinct statuses: DLT observed, no DLT with incomplete follow-up, and no DLT after completed follow-up. For the weighted likelihood, the first status contributes the event term, while the latter two contribute non-event terms with their appropriate follow-up fractions. Maintaining those distinctions prevents pending patients from being treated as though their toxicity windows were complete.
Diagnostics, Alternatives, and Limitations
Before relying on a weighted update, check that each patient's dose, DLT status, follow-up start, and assessment time are recorded consistently. The follow-up fraction must satisfy \(0\leq w_i\leq1\); a value above 1 usually indicates a time-unit or window-definition error. A recorded DLT should not also be entered as a pending non-event, and a patient who has completed the window without a DLT should have \(w_i=1\). Reviewing these checks can reveal data-processing mistakes before they affect dose-toxicity estimates.
The main alternatives address different trade-offs. A trial can wait for complete observation windows before updating the CRM; this avoids using fractional contributions but may slow decisions. Coding every pending patient as a full-window non-event is not a sound shortcut because it gives more evidence than the observed follow-up warrants. An explicit time-to-event model can represent event timing and censoring directly, but requires additional assumptions about the distribution of event times. Weighted TITE-CRM uses partial follow-up with a simpler rule, at the cost of relying on an approximation rather than specifying that full event-time model.
TITE-CRM does not eliminate uncertainty about patients still under observation. Its weighted likelihood cannot rescue a poorly chosen dose-toxicity model or make incomplete follow-up equivalent to completed follow-up. It can also be misleading if observation times are inaccurate, the window is not applied consistently, or patients leave follow-up in a way related to their unobserved toxicity risk. In those settings, the analysis should not treat time without a recorded DLT as automatically informative in the intended way.
Check Your Understanding
Answer each question using the weighted-likelihood convention described above. Solutions are included below.
- A patient has no DLT after 9 days of a 30-day window. What is the follow-up fraction, and what contribution does the patient make at a candidate dose with \(p_j(\theta)=0.20\)?
- A patient has a DLT on day 12 of a 30-day window. What is the patient's contribution at a candidate dose with \(p_j(\theta)=0.35\)?
- Why should a patient with no DLT after 10 days of a 30-day window not be entered as a completed non-event?
- At a dose with candidate DLT probability 0.25, compare the contributions from a pending non-event with \(w_i=0.20\) and a completed non-event.
Solutions. (1) \(w_i=9/30=0.30\), so the contribution is \(0.80^{0.30}\approx0.9352\). (2) An observed DLT contributes \(p_j(\theta)=0.35\); its contribution is not raised to a follow-up fraction. (3) Only one third of the observation window has elapsed, so a DLT could still occur. The weighted likelihood uses \(w_i=10/30\), rather than treating the patient as event-free for the full window. (4) The pending contribution is \(0.75^{0.20}\approx0.9441\); the completed non-event contribution is \(0.75^1=0.75\). The pending patient provides less evidence from the absence of a DLT so far.