Choosing Follow-Up Weights in TITE-CRM

Why Follow-Up Weights Matter

In TITE-CRM, a patient who has not experienced a dose-limiting toxicity (DLT) but has not yet completed the toxicity observation window is not treated as either a fully observed non-event or as having no information. A follow-up weight represents how much of the completed no-DLT evidence that pending patient contributes to the dose-toxicity model. The choice matters most when several patients have pending outcomes: different weights can produce different estimated toxicity probabilities and, consequently, different dose recommendations.

The weight is a modelling choice, not a fact read directly from the patient record. Elapsed time is observed; the amount of evidence that elapsed time represents depends on assumptions about when toxicity can occur. The introductory material on TITE-CRM establishes the role of partial follow-up. Here the emphasis is on interpreting and choosing the weight.

How a Weight Enters the Likelihood

Write \(\pi_j\) for the CRM-model probability of DLT by the end of the window at dose \(j\). In the usual weighted-likelihood representation, a patient with no DLT observed so far contributes \((1-\pi_j)^{w_i}\). A patient with a DLT contributes \(\pi_j\), and a patient who completes the window without DLT contributes \(1-\pi_j\). Thus a weight scales the log-likelihood contribution:

For \(0<w_i<1\), the pending patient's no-DLT contribution lies between 1 and \(1-\pi_j\). It supplies less evidence against a high DLT probability than a completed toxicity-free observation does. The exponent is the relevant scale: a weight of 0.5 means half the completed non-event contribution on the log-likelihood scale, not half the probability of toxicity or half a patient.

This representation is a weighted or approximate likelihood when used with a CRM model for the binary endpoint by time \(T\). Its interpretation depends on the rule used to set \(w_i\). In particular, a weight should not be described as an exact likelihood contribution for every possible event-time distribution.

Linear Weights and Their Assumption

A common choice is the linear elapsed-time weight \(w_i=t_i/T\). One justification comes from a constant hazard over the observation window. Suppose that, at dose \(j\), the hazard of DLT is constant at \(\lambda_j\) per unit time. Then the probability of remaining DLT-free through time \(t\) is \(\exp(-\lambda_jt)\), while the probability of avoiding DLT through the full window is \(1-\pi_j=\exp(-\lambda_jT)\). Therefore:

Under this model, the linear weight gives the exact probability of remaining event-free through \(t\), expressed as a power of the full-window survival probability. A constant hazard is a substantive assumption: it says that the instantaneous event rate does not change with time during the window, conditional on dose and the model. It is not implied by measuring elapsed time.

Other Choices Based on Elapsed Time

Linear weighting treats every additional day as adding the same amount to the exponent. A flexible alternative is a power weight \(w_i=(t_i/T)^c\), where \(c>0\) is chosen in advance. When \(c<1\), the weight is greater than the linear weight at interior times, so information is credited earlier. When \(c>1\), the weight is smaller at interior times, so information accumulates later. These transformations can represent sensitivity scenarios, but they do not by themselves establish which timing pattern is clinically correct.

A more explicitly event-time-based option starts with \(G(u)\), the assumed cumulative distribution of time to DLT conditional on a DLT occurring by \(T\), evaluated at \(u=t/T\). Under this assumption, a patient with no DLT by \(t\) has probability \(1-\pi_jG(u)\) of that observation. This probability is not generally the same as \((1-\pi_j)^u\). One can express it as a power of the full-window survival probability by defining an effective weight:

This weight usually depends on the dose-specific probability \(\pi_j\), as well as elapsed time. It equals 0 at the start and 1 at the end of the window when \(G(0)=0\) and \(G(1)=1\). For small \(\pi_j\), the approximation \(\log(1-z)\approx-z\) gives \(w_{\mathrm{eff}}\approx G(u)\). Thus an event-time distribution concentrated early assigns more information to early event-free follow-up than one concentrated late.

Choosing and Checking a Weighting Rule

A usable elapsed-time rule should be nondecreasing, remain between 0 and 1, assign 0 at no follow-up, and reach 1 at the end of the window. These properties ensure that a patient receives no more no-DLT credit at an earlier time than at a later time, and that completed toxicity-free follow-up receives its full contribution. Beyond these basic checks, choice requires a defensible view of event timing and sensitivity to uncertainty about that view.

For a fixed window, linear weighting is simple, transparent, and interpretable under a constant-hazard model. A power weight offers a transparent sensitivity analysis when earlier or later accumulation of evidence is considered plausible. An event-time-calibrated weight can encode a specified timing distribution more directly, but it adds assumptions and may vary with \(\pi_j\). A model that directly includes time to event can represent timing in the likelihood rather than approximating it with a common weight; that approach also requires its own event-time model and assumptions. No weighting rule removes the need to check that the endpoint is consistently ascertained and that censoring before the end of the window is not informative in a way the model ignores.

Useful diagnostics include plotting the proposed weight against elapsed time, checking that it has the endpoint properties above, and comparing dose updates under plausible alternative rules. If conclusions change materially across reasonable weights, pending follow-up is influential and the chosen timing assumption should be made explicit. A weight sum can be a rough descriptive measure of accumulated weighted follow-up, but it is not generally a literal effective sample size: patients contribute information through the model likelihood, not through a simple count alone.

The main practical comparison is therefore not “which weight is mathematically correct?” in isolation, but “which timing assumptions are credible, and would reasonable alternatives change the dose decision?” State the rule and its assumptions, use the same endpoint window consistently, and examine sensitivity when pending outcomes could influence the update.