Deriving the TITE-CRM Likelihood With Pending Follow-Up

Why Pending Follow-Up Changes the Likelihood

In a dose-finding trial, a patient who has not yet experienced a dose-limiting toxicity (DLT) may still be within the prespecified toxicity observation window. That patient is not equivalent to one who completed the window without a DLT: the first has only partial outcome information, while the second has a fully observed toxicity-free outcome. Treating both as toxicity-free can make the accumulated evidence appear stronger than it is.

The TITE-CRM (time-to-event continual reassessment method) incorporates such patients through a weighted likelihood. The earlier material on partially observed toxicity windows introduces the practical role of partial follow-up; here the likelihood contributions are derived and their interpretation is made explicit. In particular, the standard contribution for a pending patient is a working, weighted-likelihood contribution, not generally the exact probability of remaining toxicity-free to the current time under a time-to-event model.

Outcome, Dose Model, and Follow-Up Notation

Definition: Let $W$ be the prespecified toxicity observation-window length. For patient $k$, let $d_k$ denote the assigned dose, $t_k$ the observed follow-up time within the window, and $p_k(\alpha)=P(\text{DLT by }W\mid d_k,\alpha)$ the probability of a DLT by the end of the window under CRM parameter $\alpha$. The TITE-CRM follow-up fraction is $$ w_k=\min\left(\frac{t_k}{W},1\right). $$ A patient with a DLT during the window has an observed toxicity outcome; a patient with no DLT and $t_k=W$ has a completed toxicity-free window; a patient with no DLT and $0\le t_k<W$ is pending.

The dose-toxicity model supplies $p_k(\alpha)$, usually through a CRM model that relates dose levels to toxicity probabilities. The likelihood calculations below do not require a particular CRM curve, but they do require the probabilities to be defined for the same endpoint and window $W$. For any fixed $\alpha$, each $p_k(\alpha)$ is between 0 and 1.

The standard formulation assumes that each patient’s DLT status by $W$ is the binary endpoint of interest, that recorded DLTs within the window are correctly classified, and that a toxicity-free patient’s follow-up time is known. It also uses the elapsed fraction $w_k$ as a measure of the information contributed by a patient without an observed DLT. This last step is an approximation rather than a consequence of knowing only the binary endpoint probability $p_k(\alpha)$.

Deriving the Three Contributions

First consider a patient whose DLT status by the end of the window is observed. If $Y_k=1$ denotes a DLT and $Y_k=0$ denotes no DLT by $W$, the Bernoulli likelihood contribution is

L_k(\alpha)=p_k(\alpha)^{Y_k}\{1-p_k(\alpha)\}^{1-Y_k}.

Substituting $Y_k=1$ gives a contribution $p_k(\alpha)$. Substituting $Y_k=0$ gives $1-p_k(\alpha)$. These are the usual contributions for an observed event and a fully observed non-event, respectively.

For a pending patient, the final binary endpoint is not known. The patient has not had a DLT up to $t_k$, but might still have one before $W$. The standard TITE-CRM construction uses the full-window no-DLT likelihood contribution, $1-p_k(\alpha)$, but scales its log contribution by the observed fraction $w_k$:

\log L_k(\alpha)=w_k\log\{1-p_k(\alpha)\}, \qquad L_k(\alpha)=\{1-p_k(\alpha)\}^{w_k}.

This gives no contribution when follow-up is zero, a full no-DLT contribution when the window is complete, and an intermediate contribution otherwise. An observed DLT is different: its occurrence resolves the endpoint, so its standard contribution is $p_k(\alpha)$, not a follow-up-fraction power of $p_k(\alpha)$.

Standard TITE-CRM likelihood: For independent patients conditional on the CRM parameter, with observed DLTs, completed toxicity-free windows, and pending toxicity-free follow-up, the weighted likelihood is $$ L(\alpha)= \prod_{k:\,\mathrm{DLT}}p_k(\alpha) \prod_{k:\,\mathrm{completed\ without\ DLT}}\{1-p_k(\alpha)\} \prod_{k:\,\mathrm{pending}}\{1-p_k(\alpha)\}^{w_k}. $$ Equivalently, a patient without an observed DLT contributes $\{1-p_k(\alpha)\}^{w_k}$, where a completed window has $w_k=1$. A DLT patient contributes $p_k(\alpha)$.

A Justification—and an Important Qualification

The exponent $w_k$ can be understood by starting with the full-window log-likelihood contribution for a toxicity-free outcome, $\log\{1-p_k(\alpha)\}$. If a fraction $w_k$ of the observation window has elapsed, the working likelihood assigns that patient the fraction $w_k$ of this log contribution. Exponentiating yields the power contribution. Across patients, conditional independence makes the contributions multiply, which gives the displayed product.

This justification explains the construction but does not turn it into an exact survival likelihood. The binary probability $p_k(\alpha)$ alone does not specify when a DLT would occur within the window. To see the distinction, let $T_k$ be time to DLT and define $G_k(t)=P(T_k\le t\mid T_k\le W,d_k,\alpha)$, the distribution of event time among patients who would have a DLT by $W$. If this event-time distribution is specified, the exact probability of no DLT by time $t$ is

P(T_k>t\mid d_k,\alpha)=1-p_k(\alpha)G_k(t).

This follows by separating patients who will not have a DLT by $W$, with probability $1-p_k(\alpha)$, from those who will, with probability $p_k(\alpha)$, but have not yet experienced it by $t$, with conditional probability $1-G_k(t)$. The total is $(1-p_k)+p_k(1-G_k(t))=1-p_kG_k(t)$. This exact contribution depends on the event-time model $G_k$. It is not generally equal to $\{1-p_k(\alpha)\}^{t/W}$.

For example, if event times among eventual DLTs are uniform over the window, then $G_k(t)=t/W$ for $0\le t\le W$, giving exact survival probability $1-p_k(\alpha)t/W$. The standard TITE-CRM power contribution instead uses $\{1-p_k(\alpha)\}^{t/W}$. The two can be close for some probabilities and follow-up fractions, but they are distinct models. The weighted likelihood is useful because it permits partial follow-up to inform the CRM update without requiring a complete event-time model; its interpretation should remain that of a working likelihood.

Worked Example: One Patient in Each Follow-Up State

Suppose three hypothetical patients are all treated at a dose where the candidate CRM parameter gives $p_k(\alpha)=0.25$, and the toxicity window is $W=80$ days. Patient A has a DLT on day 30. Patient B completes 80 days without a DLT. Patient C has no DLT by day 40 and is still being followed.

Patient A has an observed DLT, so the contribution is $0.25$. Patient B has a completed toxicity-free window, so the contribution is $1-0.25=0.75$. For patient C, $w_C=40/80=0.5$, so the pending contribution is

\{1-p_C(\alpha)\}^{w_C} =(1-0.25)^{0.5} =0.75^{0.5} \approx 0.8660.

The combined likelihood contribution at this candidate parameter is therefore

L(\alpha)=0.25(0.75)(0.8660)\approx 0.1624.

The pending patient contributes more than no information (a contribution of 1 would leave the likelihood unchanged) but less evidence for a toxicity-free outcome than the fully observed patient, whose contribution is 0.75.

Comparing Candidate CRM Parameters

Likelihood contributions matter because different values of $\alpha$ imply different dose-specific probabilities. The likelihood summarizes how compatible the observed outcomes and follow-up are with each candidate value. A Bayesian CRM update multiplies this likelihood by the prior density for $\alpha$ and normalizes; the likelihood by itself is not a posterior probability.

Worked Example: Comparing Two Candidate Dose Models

Consider three hypothetical patients and two candidate CRM parameter values, A and B. Patient 1 has a DLT at dose 2; patient 2 completes the window without a DLT at dose 1; patient 3 is pending at dose 2 after half the window. Thus $w_3=0.5$. Suppose the candidate probabilities are $p_1=p_3=0.40$ and $p_2=0.20$ under A, and $p_1=p_3=0.25$ and $p_2=0.10$ under B.

Under A, the event contribution is 0.40, the completed non-event contribution is $1-0.20=0.80$, and the pending contribution is $(1-0.40)^{0.5}$. Therefore,

L(A)=0.40(0.80)(0.60)^{0.5} =0.32(0.7746) \approx 0.2479.

Under B, the corresponding contributions are 0.25, $1-0.10=0.90$, and $(1-0.25)^{0.5}$, so

L(B)=0.25(0.90)(0.75)^{0.5} =0.225(0.8660) \approx 0.1949.

The likelihood ratio is $L(A)/L(B)\approx 0.2479/0.1949=1.272$. Thus these observations are about 1.27 times as likely under candidate A as under candidate B according to the weighted likelihood. If the two candidates had equal prior probabilities, their normalized posterior probabilities would be approximately $0.2479/(0.2479+0.1949)=0.5599$ for A and $0.4401$ for B. These numbers are conditional on the two-candidate illustration and equal prior probabilities; they are not probabilities that either model is true.

Worked Example: Power Contribution Versus Exact Survival

Suppose, hypothetically, that $p_k(\alpha)=0.30$ and a patient is observed without DLT through one quarter of the window, so $w_k=0.25$. The standard weighted contribution is

(1-0.30)^{0.25}=0.70^{0.25}\approx 0.9147.

If instead an exact time-to-event model assumes event times are uniform among eventual DLTs, then $G_k(t_k)=0.25$, and the exact probability of no DLT by that time is

1-p_k(\alpha)G_k(t_k) =1-(0.30)(0.25) =0.9250.

The difference is $0.9250-0.9147=0.0103$. Both calculations use the same window-end DLT probability, but they encode different assumptions about how partial follow-up translates into evidence. One must not label the power contribution an exact uniform-event-time survival probability.

Combining Patients, Checking the Data, and Choosing an Approach

For computation, it is often more stable to work with the log likelihood. Taking logarithms of the product gives a sum of event contributions and weighted no-event contributions:

\ell(\alpha)= \sum_{k:\,\mathrm{DLT}}\log p_k(\alpha) +\sum_{k:\,\mathrm{no\ DLT\ observed}}w_k\log\{1-p_k(\alpha)\}.

Here a completed toxicity-free patient has $w_k=1$; a pending patient has $0\le w_k<1$. This form also makes clear that pending follow-up changes the amount of evidence through the multiplier on the log contribution. It does not change the dose-specific probability supplied by the CRM model.

Before calculating an update, verify the following:

Classify status correctly. A DLT within the window is an event even if the patient has not completed the window. No DLT at an interim data cut is pending unless the entire window is complete.
Calculate time consistently. Use the same units for $t_k$ and $W$, and ensure the resulting fraction is between 0 and 1. Follow-up beyond $W$ is capped at $w_k=1$.
Match endpoint and window. The probability $p_k(\alpha)$ must refer to the same DLT definition and observation window used to classify the patient.
Inspect unusual observation patterns. Missing follow-up times, loss to follow-up, treatment discontinuation, or death before the window ends may require explicit censoring or competing-event assumptions. Automatically treating these situations as ordinary pending follow-up can be misleading.

The main alternative is to wait until each patient completes the window and then use the ordinary binary CRM likelihood. This avoids fractional contributions but delays updates and dose decisions. A fully specified time-to-event model can use the actual event times and censoring times through event-density and survival contributions; it can be more directly tied to follow-up time, but requires additional assumptions about the event-time distribution. The weighted TITE-CRM likelihood is a practical middle ground: it uses pending data with a simple prespecified weighting rule, while not claiming to model event times exactly.

Common Mistakes and Limitations

Wrong: “No DLT has occurred yet, so this patient contributes $1-p_k(\alpha)$.” Correction: That is the full-window contribution. For a pending patient, the standard contribution is $\{1-p_k(\alpha)\}^{w_k}$, with $w_k<1$.
Wrong: “A pending patient has a final toxicity outcome of zero.” Correction: The final outcome is not yet observed; the weighted likelihood represents partial information about that endpoint.
Wrong: “The power contribution is the exact probability of remaining toxicity-free to time $t_k$.” Correction: Not in general. An exact probability requires an event-time model, such as $1-p_k(\alpha)G_k(t_k)$.
Wrong: “An observed DLT should be downweighted if it occurred early in the window.” Correction: Under the standard binary-endpoint TITE-CRM likelihood, observing a DLT establishes the endpoint and contributes $p_k(\alpha)$.

The weighted approach is limited by the information approximation embodied in $w_k$. Its validity as an update strategy depends on appropriate endpoint definitions, reliable follow-up data, and suitability of the chosen weighting rule for the trial setting. The likelihood derivation alone does not establish that a particular weight is optimal or that all censoring patterns are harmless.

Key takeaway: A DLT contributes $p_k(\alpha)$, a completed toxicity-free window contributes $1-p_k(\alpha)$, and a pending toxicity-free observation contributes $\{1-p_k(\alpha)\}^{w_k}$. The last expression is the standard TITE-CRM weighted-likelihood contribution, not generally an exact survival probability.

Check Your Understanding

Answer each question, then compare your work with the solutions below.

A hypothetical patient completes a 60-day window without DLT. If $p_k(\alpha)=0.18$, what is the likelihood contribution?
A patient is observed without DLT for 15 days of a 60-day window. If $p_k(\alpha)=0.36$, calculate $w_k$ and the standard TITE-CRM contribution.
For a pending patient, explain why the standard contribution is not the same as the exact survival probability under a uniform event-time model. Give both expressions when $p_k(\alpha)=0.20$ and $w_k=0.5$.
A patient experiences a DLT during follow-up after 10 days of a 50-day window. What is the standard likelihood contribution, and why is it not raised to the power $10/50$?

Solutions. (1) The completed toxicity-free contribution is $1-0.18=0.82$. (2) The fraction is $w_k=15/60=0.25$, so the contribution is $0.64^{0.25}\approx0.8944$. (3) The standard contribution is $(1-0.20)^{0.5}=\sqrt{0.80}\approx0.8944$. Under uniform event times among eventual DLTs, the exact survival probability is $1-(0.20)(0.5)=0.9000$; these differ because the power likelihood is a weighted binary-endpoint likelihood, not that event-time model. (4) The contribution is $p_k(\alpha)$, because the DLT has been observed and resolves the binary endpoint. The standard likelihood does not discount an observed endpoint according to the time at which it occurred.