Deriving TITE-CRM Weights From Partial Follow-Up

Why Partial Follow-Up Needs a Weight

In a dose-finding trial, a patient who has not experienced a dose-limiting toxicity (DLT) may still be within the DLT assessment window. Treating that patient as if the entire window had been observed would overstate the evidence for safety. Ignoring the patient until the window ends avoids that error but discards information from the time already observed. TITE-CRM addresses this trade-off by assigning a partially followed patient a weight between zero and one.

The weight is not an estimated probability that the patient will experience a DLT. It represents the fraction of the assessment window completed. To understand what that fraction means statistically, specify how DLT events are assumed to occur within the window. The derivation below first uses uniform event timing conditional on a DLT by the end of the window, then distinguishes the exact likelihood under that assumption from the conventional TITE-CRM power-likelihood contribution.

The cap matters: once a patient has completed the assessment window, additional follow-up does not make the weight exceed one. The definition also assumes a common, relevant assessment window. If the protocol defines different windows for different DLT types or patients, the weight must be tied to the window used for the corresponding endpoint.

Uniform Event Timing and the Exact Partial-Follow-Up Likelihood

For the first derivation, suppose that, conditional on experiencing a DLT by \(T\), the event time is uniformly distributed over the window. In symbols, if \(\tau_i\) is the DLT time conditional on a DLT occurring by \(T\), then \(P(\tau_i\leq u\mid \text{DLT by }T)=u/T\) for \(0\leq u\leq T\). This is an assumption about when events occur among patients who experience one; it does not say that every patient has a DLT.

The chance that a DLT has occurred by time \(u_i\) is the chance of a DLT by \(T\), multiplied by the conditional chance that its time falls by \(u_i\). Therefore the probability of no DLT observed by \(u_i\) is:

Thus, under this uniform-timing assumption, the exact contribution to the likelihood from a patient with no DLT observed through \(u_i\) is \(1-p_iw_i\). A patient with an observed DLT by the end of the window contributes \(p_i\) to a likelihood that uses only the binary window outcome. If the event time itself is included in the likelihood, its timing density must also be specified; that is a different likelihood formulation.

The conventional TITE-CRM weighted likelihood instead uses a fractional power of the full-window no-DLT contribution:

For a patient with a DLT, \(y_i=1\), so the contribution is \(p_i\). For a patient without an observed DLT, \(y_i=0\), it is \((1-p_i)^{w_i}\). A fully followed patient without a DLT has \(w_i=1\) and contributes \(1-p_i\); a patient observed for no time has \(w_i=0\) and contributes 1. This last case adds no evidence from an unobserved window.

Why the Conventional Weight Is an Approximation Under Uniform Timing

The uniform-timing calculation gave \(1-p_iw_i\), not \((1-p_i)^{w_i}\). The conventional weighted likelihood is therefore not the exact censored-data likelihood under uniform conditional event timing. It is a fractional-likelihood approximation: it gives a fraction \(w_i\) of the log-likelihood contribution from a completed, DLT-free window.

The connection can be seen by expanding both log contributions when \(p_i\) is small. The exact uniform-timing log contribution is \(\log(1-p_iw_i)\), while the conventional contribution is \(w_i\log(1-p_i)\). Their expansions are:

Both begin with the same term, \(-p_iw_i\), so they agree to first order in the DLT probability. Their second-order terms differ unless \(w_i=0\), \(w_i=1\), or \(p_i\) is negligible. More explicitly, the exact likelihood contribution minus the power contribution is generally not zero; for modest \(p_i\), the difference may be small, but it can matter at higher risks or intermediate follow-up fractions.

There is another event-time assumption under which the power contribution is exact. Suppose the event hazard is constant over the window, with hazard \(\lambda_i\), and calibrate it so that the probability of an event by \(T\) is \(p_i\). Then \(1-p_i=\exp(-\lambda_i T)\), and survival without a DLT to \(u_i\) is:

This exact result follows from a constant hazard, not from uniform event timing conditional on a DLT. The two assumptions describe different event-time distributions. It is important to state which interpretation is intended rather than presenting the power contribution as an exact consequence of every partial-follow-up model.

Combining Contributions Across Patients

For independent patients conditional on the dose-toxicity model, the likelihood is the product of their individual contributions. Under the conventional TITE-CRM formulation, this gives:

where \(p_i(\theta)\) is the modelled DLT probability at patient \(i\)'s dose. This expression shows how the time weight enters: only a patient with no DLT observed has their no-DLT factor raised to a fractional power. The likelihood does not turn that patient into a fractional number of people, and the weight is not a replacement for the dose-toxicity model.

Assumptions, Diagnostics, and Alternatives

The weight is useful only when its underlying setup matches the trial. At minimum, the analyst should verify that the assessment window is defined consistently, follow-up begins at the relevant time origin, DLT status is correctly recorded, and patients’ observation times are known. The simple product likelihood also relies on conditional independence of patients given the model parameters. The dose-risk probabilities must refer to the same DLT endpoint and assessment window used to define the weights.

Several checks can reveal problems before interpreting a model fit:

• Check bounds and timing. Confirm \(0\leq u_i\leq T\), \(0\leq w_i\leq1\), and that a patient beyond day \(T\) is capped at \(w_i=1\).
• Check event records. A patient with an observed DLT should not be coded as DLT-free merely because the window is incomplete. The binary DLT contribution is \(p_i\), not a fractional no-DLT contribution.
• Inspect timing plausibility. If DLTs tend to occur late, uniform conditional timing can overstate how much early DLT-free follow-up says about eventual risk. If events cluster early, it can understate that evidence.
• Compare formulations when risk is not small. The exact uniform-timing factor \(1-p_iw_i\) and the power factor \((1-p_i)^{w_i}\) can differ more as \(p_i\) grows. Sensitivity to this choice is a useful diagnostic.
• Look for informative observation times. If follow-up ends early because of toxicity, withdrawal, or a clinical decision related to risk, treating censoring as ordinary partial follow-up may be inappropriate without additional modelling.

The alternatives answer different modelling needs. The conventional TITE-CRM power likelihood is computationally simple and assigns a fractional no-DLT contribution. The exact uniform-timing likelihood uses \(1-p_iw_i\) for a DLT-free partial observation. A constant-hazard model yields the power contribution exactly, but assumes a particular event-time pattern. More detailed time-to-event models can represent richer timing patterns, but require additional assumptions and parameters. The choice should follow the design and event-time information, not a preference for one algebraic form.

Common Pitfall and Practical Limits

A common mistake is to say, “The patient has completed 75% of the window, so there is a 75% probability that the patient has already had a DLT,” or to treat \(w_i\) as a toxicity probability. That is incorrect. The weight is the observed time fraction. Under uniform conditional event timing, \(w_i\) is the conditional fraction of DLT times expected to have occurred by that point among patients who will have a DLT by \(T\); the unconditional probability of a DLT by then is \(p_iw_i\). In the conventional likelihood, \(w_i\) instead controls the exponent on a no-DLT contribution.

A further limitation is that a time fraction treats equal fractions of the window as comparable evidence. That is natural under a constant hazard on the cumulative-hazard scale and can be a useful approximation under other timing patterns, but it need not reflect the actual hazard profile. A weight does not correct for missed assessments, uncertain event dates, competing events, or endpoint definitions that vary across patients. Such features may call for explicit modelling rather than a simple fraction of time.