Time-to-Event Models for Dose Escalation

Why Use Event Times?

In dose escalation, a patient who has not yet experienced a dose-limiting toxicity (DLT) may still be under observation. Treating that patient as if the assessment window were complete discards information and can misclassify pending outcomes as non-events. A time-to-event likelihood uses both the length of observed follow-up and whether a DLT occurred during it. It therefore provides a direct way to estimate toxicity risk at each dose while follow-up is incomplete.

This tutorial uses a simple exponential model at each dose. It assumes a constant DLT hazard during the assessment window. That assumption makes the likelihood transparent; it is also consequential, because real toxicity hazards need not be constant. The estimated risk is conditional on this model and on the specified assessment horizon.

Definition: At dose \(j\), let \(T_i\) be patient \(i\)'s time to DLT, measured from a common treatment start. Let \(C_i\) be the time at which observation ends for a reason other than DLT, and let \(\tau\) be the assessment-window length. Record \(Y_i=\min(T_i,C_i,\tau)\), the observed follow-up time, and \(\delta_i=1\) if a DLT is observed by \(Y_i\), or \(\delta_i=0\) otherwise. Under an exponential time-to-DLT model with dose-specific hazard \(\lambda_j\), the toxicity risk by time \(\tau\) is \(p_j(\tau)=1-\exp(-\lambda_j\tau)\).

Likelihood and Assumptions

For an exponential time-to-DLT distribution, the density of an observed event time \(t\) is \(f(t\mid\lambda_j)=\lambda_j\exp(-\lambda_jt)\), and the probability of remaining free of DLT beyond time \(t\) is \(S(t\mid\lambda_j)=\exp(-\lambda_jt)\). An observed DLT contributes the density; a patient without an observed DLT contributes the probability of surviving event-free through their observed follow-up.

L_j(\lambda_j) =\prod_{i:\,dose_i=j} \left[\lambda_j^{\delta_i}\exp(-\lambda_jY_i)\right] =\lambda_j^{D_j}\exp(-\lambda_jE_j), \qquad D_j=\sum_{i:\,dose_i=j}\delta_i,\quad E_j=\sum_{i:\,dose_i=j}Y_i.

Here \(D_j\) is the number of observed DLTs and \(E_j\) is the total observed time at risk, in patient-time units. An event patient's exposure ends at the event time; a censored patient's exposure ends at censoring or at \(\tau\). The likelihood is conditional on the dose assignments and observed follow-up. It assumes independent patient outcomes given dose and that censoring is non-informative for DLT time: after conditioning on the model and relevant design information, the censoring process does not selectively end observation based on an unobserved impending DLT. Administrative censoring at \(\tau\) satisfies this condition; withdrawal related to emerging toxicity may not.

The model also assumes a constant hazard within the window at a given dose, so \(P(T_i\leq t\mid dose_i=j)=1-\exp(-\lambda_jt)\). Patients at the same dose share \(\lambda_j\). Separate dose-specific estimates do not, by themselves, impose that toxicity risk increases with dose. Exact event times and correctly recorded follow-up times are assumed; rounding times can affect the likelihood if it is substantial.

Deriving the Rate and Risk Estimates

Taking logarithms gives \(\ell_j(\lambda_j)=D_j\log(\lambda_j)-\lambda_jE_j\). When \(D_j>0\), differentiating and setting the score to zero gives the maximum-likelihood estimate below. The second derivative is negative, so this stationary point is a maximum.

\frac{\partial\ell_j}{\partial\lambda_j} =\frac{D_j}{\lambda_j}-E_j=0 \quad\Longrightarrow\quad \widehat{\lambda}_j=\frac{D_j}{E_j}, \qquad \frac{\partial^2\ell_j}{\partial\lambda_j^2} =-\frac{D_j}{\lambda_j^2}<0.

The estimated risk by the assessment horizon follows by substituting the estimated hazard into the model's risk function:

\widehat{p}_j(\tau) =1-\exp(-\widehat{\lambda}_j\tau) =1-\exp\left(-\frac{D_j\tau}{E_j}\right).

The hazard estimate has units of inverse time, such as per day; the risk is a dimensionless probability. If \(D_j=0\) and \(E_j>0\), the likelihood is \(\exp(-\lambda_jE_j)\), which decreases as \(\lambda_j\) increases. Its maximum over \(\lambda_j\geq0\) is at \(\widehat{\lambda}_j=0\), yielding a boundary estimate of zero risk. If \(D_j=0\) and \(E_j=0\), the likelihood is constant at 1 for every \(\lambda_j\geq0\), so the hazard is not identified by the data. This is a feature of the unregularized likelihood, not evidence that the true risk is zero. With few patients or no events, uncertainty can be large and the boundary estimate should not be read as reassurance.

Worked Example: Estimating Risk With Events and Censoring

Suppose five hypothetical patients receive the same dose, with an assessment window of \(\tau=28\) days. Two have DLTs on days 7 and 18. The other three remain DLT-free until days 28, 28, and 14; the last patient is lost to follow-up at day 14. Thus the observed times are \(Y_i=(7,18,28,28,14)\), the indicators are \((1,1,0,0,0)\), and censoring is assumed non-informative.

There are \(D=2\) DLTs, and the total exposure is \(E=7+18+28+28+14=95\) patient-days. Therefore,

\widehat{\lambda}=\frac{2}{95}=0.02105\ \text{per day}, \qquad \widehat{p}(28)=1-\exp\left(-\frac{2(28)}{95}\right) =1-\exp(-0.5895)\approx0.4454.

Under the fitted constant-hazard model, the estimated probability of DLT by day 28 is about 44.5%. The two event patients contribute exposure only up to their event times, not 28 days each. The censored patient contributes 14 days of event-free observation, not a completed 28-day non-event.

Estimating Risk Separately at Several Doses

For dose-specific estimation, apply the same likelihood within each dose group. The joint likelihood is the product of the dose-specific likelihoods if patient event times are independent conditional on dose. Each estimate uses that dose's event count and total exposure. Sparse groups can yield unstable estimates, and separate fitting does not borrow information across doses or enforce a monotone dose-risk curve.

Worked Example: Comparing Two Dose-Specific Estimates

Consider hypothetical data from two dose groups, each with a 28-day window. At dose A, four patients have observed times of 10 days to DLT, followed by censoring at 28, 28, and 20 days. At dose B, four patients have DLTs on days 5 and 16, followed by censoring at 28 and 28 days. Assume independent outcomes and non-informative censoring at both doses.

For dose A, \(D_A=1\) and \(E_A=10+28+28+20=86\) patient-days. Thus \(\widehat{\lambda}_A=1/86=0.01163\) per day, and

\widehat{p}_A(28)=1-\exp\left(-\frac{28}{86}\right) =1-\exp(-0.3256)\approx0.2779.

For dose B, \(D_B=2\) and \(E_B=5+16+28+28=77\) patient-days. Thus \(\widehat{\lambda}_B=2/77=0.02597\) per day, and

\widehat{p}_B(28)=1-\exp\left(-\frac{2(28)}{77}\right) =1-\exp(-0.7273)\approx0.5168.

The fitted risk is higher at dose B in this example. These estimates are based on small groups and are not precise simply because they have been calculated. Separate estimates can also be nonmonotone in another data set; monotonicity would require an additional dose-response structure rather than following from this likelihood.

Why Event-Time and Binary Analyses Differ

If the analysis uses only whether each patient had a DLT by \(\tau\), and all patients have complete status at that horizon, the data follow a binomial model with probability \(p(\tau)\). Its maximum-likelihood estimate is the observed proportion \(D/n\). That is a binary analysis: it does not use when events occurred or how much follow-up contributed before an event.

The event-time likelihood instead uses \(E=\sum_iY_i\), including only the time until DLT for patients who experience one. Its exponential-model estimate is \(1-\exp(-D\tau/E)\), not generally \(D/n\). For example, in the first worked example, the binary proportion is \(2/5=0.4000\), whereas the event-time estimate is about 0.4454. The difference is not an arithmetic contradiction: the estimates use different likelihoods and information.

Even if only complete binary statuses are retained, one can express the binary likelihood in terms of an exponential hazard through \(p(\tau)=1-\exp(-\lambda\tau)\). Its hazard estimate, when \(0<D<n\), is \(-\log(1-D/n)/\tau\), not \(D/(n\tau)\). Conversely, if exact event times are used, observed DLT patients contribute their event times, so it is incorrect to assign them \(\tau\) days of exposure. “Complete follow-up” means patients are observed until DLT or the horizon; an event stops their time at risk.

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

At a hypothetical dose, four patients are followed in a 28-day assessment window. One experiences DLT on day 12, one is lost to follow-up without DLT on day 8, and two complete 28 days without DLT. Assume the early censoring is non-informative.

The event count is \(D=1\); the total exposure is \(E=12+8+28+28=76\) patient-days. The event-time estimate is

\widehat{\lambda}=\frac{1}{76}=0.01316\ \text{per day}, \qquad \widehat{p}(28)=1-\exp\left(-\frac{28}{76}\right) =1-\exp(-0.3684)\approx0.3082.

If all four patients were incorrectly treated as having known 28-day status, the crude event proportion would be \(1/4=0.2500\). That calculation wrongly labels the day-8 censor as a completed non-event and ignores event timing. The event-time estimate uses the actual exposure and the assumed exponential model; the two values answer different analyses and should not be substituted for one another.

Diagnostics, Alternatives, and Limitations

Before using an estimate in a dose decision, verify the data and assumptions. Check that time origins, event definitions, assessment horizons, and censoring reasons are consistently recorded. Recalculate \(D_j\) and \(E_j\) from patient-level records, confirming that event patients contribute time to event and event-free patients contribute time to censoring or \(\tau\). A time or indicator miscoding changes both the event count and exposure.

Few events or no events. A zero-event dose group produces a zero maximum-likelihood hazard, but provides limited evidence about a small or moderate risk. Report the exposure and event count and recognize the boundary estimate.
Informative censoring. If patients leave observation because of symptoms that predict DLT, treating their times as ordinary independent censoring can bias risk estimates. Investigate reasons for withdrawal; a different model may be needed.
Nonconstant hazard. The exponential model imposes a constant hazard within the window. Patterns of event times that are inconsistent with this simplification, or clinically plausible time-varying risk, limit interpretation of its risk estimates. A constant-hazard fit should not be presented as proof that risk is constant over time.
Dose groups are small. Separate estimates can be highly variable and can violate an expected ordering across doses. This method alone provides neither dose-response smoothing nor a monotonicity constraint.

A binary binomial analysis is a natural alternative when the scientific question concerns only DLT status at a fixed horizon and complete statuses are available; it is simple, but discards event timing. A nonparametric survival estimate can represent the event-free curve without the exponential constant-hazard assumption, but sparse data within dose groups may make it unstable. Time-to-event likelihood methods use partial follow-up and event times through an explicit model. Earlier material on TITE-CRM uses partial follow-up in a weighted escalation approach; that weighting approach is not the same likelihood as the exponential model derived here. Choice among these approaches depends on the estimand, data available, and assumptions that are defensible for the trial.

A common mistake is to set \(E=n\tau\) whenever every patient was intended to be followed to \(\tau\), and then to interpret the event-time estimate as \(D/n\). For patients with an observed DLT before \(\tau\), exposure ends at the event, so generally \(E<n\tau\). If one nevertheless substitutes \(E=n\tau\) algebraically into the exponential risk expression, the result is \(1-\exp(-D/n)\), not \(D/n\); neither expression repairs the incorrect exposure assignment. The correction is to distinguish the event-time likelihood, with actual \(Y_i\), from the binomial likelihood based on complete binary outcomes.

Key takeaway: Under a dose-specific exponential time-to-DLT model with non-informative right censoring, the event-time likelihood depends on the DLT count \(D_j\) and total observed exposure \(E_j\). Estimate \(\lambda_j\) as \(D_j/E_j\) when events occur, then transform it to horizon-specific risk using \(1-\exp(-\widehat{\lambda}_j\tau)\). State the model and censoring assumptions whenever interpreting that risk.

Check Your Understanding

Use the exponential time-to-DLT model and show how event counts and observed exposure enter each answer.

At one dose, two hypothetical DLTs occur at days 4 and 10; two other patients are censored event-free at days 20 and 28. For \(\tau=28\), calculate \(D\), \(E\), \(\widehat{\lambda}\), and \(\widehat{p}(28)\).
For \(n\) patients with complete binary outcomes at \(\tau\), what is the binomial maximum-likelihood estimate of risk? Why is it not generally the event-time estimate?
What does the exponential likelihood estimate when \(D=0\), and why should that result not be interpreted as proof of no toxicity risk?
A patient with DLT on day 9 is followed through a 28-day window. How much exposure does this patient contribute to the event-time likelihood?

Solutions. (1) \(D=2\), \(E=4+10+20+28=62\) patient-days, \(\widehat{\lambda}=2/62=0.03226\) per day, and \(\widehat{p}(28)=1-\exp[-2(28)/62]=1-\exp(-0.9032)\approx0.5947\). (2) The binomial estimate is \(D/n\), the observed proportion with DLT by \(\tau\). It uses binary statuses only, while the event-time estimate also uses event times and each patient's observed exposure. (3) The unregularized maximum-likelihood estimate is the boundary value \(\widehat{\lambda}=0\), hence fitted risk zero. With no observed events, the data may still be compatible with positive risk; the estimate does not quantify that uncertainty. (4) The patient contributes \(Y=9\) days of exposure and \(\delta=1\), so the likelihood contribution is \(\lambda\exp(-9\lambda)\), not \(\lambda\exp(-28\lambda)\).