Dose Selection on Discrete Grids and Ties

Choosing Among Doses on a Finite Grid

In model-based dose finding, a fitted model yields a posterior distribution for the probability of dose-limiting toxicity (DLT) at each dose. A dose recommendation must turn those distributions into one choice from the doses available in the trial. This is a decision rule, not a claim that the selected dose has exactly the target toxicity probability. The rule should say both how closeness to the target is measured and what happens when the best values are tied.

Let \(D\) denote the observed trial data, let \(\phi\) be the prespecified target DLT probability, and let \(d_1<d_2<\cdots<d_K\) be the finite dose grid under consideration. Write \(p_j\) for the model’s unknown DLT probability at dose \(d_j\). The posterior distribution of \(p_j\), conditional on \(D\), reflects uncertainty about the toxicity probability at that dose under the fitted model.

Definition: A posterior target-dose criterion assigns a score to each dose using the posterior distribution of its DLT probability and the target \(\phi\). A selection rule chooses a dose that optimizes that score over the prespecified finite grid. For a closeness criterion, the selected dose minimizes the score; for a probability-of-being-close criterion, it maximizes the score.

The comparison assumes that the target \(\phi\), dose grid, posterior model, and set of doses eligible for this particular comparison have been specified. The posterior probabilities must be proper and computed from the same observed data and model. If the model imposes an ordered toxicity curve, that ordering is an assumption of the model, not something guaranteed by observing a finite sample. These criteria do not decide which model or grid is appropriate; they compare doses conditional on those choices.

Posterior Criteria and What They Reward

A common rule first calculates the posterior mean DLT probability for each dose and selects the dose whose mean is closest to \(\phi\). Its score is the absolute distance between the posterior mean and the target. This is straightforward to communicate, but compresses each posterior distribution to one number.

m_j=E(p_j\mid D),\qquad L_{\text{mean}}(d_j)=|m_j-\phi|,\qquad \widehat d_{\text{mean}}\in\arg\min_{d_j} L_{\text{mean}}(d_j).

Other rules preserve different features of the posterior. A posterior-median rule uses the median of \(p_j\mid D\) in place of its mean. A posterior expected absolute-loss rule averages the absolute distance from the target across the posterior distribution. A squared-loss rule averages squared distance. A target-band rule instead selects the dose with the greatest posterior probability of being within a prespecified tolerance \(\delta\) of the target.

\begin{aligned} L_{\text{median}}(d_j)&=|\operatorname{Median}(p_j\mid D)-\phi|,\\ L_{\text{abs}}(d_j)&=E\bigl(|p_j-\phi|\mid D\bigr),\\ L_{\text{sq}}(d_j)&=E\bigl((p_j-\phi)^2\mid D\bigr),\\ Q_{\delta}(d_j)&=P\bigl(|p_j-\phi|\leq\delta\mid D\bigr). \end{aligned}

The median rule focuses on the middle of the posterior distribution. Expected absolute loss accounts for posterior spread but penalizes distance linearly. Squared loss penalizes large deviations more heavily. The target-band rule asks a different question: how much posterior probability lies in an acceptable neighborhood of the target? Its answer depends on the chosen \(\delta\), which must have a clinically interpretable meaning and be fixed before looking at the comparison.

Important distinction: Minimizing \(|E(p_j\mid D)-\phi|\) is not generally the same as minimizing \(E(|p_j-\phi|\mid D)\) or \(E((p_j-\phi)^2\mid D)\). The first compares posterior means; the latter two include posterior uncertainty as well as location.

Worked Example: A Tie Under the Posterior-Mean Rule

Suppose the target is \(\phi=0.20\), and three hypothetical doses have posterior mean DLT probabilities of 0.18, 0.22, and 0.31. Apply the posterior-mean rule, using lower dose as the prespecified tie-breaker.

The absolute distances are \( |0.18-0.20|=0.02\), \( |0.22-0.20|=0.02\), and \( |0.31-0.20|=0.11\). Thus the first two doses attain the same minimum score, 0.02. The lower dose is selected by the stated tie-breaker.

This recommendation means that the first dose is preferred under this particular criterion and tie rule. It does not mean its true DLT probability is 0.20, nor that the posterior probability of being exactly at the target is high.

Why the Criteria Can Disagree

Posterior location and posterior uncertainty are distinct. Two doses can have the same posterior mean but different posterior spreads, or a dose with a mean exactly at target can have substantial posterior mass far from target. The expected-loss criteria make that uncertainty consequential in the score; the posterior-mean rule does not.

For squared loss, the relationship is especially clear. Let \(m_j=E(p_j\mid D)\) and \(v_j=\operatorname{Var}(p_j\mid D)\). Expanding the square and using \(E(p_j-m_j\mid D)=0\) gives

\begin{aligned} E\bigl((p_j-\phi)^2\mid D\bigr) &=E\bigl(((p_j-m_j)+(m_j-\phi))^2\mid D\bigr)\\ &=E\bigl((p_j-m_j)^2\mid D\bigr) +2(m_j-\phi)E(p_j-m_j\mid D)+(m_j-\phi)^2\\ &=v_j+(m_j-\phi)^2. \end{aligned}

Therefore, minimizing posterior expected squared deviation over a finite set of doses weighs both squared distance of the posterior mean and posterior variance. It need not select the dose with the closest posterior mean. This is different from choosing a freely adjustable numerical estimate of one probability: on a dose grid, the available actions are discrete doses, each with its own posterior distribution.

Worked Example: Posterior Mean and Expected Absolute Loss Disagree

Consider a target \(\phi=0.25\) and two hypothetical doses. At dose A, the posterior DLT probability is 0.20 with probability 0.75 and 0.40 with probability 0.25. At dose B, it is 0.27 with probability 1. These discrete distributions are illustrative posterior summaries.

For dose A, the posterior mean is \(0.75(0.20)+0.25(0.40)=0.15+0.10=0.25\). For dose B, it is 0.27. The posterior-mean distances are 0 for A and \(|0.27-0.25|=0.02\) for B, so the posterior-mean rule selects A.

For expected absolute loss, dose A has score \(0.75|0.20-0.25|+0.25|0.40-0.25|=0.75(0.05)+0.25(0.15)=0.075\). Dose B has score \(|0.27-0.25|=0.02\). The expected-absolute-loss rule therefore selects B. Although A’s posterior mean is exactly at target, its posterior mass is spread on both sides of the target; B is consistently closer in absolute distance.

For completeness, squared-loss scores are \(0.75(0.05^2)+0.25(0.15^2)=0.0075\) for A and \(0.02^2=0.0004\) for B. The squared-loss rule also selects B. Each choice follows its stated criterion; the criteria express different ways of valuing uncertainty and distance.

Computing Criteria From Posterior Draws

In practice, posterior draws can approximate the required summaries. Suppose \(p_j^{(1)},\ldots,p_j^{(M)}\) are draws from the posterior distribution of the DLT probability at dose \(d_j\). Approximate the posterior mean by the sample average, expected absolute loss by the average absolute deviation from \(\phi\), and target-band probability by the fraction of draws within the band. Use a consistent set of posterior draws and the same \(M\) across doses when possible.

\widehat m_j=\frac{1}{M}\sum_{r=1}^{M}p_j^{(r)},\qquad \widehat L_{\text{abs}}(d_j)=\frac{1}{M}\sum_{r=1}^{M}|p_j^{(r)}-\phi|,\qquad \widehat Q_{\delta}(d_j)=\frac{1}{M}\sum_{r=1}^{M} I\{|p_j^{(r)}-\phi|\leq\delta\}.

Worked Example: Comparing a Target-Band Rule

Let \(\phi=0.25\) and \(\delta=0.05\), so the target band is from 0.20 to 0.30, including both endpoints. Suppose four hypothetical posterior draws at dose A are 0.18, 0.22, 0.27, and 0.34; at dose B they are 0.21, 0.24, 0.29, and 0.31.

At A, two of four draws are in the band (0.22 and 0.27), so \(\widehat Q_{\delta}(A)=2/4=0.50\). At B, three of four draws are in the band (0.21, 0.24, and 0.29), so \(\widehat Q_{\delta}(B)=3/4=0.75\). The target-band rule selects B.

This calculation treats each posterior draw equally and estimates a posterior probability; it is not the observed proportion of patients with DLT. With only four draws, the approximation is coarse. Actual computations should use enough draws to make Monte Carlo error small relative to the score differences that drive the decision.

Making Ties Reproducible

A complete rule specifies how to handle ties in the primary score, not merely what score to calculate. A practical deterministic convention is: first optimize the stated primary criterion; if two or more doses tie, choose the lower dose. This is one possible protocol choice, not a universal statistical requirement. A different tie-breaker may be justified, but it must be specified in advance and applied consistently.

For a more elaborate rule, the protocol can state a fixed sequence of criteria: optimize the primary score, then use a named secondary score among tied doses, and finally select the lower dose if a tie remains. Avoid vague instructions such as “use clinical judgment” as the only tie rule when reproducibility is required. Document the grid, target, criterion, direction of optimization, tolerance for numerical ties, and final tie-breaker.

Computer calculations can produce values that differ by tiny amounts because of rounding or Monte Carlo error. Do not decide ties by comparing rounded display values. Compare unrounded scores and set a numerical tolerance in advance, appropriate to the scale and precision of the computation. For example, a protocol could treat two scores as tied if their absolute difference is at most \(10^{-8}\), while recognizing that a Monte Carlo-based score may require a different assessment of numerical uncertainty. The tolerance should not be chosen after seeing which dose it favors.

Define the comparison.
Record the posterior model, observed data, target \(\phi\), candidate grid, and primary criterion.

Calculate all dose scores.
Use unrounded posterior summaries or draws, with a consistent computation across doses.

Find the optimum and identify ties.
Minimize a loss or maximize a probability, applying the prespecified numerical tolerance.

Apply the tie-breaker and report the result.
State the selected dose, the primary score, any tied doses, and how the tie was resolved.

Diagnostics, Alternatives, and Limitations

Before interpreting a recommendation, check that the target and dose grid were defined correctly, posterior summaries correspond to the same fitted model, and any posterior-draw calculation used enough draws. Inspect the full posterior distributions, not only their means. A recommendation may be driven by a small score difference, a wide posterior, or an asymmetric distribution. Report that context so the numerical result is not mistaken for certainty.

The criteria are alternatives, not interchangeable labels for one rule. The posterior-mean rule is easy to explain and focuses on estimated location. The median rule is less influenced by extreme posterior values but can conceal spread. Expected absolute loss incorporates uncertainty with linear penalties, while expected squared loss gives greater influence to large deviations. A target-band rule has a direct probability interpretation, but depends on the prespecified width of the band. Comparing several criteria can reveal sensitivity; it does not justify selecting whichever criterion produces a preferred dose after seeing the results.

Every rule is limited by the model, data, target, and finite grid. The dose with the best score may still be far from the target, because a finite grid may contain no close option. The rule does not establish that the model is well calibrated or that the target itself is appropriate. Nor does a posterior criterion alone specify all trial conduct decisions. These are reasons to report the score and uncertainty alongside the selected dose, rather than presenting the selection as an unconditional fact.

Common Mistakes

Treating posterior means as the whole posterior. Two doses with similar means can have very different uncertainty. Check the distributions or use a criterion that explicitly incorporates spread.
Calling distinct rules equivalent. Closest posterior mean and minimum posterior expected absolute loss can choose different doses, as the worked example shows.
Ignoring ties or relying on rounded output. State a tie rule in advance and compare scores at adequate precision.
Interpreting the recommendation as certainty. A selected dose is optimal under the chosen criterion and model among the grid options; it need not have toxicity probability exactly equal to \(\phi\).

Key takeaway: A finite-grid target-dose recommendation requires two explicit choices: how posterior closeness to the target is scored and how equal scores are resolved. Posterior uncertainty can change the choice, so report the criterion, tie rule, and relevant posterior context.

Check Your Understanding

For each problem, calculate the requested result and compare it with the stated rule.

The target is 0.30. Two doses have posterior means 0.27 and 0.33, and a third has mean 0.39. Which dose is selected by the closest-posterior-mean rule if ties are resolved toward the lower dose?
Solution: The distances are 0.03, 0.03, and 0.09. The first two tie for the minimum, so the lower dose is selected.
At dose A, posterior probability is 0.10 with probability 0.5 and 0.30 with probability 0.5. At dose B, it is 0.19 with probability 1. The target is 0.20. Which dose minimizes expected absolute loss?
Solution: A’s score is \(0.5|0.10-0.20|+0.5|0.30-0.20|=0.10\). B’s score is \(|0.19-0.20|=0.01\). The expected-absolute-loss rule selects B.
Explain why minimizing posterior expected squared deviation need not select the dose with the closest posterior mean.
Solution: At dose \(d_j\), expected squared deviation equals \(\operatorname{Var}(p_j\mid D)+(E(p_j\mid D)-\phi)^2\). The variance differs by dose, so a dose with a slightly closer mean can have a larger total score.
With target 0.25 and band width \(\delta=0.04\), four draws at dose A are 0.20, 0.23, 0.28, and 0.31. What is the estimated target-band probability?
Solution: The band is [0.21, 0.29]. Two draws, 0.23 and 0.28, are in the band, so the estimate is \(2/4=0.50\).
Two computed primary scores differ by less than the prespecified tie tolerance. What should the analyst do?
Solution: Treat the scores as tied under the stated rule, then apply the predeclared tie-breaker. Do not round selectively or invent a new rule after observing the result.