Why a Pending Patient Still Informs the Update
In a time-to-event continual reassessment method (TITE-CRM), patients need not all complete the toxicity observation window before the model is updated. A patient with incomplete follow-up contributes information, but not the same information as a patient whose full window was toxicity-free. The update therefore depends on distinguishing three records: toxicity observed, no toxicity by the end of the window, and no toxicity observed yet while follow-up remains incomplete.
This tutorial works through one numerical update and a later change to that update when a pending patient completes follow-up without toxicity. The dose recommendation is calculated only after the likelihood and posterior have been updated. As in earlier material on applying dose restrictions, any safety or dose-skipping rules still determine which doses are admissible; the posterior calculation does not override them.
Likelihood Contributions and Assumptions
For an observed toxicity, the patient’s contribution to the likelihood is \(q_j(\theta)\). For a patient who completes the full window without toxicity, it is \(1-q_j(\theta)\). For a patient with no observed toxicity at a fraction \(w_i\) of the window, the working TITE-CRM contribution is \(1-w_iq_j(\theta)\). Thus, for independent patients conditional on \(\theta\), the likelihood is
For a toxicity-free patient whose window is complete, \(w_i=1\), so the second factor becomes \(1-q_{d_i}(\theta)\). The contribution for a patient with observed toxicity does not use the partial follow-up weight: the event has already occurred.
The pending-patient factor follows from a working timing assumption. Conditional on toxicity occurring within the window, suppose the event-time distribution is uniform over that window. If the end-of-window toxicity probability is \(q_j(\theta)\), then by fraction \(w_i\) of the window the probability of having had toxicity is \(w_iq_j(\theta)\). Consequently, the probability of no toxicity yet is \(1-w_iq_j(\theta)\). This formulation also assumes that censoring or delayed observation of follow-up does not depend on a patient’s unobserved toxicity risk, conditional on the model.
These assumptions matter: a weight is not a substitute for knowing the follow-up history, and the linear relationship between follow-up fraction and event probability is a modeling choice. The update also assumes the dose skeleton is ordered, the model adequately represents dose-related toxicity, and patients’ outcomes are conditionally independent given the shared parameter. A posterior distribution is obtained by multiplying this likelihood by the prior and normalizing.
Worked Example: Constructing the Likelihood With a Pending Patient
Hypothetical data. Consider four doses with skeleton values \(\pi_1=0.05\), \(\pi_2=0.12\), \(\pi_3=0.25\), and \(\pi_4=0.40\). Three hypothetical patients have been treated: one at dose 1 with observed toxicity, one at dose 2 who completed the window without toxicity, and one at dose 3 who has completed half the window without observed toxicity. Thus their follow-up weights are 1 for the completed patient and \(w=0.5\) for the pending patient.
Use \(q_j(\theta)=\pi_j^{\exp(\theta)}\). The likelihood contribution from the observed toxicity at dose 1 is \(q_1(\theta)\). The completed toxicity-free patient contributes \(1-q_2(\theta)\), and the pending patient contributes \(1-0.5q_3(\theta)\). Therefore,
For example, at \(\theta=0\), \(\exp(\theta)=1\), so the model probabilities equal the skeleton values. Substitution gives
This is a likelihood value, not the probability that \(\theta=0\) is true. The pending patient’s contribution is \(0.875\), rather than \(1-0.25=0.75\), because only half the toxicity window has been observed without toxicity.
A Fully Worked Posterior Update
For transparent arithmetic, use a discrete prior with probability 0.25 at \(\theta=-\log 2\), probability 0.50 at \(\theta=0\), and probability 0.25 at \(\theta=\log 2\). These parameter values make \(\exp(\theta)\) equal to 0.5, 1, and 2. This is a hypothetical prior chosen to show the calculation; practical analyses may use a continuous prior and numerical integration.
The dose probabilities at each of the three parameter values follow directly from the model. For instance, at dose 1 they are \(0.05^{0.5}=0.2236\), \(0.05^1=0.0500\), and \(0.05^2=0.0025\). Calculate the likelihood for the three records in the previous example, then multiply each likelihood by its prior probability.
| \(\theta\) | \(q_1(\theta)\) | \(q_2(\theta)\) | \(q_3(\theta)\) | \(L(\theta)\) | Prior probability | Unnormalized posterior |
|---|---|---|---|---|---|---|
| \(-\log 2\) | 0.2236 | 0.3464 | 0.5000 | 0.1096 | 0.2500 | 0.02740 |
| 0 | 0.0500 | 0.1200 | 0.2500 | 0.03850 | 0.5000 | 0.01925 |
| \(\log 2\) | 0.0025 | 0.0144 | 0.0625 | 0.002387 | 0.2500 | 0.0005968 |
The unnormalized posterior entries sum to \(0.04725\) (rounding). Divide each entry by that sum to obtain posterior probabilities of approximately 0.5800, 0.4074, and 0.01263. The relatively high probability at \(\theta=-\log 2\) reflects that, among these prior values, it assigns substantially greater toxicity probabilities and hence makes the observed toxicity at dose 1 more plausible. The completed toxicity-free record and pending record also affect this balance.
Worked Example: Turning the Posterior Into a Dose Recommendation
Suppose the target toxicity probability is \(\phi=0.25\), all four doses are admissible, and the specified recommendation rule is to select the dose whose posterior mean toxicity probability is closest to \(\phi\). The posterior mean at dose \(j\) is the sum of \(q_j(\theta)\) multiplied by the posterior probability at each parameter value.
At dose 2, for example, the posterior mean is
Applying the same calculation at each dose gives posterior means of approximately 0.1501, 0.2500, 0.3926, and 0.5318 for doses 1 through 4. Dose 2 is closest to the target 0.25, so it is recommended under the stated rule. This is a model-based estimate and recommendation, not a claim that the true toxicity probability at dose 2 is exactly 0.25. If a dose is excluded by a restriction, the recommendation must instead be selected from the admissible set.
What Happens When Follow-Up Matures?
A pending record is not permanently assigned its partial weight. If the patient reaches the end of the observation window without toxicity, that patient’s contribution changes from \(1-wq_j(\theta)\) to \(1-q_j(\theta)\). If toxicity is observed before the window ends, the record contributes the toxicity likelihood factor instead. The posterior should be recalculated using the updated record rather than treating the earlier partial contribution as fixed.
Worked Example: Updating After a Pending Patient Completes Follow-Up
Return to the hypothetical three-patient dataset, and suppose the dose 3 patient completes the full window without toxicity. The likelihood is now \(q_1(\theta)\{1-q_2(\theta)\}\{1-q_3(\theta)\}\). Relative to the earlier likelihood, the factor for each parameter value is multiplied by \(\{1-q_3(\theta)\}/\{1-0.5q_3(\theta)\}\).
For \(\theta=-\log 2\), this ratio is \(0.5/0.75=0.6667\); for \(\theta=0\), it is \(0.75/0.875=0.8571\); and for \(\theta=\log 2\), it is \(0.9375/0.96875=0.9677\). Multiplying the earlier posterior probabilities by these ratios gives unnormalized values of approximately 0.3866, 0.3492, and 0.01222. Normalizing by their sum, 0.7481, gives updated posterior probabilities of approximately 0.5168, 0.4668, and 0.01634.
The updated posterior mean toxicity probability at dose 2 is therefore
The completed toxicity-free follow-up shifts probability away from parameter values predicting greater toxicity at dose 3. The mean at dose 2 is now about 0.2353 rather than 0.2500. Under the same target and recommendation rule, dose 2 remains closest to 0.25, but the posterior has changed.
Diagnostics, Alternatives, and Limitations
A numerical update should be checked at the level of both records and calculations. Confirm that each patient’s dose, event status, and follow-up time are correct; that a pending patient has \(0\leq w_i<1\); and that a completed toxicity-free window uses \(w_i=1\). Check that observed toxicities contribute \(q_j(\theta)\), not a toxicity-free factor, and verify that posterior probabilities sum to one after normalization. Also inspect the fitted dose probabilities across parameter values: the ordered skeleton and positive \(\exp(\theta)\) preserve the dose ordering in this model.
The TITE-CRM update allows dose decisions before every patient has a complete window, but it does not eliminate uncertainty from pending outcomes. Its estimate can be sensitive to the skeleton, prior, choice of follow-up weight, or timing assumption. If event risk is not approximately spread in the assumed way over the window, the linear weight \(w_i\) may misrepresent the information in partial follow-up. With very short follow-up, the pending contribution is close to 1 for a broad range of toxicity probabilities and therefore supplies little information.
Two alternatives clarify the trade-off. Waiting for all patients to finish their windows avoids using the partial-follow-up likelihood, but delays updates and leaves interim information unused. Treating every pending patient as if the window were complete and toxicity-free is not a valid shortcut: it uses \(1-q_j(\theta)\) too soon and can make higher toxicity probabilities appear less plausible than the observed follow-up supports. More detailed time-to-event models can represent nonuniform event-time patterns, but require additional modeling assumptions and information; they are not interchangeable with the simple weighted likelihood used here.
The key operational point is to keep the data update and the dose rule distinct. First encode each patient’s observed status and calculate the posterior under the stated model. Then apply the target-based selection rule and any restrictions to the resulting posterior. The posterior cannot correct a data-timing error or compensate for a poorly calibrated skeleton, so sensitivity to plausible priors, skeletons, and follow-up assumptions remains important.
Check Your Understanding
Answer each question using the likelihood and conventions given above. Solutions follow the questions.
- At a dose with \(q_j(\theta)=0.30\), a patient has no observed toxicity after 40% of the window. What likelihood contribution does that record make?
- State the likelihood contributions for an observed toxicity, a completed toxicity-free window, and a pending toxicity-free follow-up period.
- Suppose posterior probabilities at three parameter values are 0.58, 0.407, and 0.013, and the corresponding toxicity probabilities at a particular dose are 0.20, 0.30, and 0.50. Calculate the posterior mean toxicity probability, to three significant figures.
- Why is replacing a pending patient’s contribution by a completed toxicity-free contribution generally incorrect?
Solutions. (1) Here \(w=0.40\), so the contribution is \(1-wq_j(\theta)=1-(0.40)(0.30)=0.88\). (2) The contributions are \(q_j(\theta)\), \(1-q_j(\theta)\), and \(1-w_iq_j(\theta)\), respectively. (3) The posterior mean is \(0.58(0.20)+0.407(0.30)+0.013(0.50)=0.116+0.1221+0.0065=0.2446\), or 0.245 to three significant figures. (4) A pending patient has not yet been observed through the full window, so the no-event information applies only to the follow-up observed so far. Using \(1-q_j(\theta)\) assigns the patient a full-window non-event before that outcome is known.