Deriving the Posterior Target-Dose Rule

From a Posterior Distribution to a Dose Recommendation

The continual reassessment method (CRM) represents toxicity probabilities through a dose-toxicity model whose unknown parameter is updated as trial outcomes accrue. That posterior distribution describes uncertainty about the model parameter, but a dose recommendation requires a decision for each available dose. The posterior target-dose rule makes that decision by comparing each dose’s posterior mean toxicity probability with the prespecified target rate.

This comparison is not the same as substituting a single parameter estimate into the toxicity model. Because the model is generally nonlinear in its parameter, averaging the toxicity probability over the posterior can differ from evaluating it at the posterior mean or mode of the parameter. The distinction matters most when the posterior is broad or skewed.

Definition: Let $s_j$ be the skeleton probability for dose $j$, and let $a>0$ be the positive power parameter in the one-parameter CRM, so that the modeled toxicity probability is $q_j(a)=s_j^a$. Let $\pi(a\mid D)$ denote the posterior distribution of $a$ given the accumulated trial data $D$. The posterior mean toxicity probability at dose $j$ is $$ m_j=E\{q_j(a)\mid D\}=\int s_j^a\,\pi(a\mid D)\,da. $$ For target toxicity rate $\phi$, the posterior target-dose rule selects a dose minimizing $|m_j-\phi|$ over the eligible candidate doses.

Here $s_j$ is between zero and one, and the skeleton probabilities are ordered by dose. The parameterization $q_j(a)=s_j^a$ is equivalent to a log-parameter form with $a=\exp(\alpha)$. For fixed $a>0$, increasing the skeleton probability increases the modeled toxicity probability. The posterior mean $m_j$ is a probability, not a parameter estimate or a guarantee that the true toxicity rate equals that value.

Why the Rule Uses a Posterior Mean

The model and likelihood supply a posterior distribution, rather than a single known toxicity probability. Under the Bernoulli outcome model, if $y_j$ toxicities are observed among $n_j$ patients treated at dose $j$, the likelihood contribution at that dose is proportional to $q_j(a)^{y_j}\{1-q_j(a)\}^{n_j-y_j}$. Assuming conditionally independent patient outcomes and a common modeled probability at each dose, the posterior is proportional to the product of these contributions and the prior density. The posterior mean at each dose is obtained by averaging $q_j(a)$, not $a$, with respect to that posterior.

The “closest” part of the decision rule can be written in either absolute or squared distance form. Since absolute distance is nonnegative, and squaring preserves the ordering of nonnegative values, both criteria choose the same dose:

\arg\min_j |m_j-\phi| = \arg\min_j (m_j-\phi)^2.

This equivalence describes the stated rule. It does not establish that the rule is optimal under every possible clinical utility function. In particular, minimizing the squared distance between the posterior mean and the target differs in general from minimizing posterior expected squared loss in the unknown toxicity probability:

E\{(q_j(a)-\phi)^2\mid D\} = (m_j-\phi)^2+\operatorname{Var}\{q_j(a)\mid D\}.

The equality follows by writing $q_j(a)-\phi=(q_j(a)-m_j)+(m_j-\phi)$, squaring, and taking the posterior expectation. The cross-product has expectation zero because $E\{q_j(a)-m_j\mid D\}=0$. Posterior expected squared loss therefore adds a variance term that the posterior-mean target rule does not include. The rule is a specified CRM decision criterion, not a universal theorem that uncertainty is irrelevant.

Worked Example: Comparing Posterior Means With the Target

Consider three candidate doses with skeleton probabilities $s_1=0.10$, $s_2=0.25$, and $s_3=0.45$. Suppose, hypothetically, that accumulated trial data produce a posterior supported on $a=1$ and $a=2$, with probabilities $64/189$ and $125/189$, respectively. Let the target rate be $\phi=0.30$. Find the rule’s recommendation.

For any candidate skeleton value $s_j$, the posterior mean is $m_j=(64/189)s_j+(125/189)s_j^2$, because $q_j(1)=s_j$ and $q_j(2)=s_j^2$.

At dose 1, $m_1=\{64(0.10)+125(0.10^2)\}/189=(6.4+1.25)/189=0.04048$. Its signed deviation from the target is $m_1-\phi=0.04048-0.30=-0.25952$, so its absolute distance is $0.25952$.

At dose 2, $m_2=\{64(0.25)+125(0.25^2)\}/189=(16+7.8125)/189=0.12599$. Its signed deviation is $0.12599-0.30=-0.17401$, and its absolute distance is $0.17401$.

At dose 3, $m_3=\{64(0.45)+125(0.45^2)\}/189=(28.8+25.3125)/189=54.1125/189=0.28631$. Its signed deviation is $0.28631-0.30=-0.01369$, and its absolute distance is $0.01369$. All three posterior means are below the target, and dose 3 is closest. The rule therefore recommends dose 3 among these candidates.

A Short Derivation From the Model and Posterior

The calculation has two stages. First, the observed outcomes update the parameter distribution. Second, each modeled dose probability is averaged under that updated distribution. For a finite posterior distribution with support $a_1,\ldots,a_K$ and posterior probabilities $w_1,\ldots,w_K$, where $\sum_{k=1}^{K}w_k=1$, the integral becomes a weighted sum:

m_j=\sum_{k=1}^{K}w_k q_j(a_k) =\sum_{k=1}^{K}w_k s_j^{a_k}.

This follows directly from the definition of expectation for a discrete random variable: each possible toxicity probability $s_j^{a_k}$ contributes in proportion to its posterior probability $w_k$. For a continuous posterior, the same calculation is an integral, often evaluated numerically. Once all $m_j$ values have been computed, the rule compares their distances to $\phi$. No estimate of $a$ needs to be chosen first.

If the skeleton is strictly increasing across candidate doses, then for every $a>0$, $s_j<s_{j+1}$ implies $s_j^a<s_{j+1}^a$. Taking posterior expectations preserves the inequality, so $m_j<m_{j+1}$. This gives an important diagnostic: under the stated model and ordered skeleton, posterior mean toxicity probabilities should be ordered by dose. A reversed order indicates a calculation error, a different model or candidate set, or a failure to meet the assumptions just stated.

Worked Example: Updating a Two-Point Posterior and Applying the Rule

Suppose the positive power parameter has prior probability $1/2$ at $a=1$ and $1/2$ at $a=2$. At a dose with skeleton probability $s=0.50$, a hypothetical cohort of two patients has one toxicity and one non-toxicity. Under the Bernoulli model, the likelihood at $a=1$ is $0.50(1-0.50)=0.25$; at $a=2$, it is $0.25(1-0.25)=0.1875$. The common binomial coefficient, if outcomes are recorded as a count, cancels from the posterior ratio.

The unnormalized posterior weights are $0.5(0.25)=0.125$ and $0.5(0.1875)=0.09375$. Normalizing gives posterior probabilities $0.125/(0.21875)=4/7$ for $a=1$ and $0.09375/(0.21875)=3/7$ for $a=2$. The stated update assumes the cohort outcomes are fully observed and conditionally independent given $a$.

Now suppose the candidate skeleton probabilities are $0.10$, $0.25$, and $0.50$, with target $\phi=0.20$. At $s=0.10$, $m_1=\{4(0.10)+3(0.01)\}/7=0.43/7=0.06143$, giving deviation $-0.13857$.

At $s=0.25$, $m_2=\{4(0.25)+3(0.0625)\}/7=1.1875/7=0.16964$, giving deviation $-0.03036$. At $s=0.50$, $m_3=\{4(0.50)+3(0.25)\}/7=2.75/7=0.39286$, giving deviation $+0.19286$. The middle candidate has the smallest absolute deviation from 0.20, so the rule recommends the dose with skeleton probability 0.25.

Notice that the recommendation is based on posterior probabilities for $a$, after using the observed outcomes, and not on the prior or on the toxicity rate observed in this small cohort alone. The arithmetic illustrates the calculation; it does not imply that a two-patient cohort would provide precise information in a real trial.

Posterior Averaging Is Not Plug-In Estimation

Worked Example: Integrating the Curve Rather Than Plugging In a Parameter

Suppose a hypothetical posterior places probabilities 0.20, 0.50, and 0.30 on $a=0.5$, $a=1$, and $a=2$, respectively. Consider skeleton probabilities 0.10, 0.25, and 0.40, with target $\phi=0.20$. For $s=0.25$, the posterior mean toxicity probability is $m=\{0.20(0.50)+0.50(0.25)+0.30(0.0625)\}=0.24375$. Here $0.50=\sqrt{0.25}$, while $0.0625=0.25^2$.

The posterior mean of the parameter is $E(a\mid D)=0.20(0.5)+0.50(1)+0.30(2)=1.20$. Plugging that value into the model instead gives $0.25^{1.20}$, approximately $0.1895$, which is not the posterior mean toxicity probability $0.24375$.

Applying the posterior-mean rule requires averaging separately at each candidate. At $s=0.10$, $m_1=0.20(0.31623)+0.50(0.10)+0.30(0.01)=0.11625$, with deviation $-0.08375$. At $s=0.25$, $m_2=0.24375$, with deviation $+0.04375$. At $s=0.40$, $m_3=0.20(0.63246)+0.50(0.40)+0.30(0.16)=0.37449$, with deviation $+0.17449$. The closest posterior mean is $0.24375$, so this rule selects the candidate with skeleton probability 0.25.

The plug-in values are not the quantities specified by the rule. Nonlinearity means $E\{s^a\mid D\}$ generally does not equal $s^{E(a\mid D)}$. A parameter estimate can be useful for other purposes, but substituting it here changes the decision calculation.

Assumptions, Diagnostics, and Alternatives

The interpretation of the calculation depends on a correctly specified analysis model and data. The standard setup assumes that the toxicity outcome is defined consistently for every patient, that outcomes used in the likelihood are observed for the relevant assessment window, and that patient outcomes are conditionally independent given the model parameter and assigned doses. It also assumes that the one-parameter CRM and its skeleton adequately describe the dose-toxicity relationship for the purpose of the analysis. The posterior itself depends on the chosen prior as well as the likelihood.

Before interpreting a recommendation, check the following:

Posterior calculation: posterior weights or numerical integration weights must be nonnegative and sum to one, up to rounding.
Probability scale: each $q_j(a)$ and $m_j$ must lie between zero and one.
Dose ordering: with an increasing skeleton and positive $a$, posterior mean toxicity probabilities should increase across doses.
Target comparison: report both the signed deviation $m_j-\phi$ and its absolute value when useful. The sign states whether the posterior mean is below or above target; the absolute value determines closeness.
Posterior adequacy: a diffuse or highly skewed posterior can make recommendations sensitive to prior and model choices. A single closest mean does not communicate the full posterior uncertainty.

The posterior-mean rule is one of several ways to summarize a dose-toxicity posterior for a decision. A plug-in approach evaluates the curve at a point estimate of $a$; it is simpler but does not average over parameter uncertainty. A posterior-median toxicity estimate uses a different summary and need not produce the same recommendation. A probability-based criterion could instead assess how much posterior mass lies above or below the target. Those approaches answer different decision questions, so they should not be treated as algebraic substitutes for the stated rule.

A further limitation is that the rule ranks candidate doses by closeness alone. It does not, by itself, encode the clinical consequences of different toxicity severities, uncertainty penalties, or restrictions on which doses may be considered. Those considerations require additional design specifications. Nor does a posterior mean exactly at the target establish that the true toxicity probability equals the target; it is a posterior summary conditional on the model, prior, and accumulated data.

Common Pitfall

Common mistake: “The posterior mean toxicity probability is 0.286 and the target is 0.30, so the deviation is $+0.014$.” This loses the direction of the comparison. The signed deviation is $0.286-0.30=-0.014$, meaning the posterior mean is below target. Its absolute distance, about $0.014$, is used to compare closeness. Another related mistake is to use the posterior mean of $a$ in place of the posterior mean of $q_j(a)$; the nonlinear model generally makes those calculations different.

In practice, keep the decision calculation transparent: state the target, list each candidate’s posterior mean toxicity probability, calculate signed deviations, and compare absolute distances. This makes it possible to audit the recommendation and to distinguish “closest to target” from “below target” or “safest.”

Check Your Understanding

Answer each question, then compare with the solutions below.

If the posterior mean toxicity probability at a candidate dose is 0.18 and the target is 0.25, what are the signed deviation and absolute distance? What does the sign mean?
For a discrete posterior with weights 0.6 at $a=1$ and 0.4 at $a=2$, calculate the posterior mean toxicity probability when $s=0.30$.
Why can $E\{s^a\mid D\}$ differ from $s^{E(a\mid D)}$?
If every candidate posterior mean is below the target, does the rule select a candidate above target? Explain what it selects instead.

Solutions. 1. The signed deviation is $0.18-0.25=-0.07$, and the absolute distance is $0.07$. The negative sign means the posterior mean is below target. 2. The posterior mean is $0.6(0.30)+0.4(0.30^2)=0.18+0.036=0.216$. 3. The function $s^a$ is nonlinear in $a$, so evaluating it at the mean parameter generally does not equal averaging its values over the posterior. 4. No. It selects the eligible candidate with the smallest absolute distance to target, even if that candidate’s posterior mean remains below target; selection by this criterion does not guarantee a candidate is at or above target.