Why Cohort Outcomes Update the CRM
A continual reassessment method (CRM) uses a dose-toxicity model to learn about toxicity risk as trial outcomes accrue. Before a cohort is treated, the model represents uncertainty about the dose-toxicity curve through a prior distribution on its parameter. Once that cohort’s dose-limiting toxicity (DLT) outcomes are fully assessed, its DLT count contributes a likelihood. Bayes’ rule combines that likelihood with the prior to give the updated posterior. This tutorial focuses on that update and on calculating the resulting toxicity estimates at all candidate doses.
The CRM skeleton construction covered in the earlier material on constructing CRM skeletons supplies a set of increasing reference probabilities. The skeleton alone is not the current toxicity estimate: the estimate also depends on the parameter values allowed by the model and on the posterior weight assigned to each value after observing outcomes.
A common implementation places a discrete prior on a finite grid \(a_1,\ldots,a_M\), with prior weights \(\pi_m=P(a=a_m)\), where \(\pi_m\geq 0\) and \(\sum_{m=1}^M\pi_m=1\). This makes the update a finite set of likelihood calculations and normalizations. A continuous prior can also be used, in which case the same Bayes’ rule applies but posterior calculations require integration or numerical approximation.
The cohort-count likelihood assumes that, conditional on the parameter and the assigned doses, each patient’s DLT outcome is Bernoulli with the modelled probability at that dose, and outcomes are conditionally independent. It also assumes the outcomes included in the update are assessed for the relevant DLT window. The binomial count likelihood follows because the sum of \(n_k\) such outcomes has a binomial distribution. If patients have different follow-up or DLT definitions, or if dependence is material, this simple likelihood may not be appropriate.
Likelihood and Posterior Update
For one completed cohort at dose \(d_j\), with \(y\) DLTs among \(n\) patients, the probability of the observed count under parameter \(a_m\) is
For several cohorts, conditional independence gives a product of their likelihood contributions. Thus the posterior weight on grid value \(a_m\) is
The binomial coefficients are the same for every parameter value, so they cancel between numerator and denominator. They may be retained when writing the full likelihood, or omitted when calculating posterior weights. What must not be omitted is the probability contribution for each observed DLT and each non-DLT.
Worked Updates
Worked Example: One Cohort and Three Candidate Doses
Suppose the candidate-dose skeleton is \(q=(0.05,0.15,0.30)\). Use the parameter grid \(a=(0.5,1,2)\) with prior weights \((1/3,1/3,1/3)\). A hypothetical cohort of three patients is treated at dose \(d_2\), and one patient has a DLT. Calculate the posterior weights and posterior mean toxicity at all three doses.
At \(d_2\), the probabilities under the three parameter values are \(0.15^{0.5}=0.3873\), \(0.15^1=0.1500\), and \(0.15^2=0.02250\). For \(y=1\) and \(n=3\), the likelihood is \(\binom{3}{1}p(1-p)^2\). The common factor 3 cancels in the posterior-weight calculation, so use \(p(1-p)^2\):
For \(a=0.5\), the likelihood factor is \(0.3873(0.6127)^2\approx0.1454\). For \(a=1\), it is \(0.1500(0.8500)^2=0.108375\). For \(a=2\), it is \(0.02250(0.9775)^2\approx0.02150\). Multiplying by equal prior weights does not change the normalized ratios. The sum is approximately \(0.2753\), giving posterior weights approximately \(0.5281,\ 0.3936,\ 0.0781\), respectively.
At dose \(d_1\), the three parameter-specific probabilities are \(0.05^{0.5}=0.2236\), \(0.0500\), and \(0.002500\). Therefore the posterior mean is \(0.5281(0.2236)+0.3936(0.0500)+0.0781(0.002500)\approx0.1380\).
At dose \(d_2\), use the probabilities \(0.3873,\ 0.1500,\ 0.02250\): \(0.5281(0.3873)+0.3936(0.1500)+0.0781(0.02250)\approx0.2653\). At dose \(d_3\), the probabilities are \(0.30^{0.5}=0.5477,\ 0.3000,\ 0.09000\): \(0.5281(0.5477)+0.3936(0.3000)+0.0781(0.09000)\approx0.4143\). The model-based posterior mean toxicity estimates are therefore about 0.1380, 0.2653, and 0.4143 across the candidate doses. They differ from the observed cohort proportion \(1/3\), which describes only the cohort treated at \(d_2\).
Worked Example: A Single DLT With Two Possible CRM Parameters
Consider a hypothetical skeleton \(q=(0.05,0.20,0.50)\), a grid \(a=(0.5,1)\), and prior weights \((0.5,0.5)\). One fully assessed patient treated at \(d_2\) has a DLT. Find the posterior weights and posterior mean toxicity at each candidate dose.
At \(d_2\), the DLT probabilities are \(0.20^{0.5}=\sqrt{0.20}\approx0.4472\) and \(0.20^1=0.2000\). With one patient and one DLT, the likelihood is the probability itself. The unnormalized posterior weights are \(0.5(0.4472)=0.2236\) and \(0.5(0.2000)=0.1000\), with sum \(0.3236\). Normalizing gives weights \(0.2236/0.3236\approx0.6910\) and \(0.1000/0.3236\approx0.3090\).
At \(d_1\), the probabilities are \(0.05^{0.5}=0.2236\) and \(0.0500\), so the posterior mean is \(0.6910(0.2236)+0.3090(0.0500)\approx0.1700\). At \(d_2\), it is \(0.6910(0.4472)+0.3090(0.2000)\approx0.3708\). At \(d_3\), the probabilities are \(0.50^{0.5}=0.7071\) and \(0.5000\), giving \(0.6910(0.7071)+0.3090(0.5000)\approx0.6431\). The single observed DLT increases posterior weight on the parameter value that predicts the higher risk at \(d_2\), but the posterior estimates at every dose are obtained by averaging that dose’s model probabilities.
Worked Example: Sequentially Updating After Two Cohorts
Suppose a hypothetical CRM uses skeleton \(q=(0.0625,0.25,0.50)\), parameter values \(a=(1,2)\), and prior weights \((0.5,0.5)\). At \(d_2\), the first completed cohort has zero DLTs among two patients; a second completed cohort at the same dose has one DLT among two patients. Update sequentially and calculate the final posterior mean toxicity at each dose.
For the first cohort, the likelihoods are \((1-0.25)^2=0.5625\) under \(a=1\), and \((1-0.25^2)^2=(0.9375)^2=0.87890625\) under \(a=2\). Multiplying by the equal prior weights and normalizing gives \(w_1=0.5625/(0.5625+0.87890625)\approx0.39024\) and \(w_2\approx0.60976\).
For the second cohort, the count likelihood is \(\binom{2}{1}p(1-p)\). The common factor 2 cancels. Under \(a=1\), the likelihood factor is \(0.25(0.75)=0.1875\); under \(a=2\), it is \(0.0625(0.9375)=0.05859375\). Equivalently, using the full likelihoods gives 0.375 and 0.1171875. Multiplying the first-cohort weights by the full likelihoods gives unnormalized weights \(0.39024(0.375)\approx0.14634\) and \(0.60976(0.1171875)\approx0.07146\). After normalization, the final weights are approximately \(0.6719\) and \(0.3281\).
The probabilities for \(a=1\) across the three doses are \(0.0625,\ 0.25,\ 0.50\); for \(a=2\), they are \(0.00390625,\ 0.0625,\ 0.25\). Thus the final posterior means are:
At \(d_1\): \(0.6719(0.0625)+0.3281(0.00390625)\approx0.04328\). At \(d_2\): \(0.6719(0.25)+0.3281(0.0625)\approx0.1885\). At \(d_3\): \(0.6719(0.50)+0.3281(0.25)\approx0.4180\).
Sequential updating agrees with updating once using both cohorts, because the joint likelihood is the product of the two cohort likelihoods. The first cohort with no DLT shifts weight toward \(a=2\), which predicts lower toxicity, while the second cohort’s DLT shifts weight back toward \(a=1\). The final posterior reflects both contributions.
Why the Posterior Mean Formula Holds
The formula for \(\hat{p}_j\) follows directly from averaging over the mutually exclusive parameter-grid values. Conditional on \(a=a_m\), the DLT probability at dose \(d_j\) is \(q_j^{a_m}\). After observing the cohorts, the parameter value has posterior probability \(w_m\). The law of total probability therefore gives:
The first conditional probability is \(q_j^{a_m}\) because the model specifies the patient’s DLT probability at dose \(d_j\) given \(a_m\); conditional independence means the observed outcomes update the parameter but do not change that model probability once the parameter is fixed. The posterior mean toxicity is thus also the posterior predictive probability of a DLT for a new patient at that dose, conditional on the model and observed data.
Checking the Update and Its Limits
Several checks help catch implementation errors. Posterior weights must be nonnegative and sum to one; likelihood contributions must use \(y_k\) DLTs and \(n_k-y_k\) non-DLTs; and every candidate-dose estimate must be a weighted average of probabilities, hence lie between the smallest and largest parameter-specific probabilities at that dose. With \(q_j\) increasing and every \(a_m>0\), each parameter-specific curve is increasing in dose, so the posterior mean curve is also increasing. If calculations violate these checks, investigate indexing, exponentiation, missing outcomes, or normalization.
A cohort’s observed DLT proportion \(y/n\) is not interchangeable with the model-based estimate. The observed proportion contains no direct information about other doses, and at a small cohort size it can vary substantially. CRM borrows across doses through the assumed dose-response curve. That borrowing is useful, but it means conclusions depend on whether the chosen skeleton and model family can represent plausible toxicity patterns.
The binomial update also requires outcomes to be treated consistently. An unassessed patient cannot simply be counted as a non-DLT; doing so would add a likelihood contribution that the patient’s outcome does not support. The update here concerns completed cohort outcomes only. Likewise, if dose assignments or DLT assessments are incorrectly recorded, even a correctly implemented formula yields an incorrect posterior.
This one-parameter CRM update differs from estimating a separate probability at each dose: the latter does not impose the CRM’s shared curve structure. It also differs from a logistic dose-toxicity model, which uses a different parameterization and likelihood structure. These approaches can produce different estimates because they encode different relationships among doses and different assumptions. The CRM posterior update alone does not determine a complete escalation decision; additional trial rules may constrain which dose can be assigned.
A finite parameter grid is computationally convenient, but a grid that is too narrow or too coarse can poorly represent relevant curves. A posterior concentrated at an edge of the grid is a warning to check whether the grid and prior cover adequate possibilities. Strong posterior concentration is not proof that the model form is correct. Posterior estimates can also be misleading if the assumed skeleton is badly calibrated or if the true relationship is not adequately described by \(q_j^a\). Model checking and clinical judgment remain necessary.
Check Your Understanding
Answer each question, then compare with the solutions below.
- At a dose with model probability \(p=0.20\), write the likelihood for three patients with zero DLTs and for three patients with one DLT.
- Two parameter values have equal prior weights. At the observed dose their DLT probabilities are 0.10 and 0.30, and one patient has a DLT. What are the posterior weights?
- Why can the posterior mean toxicity at an unobserved dose differ from the observed DLT proportion at the treated dose?
- If all posterior parameter values are positive and the skeleton probabilities increase with dose, can the posterior mean toxicity decrease from one candidate dose to the next? Explain.
Solutions. (1) For zero DLTs, \(L=(1-0.20)^3=0.8^3=0.512\). For one DLT, \(L=\binom{3}{1}(0.20)(0.80)^2=3(0.20)(0.64)=0.384\). (2) The unnormalized weights are \(0.5(0.10)=0.05\) and \(0.5(0.30)=0.15\); their sum is 0.20, so the posterior weights are 0.25 and 0.75. (3) The observed proportion describes only the assessed cohort at its dose. The posterior mean at another dose averages the model probability there over the updated parameter weights. (4) No. For each \(a_m>0\), \(q_j^{a_m}\) increases with \(q_j\); a weighted average with the same nonnegative posterior weights therefore also increases with dose.