Partial Follow-Up Pitfalls in TITE-CRM

Why Partial Follow-Up Needs Care

In a time-to-event continual reassessment method (TITE-CRM), patients may begin treatment before earlier patients have completed the toxicity observation window. A patient with no dose-limiting toxicity (DLT) recorded so far is not necessarily a patient who has completed that window without a DLT. The distinction matters: treating both patients as fully toxicity-free can make a dose appear safer than the available follow-up supports.

The introductory material on TITE-CRM describes how partially observed toxicity windows can contribute to dose finding. Here the emphasis is on diagnosing three errors in those updates: coding pending patients as complete non-events, assigning weights that do not represent follow-up, and updating before the outcome window has been clearly defined.

Definition: Let \(W\) be the prespecified DLT observation-window length, measured from a patient's dose administration. Let \(t_i\) be patient \(i\)'s observed follow-up time within that window, and let \(y_i=1\) if a DLT has been observed by the analysis and \(y_i=0\) otherwise. A patient with \(y_i=0\) and \(t_i<W\) is pending; a patient with \(y_i=0\) and \(t_i\geq W\) has completed the window without an observed DLT.

Write \(p_j(\theta)\) for the CRM model's probability of a DLT by the end of the window at dose level \(j\), where \(\theta\) denotes the model parameter or parameters. The TITE-CRM update uses the observed outcomes and follow-up to update inference about these dose-specific probabilities. Whether an update is sound depends on the definition of the outcome, the window, and the contribution assigned to each patient.

The Weighted Likelihood and Its Assumptions

A common TITE-CRM working likelihood assigns each patient with no DLT observed so far a follow-up weight \(w_i\), usually \(w_i=t_i/W\) while the patient is pending. A patient with an observed DLT contributes the dose-specific DLT probability; a patient with no DLT contributes the probability of remaining free of DLT raised to the follow-up weight:

L(\theta)=\prod_i p_{j_i}(\theta)^{y_i} \left\{1-p_{j_i}(\theta)\right\}^{w_i(1-y_i)}, \qquad w_i=\min\left(\frac{t_i}{W},1\right).

Here \(j_i\) is the dose received by patient \(i\). For an observed DLT, \(y_i=1\), so the contribution is \(p_{j_i}(\theta)\); the weight does not discount an event that has already occurred. For a patient without an observed DLT, \(y_i=0\), so the contribution is \(\{1-p_{j_i}(\theta)\}^{w_i}\). A pending patient's contribution is partial, while a patient followed through the entire window has \(w_i=1\) and contributes \(1-p_{j_i}(\theta)\).

This weighted likelihood is a modeling approximation, not a universal exact likelihood for censored event times. One justification is to assume a constant event hazard over the window. If the probability of a DLT by time \(W\) is \(p\), survival through \(W\) is \(1-p\). Under a constant hazard, survival through a fraction \(w\) of the window is \((1-p)^w\), giving the no-event contribution above. This assumption is a useful motivation for the weight, but actual event-time patterns may not follow it.

Assumptions and checks: The window \(W\) must be defined for the DLT outcome being modeled, and follow-up time must be measured from the relevant patient's dose administration. The usual linear weight requires \(0\leq t_i\), \(W>0\), and \(w_i=\min(t_i/W,1)\). It treats follow-up as informative in proportion to elapsed window time; this may be a poor approximation if DLT risk is strongly concentrated early or late. Outcome ascertainment and follow-up must also be sufficiently reliable for recorded event status and time to be meaningful.

Worked Examples: Seeing What the Weight Changes

Worked Example: A Pending Patient Is Not a Completed Non-Event

Suppose, hypothetically, a patient received dose level 2 and has had no recorded DLT after 14 days. The protocol-defined window is 28 days. At a particular value of the CRM parameter, suppose \(p_2(\theta)=0.30\). Find the patient's contribution under the standard weight, then compare it with incorrectly treating the patient as having completed the window.

The patient is pending because \(14<28\). The follow-up weight is \(w=14/28=0.50\). With no DLT recorded, the weighted contribution is

(1-p_2(\theta))^w=(1-0.30)^{0.50} =0.70^{0.50}\approx 0.8367.

If the patient is incorrectly coded as a completed non-event, the contribution becomes \(0.70^1=0.7000\). That is a larger penalty to parameter values predicting \(p_2(\theta)=0.30\), even though only half the window has elapsed. Across parameter values and patients, this error can distort the model update; its effect on dose selection is not guaranteed to have the same direction in every dataset. The correct coding is no observed DLT with \(w=0.50\), not a completed non-event.

Worked Example: A Weight Must Match the Defined Window

Suppose, hypothetically, a patient's protocol-defined DLT window is 28 days and the patient has no DLT recorded after 14 days at a dose where \(p_j(\theta)=0.40\). Compare the appropriate linear weight with an inappropriate weight calculated using a 21-day window.

Using the defined window, \(w=14/28=0.50\), so the contribution is \((1-0.40)^{0.50}=0.60^{0.50}\approx0.7746\).

w_{\text{wrong}}=\frac{14}{21}=\frac{2}{3}, \qquad (1-0.40)^{2/3}=0.60^{2/3}\approx0.7114.

The 21-day denominator makes the patient appear to have accumulated more follow-up than the protocol-defined 28-day window warrants. It therefore assigns a stronger no-event contribution. The arithmetic is internally consistent, but the weight is based on the wrong window. A weight is meaningful only relative to the endpoint and time horizon that define \(p_j(\theta)\).

Worked Example: Updating Before the Window Is Fixed

Suppose, hypothetically, a team has observed 14 days without DLT for a patient and is considering whether the window should be 28 or 42 days. At a particular parameter value, assume the model's DLT probability for the eventual window is \(p_j(\theta)=0.30\). Calculate the two possible contributions and explain why neither calculation resolves the design ambiguity.

If the window is 28 days, \(w=14/28=0.50\), and the contribution is \(0.70^{0.50}\approx0.8367\). If it is 42 days, \(w=14/42=1/3\), and the contribution is \(0.70^{1/3}\approx0.8879\).

0.70^{0.50}\approx0.8367, \qquad 0.70^{1/3}\approx0.8879.

The different contributions reflect different definitions of the endpoint, not merely different arithmetic choices. The probability \(p_j(\theta)\) must refer to DLT by the same window used to calculate the weight. The window and endpoint therefore need to be specified before the update. Choosing a denominator after seeing the follow-up can change the likelihood contribution and undermine a consistent interpretation of the model.

A Short Justification for the Linear Follow-Up Weight

The usual weight can be motivated directly under a constant-hazard model. Let \(\lambda\) be a constant DLT hazard per unit time over the window. The probability of no DLT by time \(t\) is \(\exp(-\lambda t)\), so the probability of no DLT by \(W\) is \(1-p=\exp(-\lambda W)\). For \(t=wW\), it follows that:

\exp(-\lambda t) =\exp(-\lambda wW) =\{\exp(-\lambda W)\}^{w} =(1-p)^w.

Thus, with constant hazard, the probability of no DLT through observed time \(t\) has the weighted form used for a pending patient. This is a justification under a particular event-time assumption, not proof that linear weights are appropriate for every toxicity process. If risk changes sharply over time, equal fractions of elapsed follow-up need not represent equal fractions of information about the eventual window-level probability.

Diagnostics, Alternatives, and Limitations

Before an update, check patient-level status and the likelihood contribution rather than relying only on a summary count of DLTs. For each patient with no recorded DLT, verify the dose, dose date, follow-up time, window, and resulting weight. A pending patient should generally have a weight between zero and one; a completed window should have weight one. A weight above one, a negative weight, or a weight based on a different time origin signals an implementation or data-definition problem.

Also check that the endpoint is consistent. If the model's \(p_j(\theta)\) is the probability of a specific DLT by a specified time, follow-up must relate to that same event and time horizon. A different endpoint definition, a window changed during analysis, or an unrecorded interruption in observation cannot be repaired simply by inserting a convenient fraction into the formula. Missing follow-up is not automatically equivalent to observed, event-free follow-up; its handling requires defensible assumptions about why observation stopped.

The weighted approach is attractive because it can use partial follow-up without requiring a fully specified event-time distribution. Its limitation is that its weights approximate how much information partial observation contributes. An alternative is to specify an event-time model and use the likelihood for the observed event or censoring time. That can represent nonconstant hazards, but it requires stronger modeling choices and reliable event-time information. Neither approach removes the need to define the toxicity endpoint and observation window in advance.

Approach	What it uses	Main consideration
Weighted TITE-CRM	Window-level DLT status and a follow-up weight for patients without an observed event	Relatively simple; the weight is an approximation whose suitability depends on follow-up and event-time assumptions
Explicit event-time likelihood	Observed event times or censoring times under a specified time-to-event model	Can model timing more directly; requires a defensible event-time distribution and appropriate censoring assumptions
Wait for complete windows	Window-complete DLT outcomes before updating	Avoids using partial-follow-up weights but delays updates and subsequent decisions

A common mistake is to reason, “No DLT has been recorded, so this patient is toxicity-free.” The correction is to distinguish “no DLT observed so far” from “no DLT observed through the complete window.” Another mistake is to select a weight that seems intuitively conservative or convenient without checking its time origin, denominator, and relation to the modeled endpoint. A numerically plausible weight is not valid unless those definitions agree.

Key takeaway: A pending patient contributes partial information, not a completed non-event. Define the DLT window and endpoint first, calculate follow-up from the patient's dose date, and use a weight that matches both the observation and the assumptions of the chosen TITE-CRM likelihood.

Check Your Understanding

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

A patient has no recorded DLT 10 days after dosing. The defined window is 20 days. Under linear weighting, what is the follow-up weight, and what contribution applies if \(p_j(\theta)=0.25\)?
Why is it incorrect to give a patient who is halfway through the window the same no-DLT contribution as a patient who completed the window?
A weight is calculated using time since trial enrollment, although the window begins at dose administration. What is the problem?
What additional modeling feature does an explicit event-time likelihood require compared with the usual weighted TITE-CRM likelihood?

Solutions. (1) \(w=10/20=0.50\), and the contribution is \((1-0.25)^{0.50}=0.75^{0.50}\approx0.8660\). (2) The completed patient has weight 1 and contributes \(1-p_j(\theta)\); assigning that contribution to the halfway-followed patient overstates the follow-up information. (3) The time origin is wrong: elapsed time must be measured from the dose administration that starts the defined window, or the calculated weight will not represent follow-up within that window. (4) It requires a specified event-time model, together with assumptions that make the likelihood for observed event and censoring times appropriate.