Worked CRM Recommendations With Sparse DLT Data

Why Sparse DLT Data Need a Model-Based Reading

In early dose-finding trials, a cohort may contribute no dose-limiting toxicities (DLTs), or only one. The observed fraction at a dose is then a noisy summary: zero DLTs among three patients does not establish that the dose has zero toxicity risk. The continual reassessment method (CRM) instead combines each cohort’s outcomes with a prespecified dose-toxicity model and prior distribution, producing updated toxicity estimates across the candidate doses.

This tutorial follows a small hypothetical trial in which the first cohort has no DLTs, the second also has none, and the third has one. The aim is to calculate how posterior dose rankings and a target-based recommendation change after each cohort. The recommendation rule used here is specific: select the dose whose posterior mean DLT probability is closest to the target. Other rules can produce different choices.

Posterior Updating and the Recommendation Rule

Let \(y_j\) be the number of DLTs among \(n_j\) fully assessed patients treated at dose \(d_j\). Conditional on \(\beta\), the likelihood contribution from that dose is proportional to \(p_j(\beta)^{y_j}\{1-p_j(\beta)\}^{n_j-y_j}\). Contributions from different doses multiply. For a discrete prior grid \(\beta_1,\ldots,\beta_K\), the posterior weight of grid point \(k\) is therefore

The posterior mean DLT probability at dose \(d_j\) is the weighted average of the model probabilities over the posterior distribution of \(\beta\). Given target toxicity probability \(\phi\), this tutorial’s CRM recommendation is the dose minimizing the absolute difference between that posterior mean and \(\phi\).

This is a recommendation based on posterior mean risk, not a guarantee that the selected dose has risk exactly equal to the target. Because every grid value of \(\beta\) is positive and the skeleton is increasing, \(p_1(\beta)<\cdots<p_J(\beta)\) at every grid point. Averaging over the posterior preserves this ordering, although which dose is closest to the target can change as the posterior weights change.

Worked Trial: Three Cohorts and One DLT

Use three candidate doses with skeleton \((0.10,0.20,0.35)\), target \(\phi=0.25\), and a prior assigning probability \(1/3\) to each of \(\beta=0.5,1,1.5\). The skeleton values are model probabilities when \(\beta=1\); they are not observed DLT rates. The resulting probabilities at the three grid values are:

For example, at \(\beta=0.5\), the first-dose probability is \(0.10^{0.5}=0.3162\). The power is \(\beta\), not \(\exp(\beta)\). Each cohort below has three patients, and all outcomes are treated as fully observed.

What the Calculations Do—and Do Not—Establish

A short justification explains why one cohort affects estimates at every dose. The likelihood updates the distribution of the shared parameter \(\beta\), rather than fitting a separate toxicity probability for each dose. Since \(p_j(\beta)=s_j^\beta\), a change in posterior weight on \(\beta\) changes every \(p_j(\beta)\). The posterior mean at dose \(d_j\) is consequently a sum over the same updated weights, with dose-specific probabilities inside the sum. This borrowing of information across doses is useful when data are sparse, but it also makes the estimates dependent on the model and skeleton.

Several checks help reveal when a sparse-data recommendation may be fragile. Confirm that the skeleton is strictly ordered and between zero and one, that all likelihood factors match the recorded outcomes, and that posterior weights sum to one (up to rounding). Examine whether nearly all posterior weight has accumulated at one end of the chosen grid; if so, the grid or prior may inadequately represent the supported parameter values. Compare recommendations under reasonable prior or skeleton alternatives, and inspect posterior risk distributions rather than relying on posterior means alone. A mean close to the target can conceal substantial uncertainty.

The calculations have important limitations. The power CRM imposes a particular monotone curve, so apparent precision at unobserved doses comes from model assumptions as well as observed data. A discrete grid approximates a continuous prior and can affect results if it is too coarse or too narrow. The target-closest posterior-mean rule is only one decision rule: posterior overdose control, expected utility, and other criteria answer different questions. Practical escalation protocols may also restrict assignments or require stopping; the statistical dose recommendation alone does not specify those actions.

Common Pitfall: Treating Sparse Fractions as the Dose Risks

A tempting but incorrect argument is: “There were zero DLTs among three patients at \(d_2\), so its toxicity risk is zero; therefore the next dose is clearly safe.” The observed fraction is zero, but the cohort likelihood is \(\{1-p_2(\beta)\}^3\), which is positive for a range of nonzero risks. The posterior mean risk at \(d_2\) after that cohort in this example is about \(0.1439\), not zero. The correction is to distinguish observed outcomes from estimated probabilities and to report the model, prior, and uncertainty underlying the estimate.

Another error is to calculate the posterior mean using the prior weights after a new cohort. Posterior weights must be updated with that cohort’s likelihood before averaging the dose-specific probabilities. Finally, do not confuse ranking by toxicity risk with ranking by target proximity: the highest-risk dose is not automatically the best dose, and the lowest-risk dose is not automatically the target dose.