From the First Cohort to a Dose Decision
In a dose-escalation trial, the first cohort supplies the first trial-specific evidence about toxicity. The dose at which those patients were treated matters, as does the number who experienced a dose-limiting toxicity (DLT). A BLRM update combines that evidence with prior information about the dose-toxicity curve. The resulting posterior distribution—not the observed DLT rate by itself—is used to assess whether a candidate dose exceeds the trial’s toxicity target.
This worked example follows one hypothetical trial from its initial cohort through its first dose-permission decision. To make the arithmetic visible, it uses a small discrete parameter grid as a teaching approximation to a prior distribution. In practice, a BLRM commonly uses continuous priors and numerical integration; the same likelihood and posterior-probability logic applies.
The Model, Cohort Data, and Prior
For a cohort of \(n\) patients treated at dose \(d\), let \(Y\) be the number with a DLT. Conditional on the model parameters, assume \(Y\) has a binomial distribution with size \(n\) and probability \(\pi(d)\). This assumes that patients in the cohort share the same modeled probability at that dose and that their DLT outcomes are conditionally independent. The first worked example uses \(d_{\mathrm{ref}}=10\) mg and a cohort of three patients at 10 mg, with one DLT.
The hypothetical prior is a finite grid with four possible parameter pairs. The grid is deliberately simple enough to calculate by hand; its unequal weights on the two slope values allow the examples to show how posterior uncertainty in both parameters contributes to an overdose probability.
| Parameter pair \((\alpha,\beta)\) | Prior probability |
|---|---|
| \((0.10,0.8)\) | 0.375 |
| \((0.10,1.6)\) | 0.125 |
| \((0.25,0.8)\) | 0.375 |
| \((0.25,1.6)\) | 0.125 |
The prior assigns equal total probability to \(\alpha=0.10\) and \(\alpha=0.25\). Within either value of \(\alpha\), it assigns probability 0.75 to \(\beta=0.8\) and 0.25 to \(\beta=1.6\). These are prior assumptions for this hypothetical illustration, not universal BLRM settings.
Worked Example: Updating After One DLT in Three Patients
At 10 mg, the reference dose, the log-dose term is zero. Thus \(\pi(10)=\alpha/(1+\alpha)\), regardless of \(\beta\). Under \(\alpha=0.10\), the DLT probability is \(0.10/1.10=0.09091\); under \(\alpha=0.25\), it is \(0.25/1.25=0.2000\).
For \(Y=1\) among \(n=3\), the binomial likelihood at a proposed DLT probability \(q\) is $$ P(Y=1\mid q)={3\choose 1}q(1-q)^2. $$ Substituting the two probabilities gives $$ L_{0.10}=3\left(\frac{1}{11}\right)\left(\frac{10}{11}\right)^2 =\frac{300}{1331}=0.2254, \qquad L_{0.25}=3(0.2)(0.8)^2=0.3840. $$
Because the cohort was treated at the reference dose, the likelihood is the same for both slope values at a given \(\alpha\). The total prior-weighted likelihood for the two \(\alpha=0.10\) points is \(0.375(0.2254)+0.125(0.2254)=0.1127\). For the two \(\alpha=0.25\) points, it is \(0.375(0.3840)+0.125(0.3840)=0.1920\). Their sum, \(0.3047\), is the normalizing constant.
Dividing each prior-weighted likelihood by that constant gives posterior probabilities of approximately 0.2774, 0.09247, 0.4726, and 0.1575, in the order listed in the prior table. For example, the posterior probability of \((0.25,1.6)\) is \(0.125(0.3840)/0.3047=0.1575\). The posterior probability that \(\alpha=0.25\), summing its two rows, is 0.6301.
Why the Update Has This Form
Bayes’ rule says that the posterior probability of a parameter-grid point is proportional to its prior probability multiplied by the likelihood of the observed data at that point. For grid point \(j\), write its prior weight as \(w_j\) and its modeled DLT probability at the observed dose as \(\pi_j(d)\). The normalized update is
The denominator sums the prior-weighted likelihood over all parameter-grid points, so the posterior weights sum to one. In this cohort, all points with the same \(\alpha\) have the same likelihood. Their relative posterior weights therefore retain their prior ratio: within either \(\alpha\) group, the posterior slope probabilities remain 0.75 and 0.25. The cohort does update the relative support for the two \(\alpha\) values.
This calculation also illustrates an important information limit. At the reference dose, \(\log(d/d_{\mathrm{ref}})=0\), so the likelihood contains no \(\beta\). A cohort observed only at that dose cannot directly distinguish a shallow from a steep dose-response curve. The slope’s posterior distribution here remains as specified by the prior; treating the data as if they had estimated the slope would overstate what this first cohort tells us.
Posterior Overdose Probabilities and Permission
Let \(\theta\) be the target DLT probability and \(c\) the protocol’s overdose-probability cutoff. At a candidate dose \(d\), the posterior overdose probability is the posterior probability that the DLT probability exceeds the target:
For the example, suppose the target is \(\theta=0.30\), the cutoff is \(c=0.25\), and the candidate doses are 10, 15, and 20 mg. For each parameter pair, calculate \(\pi(d)\) from the BLRM. Then sum the posterior weights of precisely those pairs for which \(\pi(d)>0.30\). This is a posterior probability, not an average of the toxicity probabilities.
Worked Example: Calculating the First Overdose Probabilities
At 15 mg, the dose ratio is 1.5. The four modeled DLT probabilities, using the BLRM, are approximately 0.1215 for \((0.10,0.8)\), 0.1606 for \((0.10,1.6)\), 0.2570 for \((0.25,0.8)\), and 0.3235 for \((0.25,1.6)\). Only the last exceeds the target of 0.30. Its posterior weight is 0.1575, so $$ P\{\pi(15)>0.30\mid Y=1\}=0.1575. $$ This is below the cutoff 0.25, so 15 mg is admissible.
At 20 mg, the dose ratio is 2. The four probabilities are approximately 0.1483, 0.2326, 0.3033, and 0.4311, respectively. Both parameter pairs with \(\alpha=0.25\) exceed the target. Summing their posterior weights gives $$ P\{\pi(20)>0.30\mid Y=1\}=0.4726+0.1575=0.6301. $$ Since \(0.6301>0.25\), 20 mg is not admissible. At 10 mg, the modeled probabilities are either 0.09091 or 0.2000, so the posterior overdose probability is zero.
The first cohort was treated at 10 mg. In this hypothetical protocol, the next candidate step is 15 mg, and escalation may proceed only to an admissible dose. The calculated results permit 15 mg and do not permit 20 mg. Thus 15 mg is the next permitted dose under the stated candidate-dose and overdose-control rules.
A Second Cohort Outcome and a Diagnostic Check
A useful check is whether the update behaves sensibly when the cohort outcome changes. A cohort with no DLTs should shift posterior support toward parameter values that predict lower toxicity at the observed dose. It does not, by itself, prove that every higher dose is safe: the slope and the extrapolation to higher doses still matter.
Worked Example: Updating After Zero DLTs
Keep the same prior and cohort size, but suppose the three patients at 10 mg have zero DLTs. At the reference dose, the likelihoods are \((1-q)^3\). They are \((10/11)^3=0.7513\) for \(\alpha=0.10\), and \((0.8)^3=0.5120\) for \(\alpha=0.25\). Since the prior puts total probability 0.5 on each \(\alpha\), the posterior probability of \(\alpha=0.25\) is $$ P(\alpha=0.25\mid Y=0)= \frac{0.5(0.5120)}{0.5(0.7513)+0.5(0.5120)} =0.4053. $$
The conditional slope weights remain 0.75 and 0.25, because the likelihood at 10 mg does not depend on \(\beta\). At 20 mg, both parameter pairs with \(\alpha=0.25\) still predict a DLT probability above 0.30, so the overdose probability is 0.4053, above the 0.25 cutoff. At 15 mg, only the \((0.25,1.6)\) point exceeds the target, giving an overdose probability of \(0.4053(0.25)=0.1013\), below the cutoff. Even with zero DLTs at the starting dose, this update permits 15 mg but not 20 mg.
Assumptions, Alternatives, and Failure Modes
The calculations rely on the specified BLRM, the prior weights, the binomial cohort likelihood, and the protocol’s target, cutoff, and candidate-dose rules. They also assume DLT outcomes have been assessed consistently for all three patients. If follow-up is incomplete, a patient’s outcome is not simply equivalent to “no DLT”; the likelihood and analysis must reflect the trial’s handling of incomplete observation.
Several tempting alternatives answer different questions. The observed DLT rate in the first example is \(1/3=0.3333\), but it describes the cohort at 10 mg; it is not the probability that a higher dose exceeds the target. The posterior mean DLT probability at a candidate dose is also a different summary from the posterior probability of exceeding the target. Neither substitutes for the specified overdose rule. As the earlier material on Posterior Mean Toxicity Is Not an Overdose Rule emphasizes, the decision requires the posterior mass above the target, not just a point summary.
A continuous-prior BLRM would usually obtain posterior overdose probabilities by numerical integration or simulation rather than by summing four grid weights. The discrete grid makes the update transparent but can represent uncertainty coarsely. Its answer can change if the grid omits plausible parameter values, puts too little weight on steep curves, or places a parameter value very near the overdose boundary. In a real analysis, numerical accuracy and prior sensitivity need to be checked using the protocol’s specified model and prior.
- Confusing a cohort rate with a posterior risk. The rate \(1/3\) is not the posterior probability of overdose at 15 or 20 mg. Calculate the model probability at each candidate dose and sum posterior weights above the target.
- Using the wrong inequality. The event is \(\pi(d)>\theta\). A parameter point exactly at \(\theta\) is not counted as overdose under this strict definition; the convention should be stated and applied consistently.
- Assuming the first cohort estimates the slope. At the reference dose the slope term is zero. With outcomes at only that dose, the likelihood cannot distinguish slope values.
- Skipping model and data checks. Verify the dose units and reference dose, the DLT count and cohort size, the model’s monotonicity assumptions, and that posterior weights sum to one before making a decision.
A posterior overdose probability close to the cutoff deserves particular care: rounding could change the apparent classification. Retain adequate numerical precision when applying the rule, and follow the protocol’s specified boundary and rounding conventions. The computed permission is conditional on the model, prior, data, and decision rule; it is not a guarantee that the next cohort will or will not experience toxicity.
Check Your Understanding
Answer each question using the definitions and calculations above; concise solutions follow.
- At the reference dose, why is the likelihood independent of \(\beta\)?
- After zero DLTs in three patients at 10 mg, what is the posterior probability of \(\alpha=0.25\), to four significant figures?
- After one DLT in three patients, which parameter-grid point alone makes 15 mg an overdose, and what is the resulting overdose probability?
- Why is an observed rate of one DLT in three patients not itself the overdose probability at 20 mg?
Solutions. (1) At \(d=d_{\mathrm{ref}}\), \(\log(d/d_{\mathrm{ref}})=0\), so the slope term vanishes. (2) It is \(0.4053\), from the likelihood-weighted calculation in the zero-DLT example. (3) The point \((0.25,1.6)\) predicts \(\pi(15)\approx0.3235\); its posterior weight, and therefore the overdose probability, is 0.1575. (4) The observed rate summarizes outcomes at 10 mg. The overdose probability at 20 mg instead sums posterior probabilities for parameter values whose modeled DLT probability at 20 mg exceeds 0.30.