Why DLT timing matters
In time-to-event dose escalation, a patient who has not yet experienced a dose-limiting toxicity (DLT) is not necessarily equivalent to a patient who completed the assessment window without a DLT. The strength of the evidence depends on how long that patient has been followed and on how likely a DLT is to occur by that time, conditional on one occurring within the window. If DLTs tend to occur early, a patient with no DLT after substantial follow-up provides more reassurance than if DLTs tend to occur late.
This tutorial focuses on that conditional timing assumption. It is separate from the model for the overall probability of a DLT by the end of the assessment window. The timing assumption determines how partial follow-up contributes to the likelihood for that probability, and can therefore change posterior risk estimates and an escalation screen.
This separation is useful because the endpoint remains “DLT by \(W\),” even when some patients have less than \(W\) days of observation. It also makes the assumption visible: the design needs a plausible \(G_d\), or a set of plausible timing models for sensitivity analysis. A simple family is \(G_d(t)=(t/W)^k\). When \(k=1\), timing is uniform within the window; \(k>1\) places relatively more events later; and \(k<1\) places relatively more events earlier. This is a modeling device, not a claim that the actual event-time distribution must have that form.
Likelihood contributions from observed and pending outcomes
Suppose a patient at dose \(d\) has been followed for \(u\leq W\) days and has had no DLT. The probability of that observation is the probability that the patient will have no DLT by \(W\), plus the probability that a DLT will occur by \(W\) but after \(u\). The two possibilities are disjoint, so
If the patient has a DLT at an observed time \(t\leq W\), and \(g_d(t)\) is the density corresponding to \(G_d(t)\), the likelihood contribution is \(p_dg_d(t)\). A patient observed without DLT through the full window contributes \(1-p_d\), which follows from \(G_d(W)=1\). These expressions assume the event time is measured from a defined origin, the DLT is observed when it occurs, and censoring or loss to follow-up is non-informative for DLT timing after conditioning on modeled information.
The no-DLT expression gives a direct way to see the role of timing. At a fixed \(p_d\), larger \(G_d(u)\) means more of the DLT-prone patients would be expected to have declared by follow-up time \(u\). Therefore, seeing no DLT is less likely under that \(p_d\) and weighs more strongly against it. Conversely, if \(G_d(u)\) is small, many eventual DLTs could still be pending, and the same observation is weaker evidence about \(p_d\).
Worked Example: Likelihood for Partial Follow-Up
Consider a hypothetical 28-day assessment window. A patient at a particular dose has no DLT after 14 days. For illustration, take \(p_d=0.30\). Compare the uniform timing model \(k=1\) with an early-event model \(k=0.5\) and a late-event model \(k=2\).
Under \(G_d(t)=(t/W)^k\), the follow-up fraction is \(u/W=14/28=0.5\). The probability of no DLT through day 14 is \(1-p_dG_d(14)\).
For uniform timing, \(G_d(14)=(0.5)^1=0.5\), so the likelihood contribution is \(1-(0.30)(0.5)=0.8500\).
For early timing, \(G_d(14)=(0.5)^{0.5}=0.7071\), so the contribution is \(1-(0.30)(0.7071)=0.7879\). No DLT by day 14 is less likely when events are expected earlier; this observation is consequently stronger evidence against \(p_d=0.30\).
For late timing, \(G_d(14)=(0.5)^2=0.25\), giving \(1-(0.30)(0.25)=0.9250\). Here the same pending observation is relatively unsurprising because few eventual DLTs are expected to have occurred by day 14.
These are likelihoods for one observation under a specified \(p_d\), not posterior probabilities that the dose is safe. They show why the timing assumption matters before any posterior dose recommendation is calculated.
How timing assumptions can change a dose screen
A model-based escalation procedure combines the likelihood with a prior distribution for dose-toxicity risk and then applies its decision rule. To isolate the impact of timing, the following hypothetical calculation uses a small discrete prior and a simple illustrative overdose screen. Real designs may use continuous dose-toxicity models and more detailed decision rules.
Worked Example: A Pending Patient Changes Screen Eligibility
At dose \(d\), suppose the prior distribution assigns equal probability to \(p_d=0.10,\ 0.25,\ 0.40\). Let the target DLT probability be \(0.30\). For this example only, call the dose eligible to pass the screen when the posterior probability that \(p_d>0.30\) is no more than \(0.30\). One patient has no DLT after 14 days of a 28-day window.
For each timing model, multiply each prior probability by the no-DLT likelihood \(1-p_dG_d(14)\), then normalize the three weights. The prior probabilities are all \(1/3\), so that common factor cancels. Only the \(p_d=0.40\) state is above target.
Under uniform timing, \(G_d(14)=0.5\). The unnormalized weights are \(1-(0.10)(0.5)=0.9500\), \(1-(0.25)(0.5)=0.8750\), and \(1-(0.40)(0.5)=0.8000\). Their sum is \(2.625\). Thus,
This is slightly above the example’s 0.30 screen boundary, so the dose does not pass.
Under early timing with \(k=0.5\), \(G_d(14)=0.7071\). The weights are \(0.9293,\ 0.8232,\ 0.7172\), with sum \(2.4697\). The posterior probability above target is \(0.7172/2.4697=0.2904\), so this dose passes the stated screen.
Under late timing with \(k=2\), \(G_d(14)=0.25\). The weights are \(0.9750,\ 0.9375,\ 0.9000\), with sum \(2.8125\). The posterior probability above target is \(0.9000/2.8125=0.3200\), so the dose does not pass. The early model gives more weight to the absence of an event at day 14, while the late model gives less. This example is intentionally near the boundary: a modest modeling choice can change a screen outcome, which is a reason to assess timing sensitivity rather than hide it.
The posterior probability in this example is not a probability that the null hypothesis is true, nor does passing establish that the dose is safe. It is a model- and prior-dependent posterior quantity used by a specified decision rule. A different target, prior, boundary, or dose-toxicity model can produce a different recommendation.
What an observed event time contributes
An observed DLT has two pieces of information: that the patient had a DLT by \(W\), and when it occurred. In a separated model with a timing distribution fixed independently of \(p_d\), the event likelihood \(p_dg_d(t)\) includes a factor that depends on timing and another that depends on the DLT probability. For an observation at a fixed dose and time, the timing factor can cancel from posterior comparisons among values of \(p_d\). That cancellation does not mean timing is irrelevant in every model: it can fail when timing parameters are estimated jointly with toxicity risk or linked to dose-toxicity parameters.
Worked Example: An Event at Day 7
Suppose the same hypothetical dose has the equal prior probabilities \(1/3\) on \(p_d=0.10,\ 0.25,\ 0.40\), and one patient experiences a DLT at day 7 of a 28-day window. Use \(G_d(t)=(t/W)^k\), whose density is \(g_d(t)=kt^{k-1}/W^k\). For \(k=1\), \(g_d(7)=1/28\). For \(k=2\), \(g_d(7)=2(7)/28^2=0.01786\). In either case the likelihood for a given \(p_d\) is \(p_dg_d(7)\).
For \(k=1\), the unnormalized posterior weights are \((0.10)(1/28)\), \((0.25)(1/28)\), and \((0.40)(1/28)\). The common factor \(1/28\) cancels on normalization. The posterior probability above the 0.30 target is therefore \(0.40/(0.10+0.25+0.40)=0.5333\).
For \(k=2\), the corresponding weights are \((0.10)(0.01786)\), \((0.25)(0.01786)\), and \((0.40)(0.01786)\). The common density again cancels, giving the same \(0.5333\). Thus, with this fixed timing model and a single observed event at a known dose, changing \(k\) does not change this posterior comparison of \(p_d\) states. It does change the event-time likelihood itself, and can matter when timing is estimated, varies by dose, or is related to toxicity risk.
Assumptions, diagnostics, and alternatives
The likelihood calculation is useful only if its assumptions are reasonably supported. In particular, the assessment window and event-time origin must be consistent across patients; event times must be recorded with adequate accuracy; and censoring before \(W\) must not selectively remove patients with different unmodeled DLT risks. A patient lost to follow-up for reasons related to emerging toxicity cannot automatically be treated as an ordinary non-informatively censored patient. Also, the model assumes the endpoint is DLT by \(W\); events after \(W\) do not contribute to this endpoint unless the design explicitly defines a different window.
Diagnostics should examine the data and the implications of the assumed timing curve, not merely whether the dose-toxicity model converges. Useful checks include comparing observed event times with posterior predictive event-time distributions, tabulating event timing by dose, and checking whether a chosen model predicts implausibly few early or late events. With small samples, these checks may be inconclusive; sensitivity analyses across plausible \(G_d\) curves then become especially important. If dose-specific event times differ systematically, a single shared timing distribution may be too restrictive.
The main alternatives answer different questions. A complete-case analysis waits until every patient has finished \(W\), avoiding the need to use partial follow-up in the likelihood but potentially slowing decisions. A binary analysis that treats every patient without a completed window as a non-DLT incorrectly treats pending outcomes as known negatives. Weighting approaches such as TITE-CRM use follow-up to give partial information through a specified weighting construction; a time-to-event likelihood instead states an event-time model and derives each observation’s likelihood contribution. Neither approach removes the need to justify how much information partial follow-up provides.
A practical workflow is to specify a primary timing model, compute the likelihood using each patient’s actual follow-up or event time, and repeat the dose-decision calculation under credible alternatives. Report whether the recommendation is stable. If only an implausibly narrow timing assumption leads to a favorable screen, the apparent precision is not warranted. The model quantifies uncertainty conditional on its assumptions; it does not protect the decision from a poor timing model.
Check Your Understanding
Use the likelihood and timing distinctions above to answer the questions. Brief solutions follow each item.
- For \(W=30\) days, \(u=15\) days, \(p_d=0.20\), and uniform timing, calculate the likelihood contribution for no DLT by day 15.
- At the same follow-up fraction, which model makes no DLT by day 15 stronger evidence against a fixed \(p_d\): \(k=0.5\) or \(k=2\)? Explain.
- What likelihood contribution applies to a patient with no DLT through the complete assessment window?
- Under what condition can the timing-density factor for an observed event cancel when comparing posterior values of \(p_d\)?
Solutions. (1) Uniform timing gives \(G_d(15)=15/30=0.5\), so the contribution is \(1-(0.20)(0.5)=0.9000\). (2) The \(k=0.5\) model has \(G_d(15)=(0.5)^{0.5}=0.7071\), greater than the \(k=2\) value of \(0.25\). No event is less likely under the early model at fixed \(p_d\), so it is stronger evidence against that risk. (3) Set \(u=W\), so \(G_d(W)=1\); the contribution is \(1-p_d\). (4) It cancels when the timing density at the observed time is a common factor across the compared \(p_d\) values—for example, when timing is fixed independently of \(p_d\) in the separated model \(p_dg_d(t)\).