Hand Calculation of a CRM Dose Recommendation

From Cohort Outcomes to a Dose Recommendation

A CRM dose recommendation is the end of a short chain of calculations: specify the skeleton and parameter prior, evaluate how well each candidate model fits the observed outcomes, update the parameter weights, and compare the resulting toxicity estimates with the target rate. Hand calculation makes each link visible. It is also a useful check that software is using the intended likelihood, prior, and dose-selection rule.

The calculations below use the one-parameter power CRM and a discrete prior on a small set of parameter values. A discrete prior keeps the arithmetic transparent; the same posterior principle applies when the parameter has a continuous prior, in which case integration or numerical approximation replaces finite sums.

Definition: Let \(q_j\) be the skeleton toxicity probability at dose level \(j\), with \(0<q_1<\cdots<q_J<1\). In the power CRM, the probability of dose-limiting toxicity (DLT) at dose \(j\), conditional on parameter \(a\), is \(p_j(a)=q_j^{\exp(a)}\). Write \(\gamma=\exp(a)>0\), so a candidate model has toxicity probability \(q_j^\gamma\). The target toxicity rate is \(\phi\).

The skeleton gives an ordered set of starting probabilities, not observed toxicity rates. The power parameter changes their shape: \(\gamma=1\) reproduces the skeleton, while values below or above 1 raise each \(q_j\) to a power that respectively increases or decreases it. Because \(\gamma>0\), the model preserves dose ordering: if \(q_j<q_k\), then \(q_j^\gamma<q_k^\gamma\).

Likelihood and Posterior Weights

Suppose the data consist of \(n_j\) patients treated at dose \(j\), of whom \(y_j\) experienced DLT in the defined assessment window. Conditional on a candidate parameter value, assume the patient outcomes are independent Bernoulli observations with probability \(q_j^\gamma\) at dose \(j\). Assume each outcome is fully observed and correctly classified for that window. These assumptions give the likelihood, apart from binomial coefficients that do not depend on \(\gamma\):

L(\gamma)\ \propto\ \prod_{j=1}^{J} (q_j^\gamma)^{y_j}(1-q_j^\gamma)^{n_j-y_j}.

If the prior assigns weight \(\pi_r\) to candidate \(\gamma_r\), its posterior weight is proportional to prior weight times likelihood. Dividing by the sum of these products normalizes the weights to sum to 1.

w_r = P(\gamma=\gamma_r\mid \text{data}) = \frac{\pi_r L(\gamma_r)} {\sum_s \pi_s L(\gamma_s)}.

Once the posterior weights are available, the posterior mean toxicity at dose \(j\) is the weighted average of the candidate probabilities. The usual posterior target-dose rule recommends the grid dose whose posterior mean toxicity is closest to \(\phi\):

\widehat p_j = E[p_j(\gamma)\mid\text{data}] = \sum_r w_rq_j^{\gamma_r}, \qquad j^*=\arg\min_{1\leq j\leq J}|\widehat p_j-\phi|.

This rule uses the posterior mean toxicity at each dose, not the toxicity probability evaluated at the posterior mean of \(\gamma\). In general, averaging after transforming the parameter is not the same as transforming after averaging it.

Worked Calculation: Updating from a Skeleton and Prior

Worked Example: A Cohort History and a Three-Point Prior

Hypothetical data. Consider four dose levels with skeleton \(q=(0.05,0.12,0.25,0.40)\), target \(\phi=0.25\), and candidate power values \(\gamma=(0.5,1,2)\) with prior weights \((0.20,0.50,0.30)\). Three patients were treated at dose 2 and none experienced DLT; three were treated at dose 3 and one experienced DLT. Assume these outcomes are complete and independent conditional on \(\gamma\), as specified by the model.

Step 1: Calculate candidate probabilities. At dose 2, the probabilities for the three candidates are \(0.12^{0.5}=0.3464\), \(0.12\), and \(0.12^2=0.01440\). At dose 3, they are \(0.25^{0.5}=0.5000\), \(0.25\), and \(0.25^2=0.06250\). Each probability is between 0 and 1, and the probabilities increase with dose for each candidate, as the ordered skeleton and positive powers require.

Step 2: Calculate likelihoods. The observed dose-2 contribution is the probability of zero DLTs among three patients. The dose-3 contribution is the probability of exactly one DLT among three. The binomial coefficient for one DLT is the same for all candidates and can be omitted when calculating relative posterior weights.

\begin{aligned} L(0.5)&=(1-0.3464)^3(0.5000)(1-0.5000)^2 \approx 0.03490,\\ L(1)&=(1-0.12)^3(0.25)(1-0.25)^2 \approx 0.09583,\\ L(2)&=(1-0.0144)^3(0.06250)(1-0.06250)^2 \approx 0.05259. \end{aligned}

For example, the first line is \((0.6536)^3(0.5000)(0.5000)^2\), approximately \(0.03490\). It accounts for all six patients: three observed non-DLT outcomes at dose 2 and one DLT plus two non-DLT outcomes at dose 3.

Step 3: Multiply by prior weights and normalize. The unnormalized values are \(0.20(0.03490)=0.006980\), \(0.50(0.09583)=0.04792\), and \(0.30(0.05259)=0.01578\). Their sum is approximately \(0.07067\), giving:

(w_{0.5},w_1,w_2) \approx \left( \frac{0.006980}{0.07067}, \frac{0.04792}{0.07067}, \frac{0.01578}{0.07067} \right) = (0.09877,\ 0.6781,\ 0.2231).

The rounded posterior weights sum to 1.0000. The data give the \(\gamma=1\) model the largest posterior weight, but that fact alone does not identify the recommended dose; the dose-specific posterior means still need to be calculated.

Step 4: Calculate posterior mean toxicity and choose a dose. At dose 3, for example, the posterior mean is \(0.09877(0.5000)+0.6781(0.2500)+0.2231(0.06250)\approx0.2329\). The means across all four doses are approximately \(0.05655,\ 0.1188,\ 0.2329,\ 0.3694\). Their absolute differences from the target \(0.25\) are approximately \(0.1935,\ 0.1312,\ 0.0171,\ 0.1194\), respectively. Dose 3 is therefore recommended by the posterior target-dose rule. This is a model-based recommendation for the next cohort, not a claim that its true DLT rate is known.

Checking the Dose-Level Averages

Posterior mean toxicity is calculated dose by dose using the same posterior weights, but a different skeleton probability at each dose. The multiplication should be shown explicitly: small arithmetic errors can change which dose appears closest to the target, especially when two doses have similar posterior means.

Worked Example: Verifying Posterior Mean Toxicities

Suppose a discrete posterior has weights \(0.2594,\ 0.3493,\ 0.3912\) on power values \(\gamma=0.5,\ 1,\ 2\), respectively. These rounded weights sum to \(0.9999\). For the skeleton values \(q_2=0.12\), \(q_3=0.25\), and \(q_4=0.40\), the candidate probabilities are \((0.3464,0.12,0.01440)\) at dose 2, \((0.5000,0.25,0.06250)\) at dose 3, and \((0.6325,0.40,0.1600)\) at dose 4.

At dose 2, the weighted sum is \(0.2594(0.3464)+0.3493(0.12)+0.3912(0.01440) =0.08985616+0.041916+0.00563328 =0.13740544\), or approximately \(0.1374\). At dose 4, it is \(0.2594(0.6325)+0.3493(0.40)+0.3912(0.16) =0.1640705+0.13972+0.062592 =0.3663825\), or approximately \(0.3664\).

At dose 3, the mean is \(0.2594(0.5000)+0.3493(0.25)+0.3912(0.06250)=0.241475\), approximately \(0.2415\). If \(\phi=0.25\), this dose is closest among doses 2, 3, and 4: its distance is \(0.008525\), compared with \(0.1126\) at dose 2 and \(0.1164\) at dose 4. Because the supplied weights are rounded, these means are approximate; the displayed arithmetic uses the weights as given rather than silently renormalizing them.

Updating After the Next Outcome

When a new outcome is observed, there is no need to reconstruct the original likelihood from scratch. The existing posterior weights serve as the new prior weights, and each is multiplied by the likelihood contribution from the new observation. This sequential update is valid because the likelihood factors into contributions from conditionally independent patient outcomes.

Worked Example: One Additional DLT at Dose 3

Continue the first example, but suppose one additional patient is treated at dose 3 and experiences DLT. The dose-3 probabilities under \(\gamma=0.5,1,2\) are \(0.5000,\ 0.2500,\ 0.06250\). Multiply the previous posterior weights \((0.09877,0.6781,0.2231)\) by these probabilities. The unnormalized updated weights are approximately \(0.04939,\ 0.1695,\ 0.01395\), with sum \(0.2329\). Normalizing gives updated weights approximately \((0.2121,0.7283,0.0599)\).

Using these weights, the posterior mean toxicity at dose 3 is \(0.2121(0.5000)+0.7283(0.2500)+0.0599(0.06250)\approx0.2919\). The corresponding means at doses 1, 2, and 4 are approximately \(0.0840,\ 0.1617,\ 0.4350\). With target \(0.25\), dose 3 remains closest: its distance is \(0.0419\), versus \(0.1660,\ 0.0883,\) and \(0.1850\) for doses 1, 2, and 4. The posterior and recommendation have changed, even though the recommended dose in this example has not.

Calculation Checks, Alternatives, and Limits

A reliable hand calculation has several simple diagnostics. First, check that each candidate probability \(q_j^\gamma\) lies between 0 and 1 and that probabilities increase with dose. Second, check that every likelihood factor matches the observed count: \(y_j\) powers of \(p_j\), and \(n_j-y_j\) powers of \(1-p_j\). Third, posterior weights must be nonnegative and sum to 1 before rounding. Finally, posterior mean toxicity at a dose must lie between the smallest and largest candidate toxicity probabilities at that dose, because it is a weighted average.

A particularly informative check is to inspect the effect of the outcomes. If a candidate assigns a very small toxicity probability to a dose where DLT was observed, its likelihood contribution should be small. Conversely, observing no DLTs tends to favor candidates assigning lower probabilities at that dose, all else equal. These are tendencies, not rules for every full dataset: outcomes at several doses and the prior jointly determine the posterior.

The main alternative in earlier material on rule-based escalation is to make decisions from prespecified local count rules, rather than to fit a dose-toxicity model and borrow information across dose levels through a common parameter. The CRM can use the shape of the whole skeleton, but that is also a limitation: a poorly chosen skeleton or an inadequate one-parameter model can yield misleading estimates. A discrete prior adds another approximation, since the posterior can only assign weight to the candidate values included in its support. Hand calculations are most useful for understanding and auditing a design; practical implementations should also be checked computationally.

Wrong reasoning: “The candidate with the largest posterior weight determines the dose.”
Correction: The candidate weight concerns the model parameter, not a dose. Calculate the posterior mean toxicity at each dose, then apply the specified target-dose rule.
Wrong reasoning: “At dose \(j\), average the skeleton probability and then raise it to the posterior mean power.”
Correction: Average the candidate dose probabilities using posterior weights: \(\sum_r w_rq_j^{\gamma_r}\). In general, this is not \(q_j^{\sum_r w_r\gamma_r}\).
Wrong reasoning: “A low observed DLT count proves that a dose is safe.”
Correction: Cohort counts are uncertain data. CRM recommendations are conditional on the model, prior, and observed outcomes; they do not establish a true rate with certainty.

Key takeaway: A hand-calculated power-CRM recommendation requires three auditable steps: evaluate the cohort likelihood under each candidate parameter, normalize prior-times-likelihood weights, and compare the posterior mean toxicity at every grid dose with the target rate.

Check Your Understanding

Try each question, then compare your reasoning with the brief solutions.

For \(q=0.20\), calculate the candidate toxicity probability when \(\gamma=0.5\) and when \(\gamma=2\).
At one dose, four patients are treated and two experience DLT. Write the likelihood contribution at candidate probability \(p\), omitting the binomial coefficient.
Prior weights are \(0.25\) and \(0.75\), and the corresponding likelihoods are \(0.08\) and \(0.04\). Calculate the posterior weights.
Two doses have posterior mean toxicity \(0.18\) and \(0.31\), with target \(\phi=0.25\). Which dose is selected by the posterior target-dose rule?

Solutions. (1) \(0.20^{0.5}=\sqrt{0.20}\approx0.4472\), and \(0.20^2=0.0400\). (2) The contribution is \(p^2(1-p)^2\), since two outcomes are DLT and two are not. (3) The unnormalized weights are \(0.25(0.08)=0.020\) and \(0.75(0.04)=0.030\); dividing by \(0.050\) gives posterior weights \(0.40\) and \(0.60\). (4) The distances from target are \(0.07\) and \(0.06\), so the dose with posterior mean toxicity \(0.31\) is selected.