Updating the BLRM After Each Cohort

Why Update After a Completed Cohort?

In a dose-escalation trial, each completed cohort contributes evidence about the dose-toxicity curve. The model used to evaluate the next assignment should reflect that evidence, rather than only the starting prior or the results from the most recent cohort. An update therefore has two linked parts: combine the new cohort outcomes with the current posterior, then recalculate quantities of interest—here, posterior overdose probabilities—at the candidate doses.

The update is not itself a dose-assignment rule. It provides the current posterior evidence for the decision process. Which dose is assigned depends on the protocol’s separate eligibility and selection rules. The focus here is how to update and report the model quantities accurately.

Model and data: Let $d$ be dose and let $p(d\mid\beta_0,\beta_1)$ denote the probability of a dose-limiting toxicity (DLT) at that dose, conditional on BLRM parameters $\beta_0,\beta_1$. Use the logistic dose-toxicity model $$ \operatorname{logit}\{p(d\mid\beta_0,\beta_1)\} =\beta_0+\beta_1\log(d/d_{\mathrm{ref}}), \qquad \beta_1>0, $$ where $d_{\mathrm{ref}}$ is a fixed reference dose. For cohort $j$, let $y_j$ be the number of participants with a DLT among $n_j$ participants treated at dose $d_j$.

The model assumes that, conditional on the parameters and dose, participants’ DLT outcomes are independent Bernoulli observations with the same probability $p(d_j\mid\beta_0,\beta_1)$ within a cohort. The cohort count then has a binomial likelihood. This likelihood also presumes that DLT outcomes are ascertained under the protocol’s definition and observation window, and that the outcomes being analyzed are available for the participants counted in $n_j$. If follow-up is incomplete, treating an unobserved outcome as “no DLT” is not a valid update.

The BLRM structure also imposes a common dose-toxicity curve and, with $\beta_1>0$, increasing modeled toxicity probability with increasing dose. These are modeling assumptions, not conclusions proved by a cohort’s observed rates. The earlier material on the BLRM establishes the model’s parameter interpretation and prior calibration; updating uses that model and the prior already specified for it.

The Cohort Update

For one cohort at dose $d_j$, the likelihood contribution is

L_j(\beta_0,\beta_1) =\binom{n_j}{y_j} p(d_j\mid\beta_0,\beta_1)^{y_j} \{1-p(d_j\mid\beta_0,\beta_1)\}^{n_j-y_j}.

If the current posterior density before this cohort is $\pi_{\mathrm{current}}(\beta_0,\beta_1)$, Bayes’ rule gives the updated density, up to its normalizing constant:

\pi_{\mathrm{updated}}(\beta_0,\beta_1) \propto L_j(\beta_0,\beta_1)\, \pi_{\mathrm{current}}(\beta_0,\beta_1).

For multiple cohorts, the likelihood contributions multiply. Thus the posterior after cohorts $1,\ldots,J$ is proportional to the starting prior times the product of their cohort likelihoods. Equivalently, one can update sequentially: use the posterior after cohort $j-1$ as the current distribution, then multiply by cohort $j$’s likelihood. This equivalence follows directly from multiplying the factors; the order of multiplication does not change the resulting posterior.

Posterior overdose probability: For a specified overdose boundary $p_{\mathrm{over}}$, the posterior overdose probability at dose $d$, after the observed cohorts, is $$ P\{p(d\mid\beta_0,\beta_1)>p_{\mathrm{over}}\mid\text{cohort data}\}. $$ It is the posterior probability that the dose-specific toxicity probability exceeds the boundary. It is not the probability that a particular participant will have a DLT, nor is it by itself a dose-assignment decision.

Why the Posterior Probability Changes

A cohort changes the relative posterior support for parameter values according to how well each value predicts the observed outcomes at the administered dose. A particle or posterior draw that assigns a relatively high probability to the observed cohort data receives more weight; one that assigns a low probability receives less. Since each parameter value implies a toxicity curve across doses, reweighting those values can change overdose probabilities at doses other than the dose just studied.

In a continuous posterior computation, posterior draws can be used to estimate the overdose probability by the fraction of draws whose dose-specific toxicity probability exceeds the boundary. With weighted particles, use the sum of the posterior weights for particles satisfying that condition. The following derivation shows why the weights are likelihood-reweighted.

Represent the current posterior.
Suppose particles $(\beta_{0,m},\beta_{1,m})$ have current weights $w_m$, with $\sum_m w_m=1$. They approximate the current posterior distribution.

Evaluate the new cohort likelihood.
For each particle, calculate $L_j(\beta_{0,m},\beta_{1,m})$ from its predicted toxicity probability at $d_j$.

Normalize the updated weights.
Bayes’ rule gives $w_m^{\mathrm{updated}}=w_m L_j(\beta_{0,m},\beta_{1,m})/\sum_r w_r L_j(\beta_{0,r},\beta_{1,r})$. The denominator makes the updated weights sum to one.

Sum weights in the overdose event.
At candidate dose $d$, sum $w_m^{\mathrm{updated}}$ over particles for which $p(d\mid\beta_{0,m},\beta_{1,m})>p_{\mathrm{over}}$.

The binomial coefficient is constant across parameter values for a fixed cohort, so it cancels when normalizing particle weights. It may be retained in the likelihood for completeness, but omitting it does not change the posterior parameter distribution.

Worked Updates With Posterior Particles

The examples use small, hypothetical particle sets to make the arithmetic visible. In practice, a fitted posterior generally uses many more draws or particles, and reported probabilities are estimates subject to simulation error. The dose-specific probabilities listed for each particle are assumed to come from BLRM curves with positive slopes.

Worked Example: No DLTs in a Three-Person Cohort

Suppose the current posterior is approximated by four equally weighted particles. At a lower cohort dose $d_L$, their predicted DLT probabilities are $0.10, 0.20, 0.40, 0.60$; at a higher candidate dose $d_C$, they are $0.25, 0.35, 0.50, 0.70$. The completed cohort at $d_L$ has $y=0$ DLTs among $n=3$. Let $p_{\mathrm{over}}=0.30$.

For zero DLTs, the likelihood at a particle with probability $p$ is $(1-p)^3$. The four likelihoods at $d_L$ are

0.90^3=0.729,\quad 0.80^3=0.512,\quad 0.60^3=0.216,\quad 0.40^3=0.064.

Their sum is $1.521$, so the updated particle weights are approximately $0.729/1.521=0.4793$, $0.512/1.521=0.3366$, $0.216/1.521=0.1420$, and $0.064/1.521=0.0421$. At $d_C$, particles 2, 3, and 4 exceed the overdose boundary. The updated overdose probability is therefore

$$ 0.3366+0.1420+0.0421=0.5207. $$

Before the cohort, three of the four equally weighted particles exceeded the boundary at $d_C$, giving probability $0.7500$. Under this particle approximation, the cohort lowers the estimated overdose probability at $d_C$ to $0.5207$, but it does not make that probability zero.

Worked Example: Two DLTs in a Three-Person Cohort

Use the same four equally weighted current particles and the same predicted probabilities at $d_L$ and $d_C$, but suppose the completed cohort at $d_L$ has $y=2$ DLTs among $n=3$. The likelihood at probability $p$ is $\binom{3}{2}p^2(1-p)=3p^2(1-p)$.

At $d_L$, the likelihoods are $3(0.10)^2(0.90)=0.027$, $3(0.20)^2(0.80)=0.096$, $3(0.40)^2(0.60)=0.288$, and $3(0.60)^2(0.40)=0.432$. Their sum is $0.843$. The normalized weights are approximately $0.0320, 0.1139, 0.3416,$ and $0.5125$.

At $d_C$, particles 2, 3, and 4 exceed $0.30$. Thus

P\{p(d_C)>0.30\mid\text{cohort data}\} \approx 0.1139+0.3416+0.5125=0.9680.

The estimated probability rises from its pre-cohort value of $0.7500$ to $0.9680$. The calculation is about posterior support for curves that exceed the boundary at $d_C$; it is not a claim that 96.80% of participants at that dose would experience a DLT.

Worked Example: Updating Sequentially After Two Cohorts

Consider four equally weighted hypothetical BLRM particles. Their predicted probabilities at doses $d_1$ and $d_2>d_1$ are, respectively, $(0.05,0.15)$, $(0.10,0.30)$, $(0.20,0.50)$, and $(0.35,0.70)$. A first cohort at $d_1$ has $0/3$ DLTs, followed by a completed cohort at $d_2$ with $1/3$ DLTs. Calculate the updated overdose probability at $d_2$ for boundary $0.30$.

After the first cohort, the likelihoods are $(1-p)^3$ at $d_1$: $0.857375, 0.729, 0.512,$ and $0.274625$. Their sum is $2.373$, giving weights approximately $0.3613, 0.3072, 0.2158,$ and $0.1157$. At $d_2$, the third and fourth particles exceed $0.30$, so the intermediate overdose probability is $0.2158+0.1157=0.3315$.

For the second cohort, the likelihood at probability $p$ is $3p(1-p)^2$. At $d_2$, these likelihoods are $0.325125, 0.441, 0.375,$ and $0.189$. Multiplying each by its current weight gives unnormalized updated weights approximately $0.11747, 0.13548, 0.08091,$ and $0.02187$, with sum $0.35573$. The normalized weights are approximately $0.3302, 0.3809, 0.2274,$ and $0.0615$.

Only particles 3 and 4 have probability strictly greater than $0.30$ at $d_2$; particle 2 is exactly at the boundary and is not included. The final estimate is $0.2274+0.0615=0.2889$. This sequential calculation gives the same result as weighting the original particles by the product of both cohort likelihoods.

Diagnostics, Alternatives, and Limitations

Before interpreting an updated overdose probability, verify the data and the computation. Check that each cohort’s dose, enrolled number, DLT count, and outcome status are correct; that the DLT window is complete for the participants analyzed; and that $0\le y_j\le n_j$. Confirm that the cohort likelihood uses the dose actually administered and that all cohorts included in the update are counted once.

For simulation-based updating, inspect convergence and mixing of posterior chains, Monte Carlo error for the overdose probability, and whether the draws cover the parameter regions that contribute to the overdose event. For importance-weighted particles, inspect weight concentration and effective sample size. If nearly all normalized weight sits on one or two particles, the numerical estimate may be unstable even though the arithmetic is correct. A fresh posterior simulation or a more appropriate sampling method may be needed.

A useful model check is to compare observed cohort outcomes with posterior predictive distributions: under the updated model, ask whether the observed counts are plausible at their administered doses. Marked discrepancies across cohorts can flag data problems, model misfit, or an assumption worth reassessing. A nonmonotone pattern in observed cohort rates alone does not establish that the BLRM update is wrong; sampling variation can produce such a pattern under a monotone curve.

Updating all cohorts together and updating cohort by cohort are mathematically equivalent when they use the same likelihood, prior, and data. Sequential updating is operationally useful because it makes clear when the posterior was refreshed. By contrast, using only the latest cohort discards earlier evidence. Replacing the model with observed DLT proportions or a point estimate also loses parameter uncertainty and does not generally produce the BLRM posterior overdose probability. These alternatives answer different, more limited questions.

Common pitfall: A learner may see $0/3$ DLTs and conclude that toxicity is zero, or may treat the observed rate $0/3$ as the posterior toxicity probability. Neither follows. The cohort contributes a likelihood that reweights the current posterior; uncertainty remains, and the posterior probability of overdose must be recomputed from the updated distribution. Likewise, a low posterior overdose probability is evidence for a dose rule to consider, not a dose assignment by itself.

The update depends on the specified model, prior, and valid cohort outcomes. A two-parameter curve may not represent all features of toxicity risk, and small cohorts can leave substantial uncertainty. Posterior probabilities are conditional on those assumptions. They should be reported with the dose, overdose boundary, included data, and computational method clearly stated.

Check Your Understanding

Answer each question, then compare with the brief solutions.

For a cohort with $y=1$ DLT among $n=2$, write its parameter-dependent likelihood at dose $d$.
If a current posterior particle has weight $0.20$ and the new cohort likelihood at that particle is $0.10$, while the normalizing sum over all weighted particles is $0.50$, what is its updated weight?
At a candidate dose, four updated particles have weights $0.10, 0.25, 0.30, 0.35$ and predicted probabilities $0.20, 0.30, 0.45, 0.60$. What is the overdose probability for boundary $0.30$, using a strict greater-than definition?
Why can a cohort change the overdose probability at a dose different from the cohort’s administered dose?
Name two checks that can reveal an unreliable update.

Solutions. (1) The likelihood is $2p(d)\{1-p(d)\}$, where $p(d)=p(d\mid\beta_0,\beta_1)$. (2) Its updated weight is $0.20(0.10)/0.50=0.0400$. (3) The second particle is exactly at the boundary and does not count; the probability is $0.30+0.35=0.65$. (4) Reweighting parameter values changes support for entire dose-toxicity curves, which determine probabilities at every dose. (5) Examples include checking cohort counts and follow-up completeness, assessing chain convergence or importance-weight concentration, and checking posterior predictive fit.