Conservative Starting Doses in CRM

Why the Starting Dose Needs Its Own Safety Check

A CRM uses accumulating toxicity data to guide dose choices, but the first participants are treated before the trial has observed any outcomes. The starting dose therefore needs a rationale based on the information available at design time: the dose-toxicity skeleton, the prior for the CRM parameter, and clinical safety constraints. A starting dose is not simply the dose with the skeleton probability closest to the target. It is an initial operational choice made under uncertainty, before the likelihood has updated the prior.

The introductory material on the CRM parameter and skeleton provides the model framework. Here the focus is how to screen candidate starting doses using prior toxicity probabilities, and how that choice shapes the first evidence the model receives. The calculations below are hypothetical and illustrate a possible protocol rule, not a universal starting-dose standard.

Definition: Let $d_1<d_2<\cdots<d_J$ denote the ordered dose levels, and let $s_j$ be the skeleton probability at dose $d_j$, with $0<s_1<\cdots<s_J<1$. Under the one-parameter power CRM, the conditional probability of a dose-limiting toxicity (DLT) at $d_j$, given parameter $a$, is $$ p_j(a)=s_j^{\exp(a)}. $$ If $a$ has prior distribution $\pi(a)$, the prior predictive DLT risk at $d_j$ is $$ \bar q_j=E_{\pi}\{p_j(a)\}=\int p_j(a)\,\pi(a)\,da. $$ For a discrete prior assigning probability $w_k$ to $a_k$, this is $\bar q_j=\sum_k w_k p_j(a_k)$.

The prior predictive risk averages over uncertainty in the model parameter. It is the prior probability of a DLT for a future participant assigned to $d_j$, under the model and prior. It is not the toxicity probability at the average value of $a$: in general, $E\{p_j(a)\}\ne p_j\{E(a)\}$. A separate safety summary is the prior probability that the dose-specific risk exceeds a clinically chosen limit $c$, namely $P_{\pi}\{p_j(a)>c\}$. That probability describes uncertainty about the underlying risk, rather than the probability of a DLT for one participant.

Turning Clinical Limits into a Starting-Dose Rule

A protocol can state one or more constraints before the trial begins. For example, it can require the prior predictive risk at the initial dose to be no greater than a prespecified bound $b$, and require the prior probability that the true dose-specific risk exceeds a concerning level $c$ to be no greater than $\gamma$. The values of $b$, $c$, and $\gamma$ are clinical and design choices; they are not supplied automatically by the CRM model.

One transparent rule is to identify all doses satisfying every prespecified constraint and choose the highest dose in that eligible set. This rule is conservative relative to unconstrained selection by proximity to the target, because an otherwise attractive dose is excluded if it fails a safety screen. A protocol may instead start at the lowest dose, or choose a dose below the highest eligible level. The important requirement is to specify the policy and clinical rationale in advance rather than improvise after seeing trial outcomes.

Worked Example: Screening a Starting Dose with Two Safety Limits

Suppose a hypothetical trial has four doses with skeleton $(0.05,0.12,0.25,0.35)$, and a discrete prior giving equal probability to $a=-0.5,0,0.5$. The protocol requires both $\bar q_j\le 0.10$ and $P_{\pi}\{p_j(a)>0.25\}\le 0.10$. Calculate the prior risks and apply the rule.

For the first dose, the three conditional risks are $0.05^{\exp(-0.5)}\approx0.1625$, $0.05^{\exp(0)}=0.0500$, and $0.05^{\exp(0.5)}\approx0.0072$. Thus,

\bar q_1\approx\frac{0.1625+0.0500+0.0072}{3}=0.0732.

Applying the same calculation to each skeleton value gives the following approximate results. For the fourth dose, for example, the conditional risks are $0.5290, 0.3500,$ and $0.1771$, so $\bar q_4\approx(0.5290+0.3500+0.1771)/3=0.3520$.

Dose	Prior predictive risk $\bar q_j$	$P_{\pi}\{p_j(a)>0.25\}$	Eligible under both limits?
$d_1$, skeleton 0.05	0.0732	0	Yes
$d_2$, skeleton 0.12	0.1422	1/3	No
$d_3$, skeleton 0.25	0.2611	1/3	No
$d_4$, skeleton 0.35	0.3520	2/3	No

At $d_1$, none of the three conditional risks exceeds 0.25, so the exceedance probability is zero. At $d_2$, the risk for $a=-0.5$ is approximately 0.276, so one of the three equally weighted prior values exceeds 0.25; the exceedance probability is $1/3$. At $d_3$, the risk for $a=-0.5$ is approximately 0.431, and at $d_4$ the risks for $a=-0.5$ and $a=0$ exceed 0.25. Only $d_1$ satisfies both constraints, so it is the starting dose under this rule. This conclusion follows from the stated bounds and prior, not from a general claim that every CRM must start at its lowest dose.

Why the Two Safety Summaries Can Differ

The mean-risk constraint and the exceedance-probability constraint answer different questions. The first averages the risk over the prior; the second checks how much prior probability lies on parameter values that imply a risk above a clinical limit. A small average can coexist with a concerning upper tail. Conversely, a mean can exceed a chosen bound even if the prior assigns little probability to very high risks.

Worked Example: A Mean-Risk Limit Alone Can Miss Prior Uncertainty

Use the same hypothetical skeleton and prior, but suppose a protocol used only $\bar q_j\le0.16$ as its starting-dose screen. The calculated means show that $d_1$ and $d_2$ pass, while $d_3$ and $d_4$ fail. If the rule selects the highest passing dose, it would start at $d_2$, whose prior predictive risk is approximately 0.1422.

However, one third of the prior probability at $d_2$ is on $a=-0.5$, for which the conditional DLT risk is approximately $0.12^{\exp(-0.5)}=0.276$. Thus a mean-risk screen alone would not flag the fact that the prior assigns probability $1/3$ to a risk above 0.25. If the clinical concern is specifically the chance of a dose having risk above 0.25, the protocol should include an exceedance constraint as well. Neither summary is universally sufficient: the constraints should match the safety question clinicians need answered.

How the Starting Dose Shapes Early CRM Learning

The first dose determines where the initial toxicity evidence is collected. Under the model, a DLT or no DLT at that dose changes the relative support for parameter values, and therefore changes the predicted risks at all doses. Conditional on $a$, outcomes are assumed independent Bernoulli observations with probabilities $p_j(a)$. The dose assignments, binary DLT outcomes within the specified assessment window, fixed skeleton, and prior must be defined for this update to have the stated interpretation.

For observed counts $y_j$ DLTs among $n_j$ participants at dose $d_j$, the likelihood contribution, apart from factors not depending on $a$, is

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

Bayes' rule weights each prior value of $a$ in proportion to its prior probability times this likelihood. The following calculation shows why an early event at a conservative starting dose can still affect the dose recommendation.

Worked Example: One Early DLT Changes the CRM Recommendation

Continue with the four-dose skeleton and equal prior on $a=-0.5,0,0.5$. Suppose, hypothetically, that one participant starts at $d_1$ and experiences a DLT. With one DLT and no other outcomes, the likelihood at each prior value is its conditional risk at $d_1$. The unnormalized posterior weights are therefore $0.1625,0.0500,0.0072$, whose sum is 0.2197. Normalizing gives posterior weights approximately $0.7396,0.2276,0.0328$, respectively.

The posterior mean DLT risk at a dose is the weighted average of its three conditional risks. At $d_1$, this is

(0.7396)(0.1625)+(0.2276)(0.0500)+(0.0328)(0.0072)\approx0.1318.

At $d_2$, the conditional risks are approximately $0.2762,0.1200,0.0303$. The posterior mean is

(0.7396)(0.2762)+(0.2276)(0.1200)+(0.0328)(0.0303)\approx0.2326.

At $d_3$, using conditional risks approximately $0.4315,0.2500,0.1017$, the posterior mean is

(0.7396)(0.4315)+(0.2276)(0.2500)+(0.0328)(0.1017)\approx0.3793.

For target rate $\phi=0.25$, the prior means at $d_2$ and $d_3$ were approximately 0.1422 and 0.2611, so $d_3$ was closer to the target before observing outcomes. After the DLT at $d_1$, the posterior means at those doses are approximately 0.2326 and 0.3793; $d_2$ is now closer to 0.25. This is an illustration of the model-based target-dose comparison, not a complete operational assignment rule. A protocol can impose additional safety restrictions on what dose may be assigned next.

Derivation and Assumptions Behind the Screening Calculation

The prior predictive calculation follows directly from conditioning on the unknown parameter. Let $Y$ indicate whether a future participant assigned to $d_j$ has a DLT. Under the CRM model, $P(Y=1\mid a,d_j)=p_j(a)$. Averaging this conditional probability over the prior gives

P(Y=1\mid d_j)=\int P(Y=1\mid a,d_j)\pi(a)\,da =\int p_j(a)\pi(a)\,da=\bar q_j.

For a discrete prior, the integral becomes the weighted sum used in the examples. This calculation assumes the chosen model and prior adequately represent the uncertainty used for design decisions. It does not establish that the skeleton or prior is scientifically correct.

Several assumptions and checks matter in practice:

The skeleton must be ordered and have probabilities strictly between zero and one. With this power model, increasing doses then have increasing conditional risks for every fixed $a$.
The prior must be proper and specified on the CRM parameter scale being used. Its spread and location influence both the mean risks and exceedance probabilities.
The DLT definition and assessment window must be clinically specified, so a recorded outcome corresponds to the binary endpoint modeled by $p_j(a)$.
The safety thresholds and the rule for choosing among eligible doses must be fixed before outcomes are observed. A computational screen does not replace clinical review of whether the chosen limits are appropriate.

A practical diagnostic is to examine the entire vector of conditional risks over plausible prior values, not only the prior means. Check that risk increases with dose, identify dose-prior combinations that violate clinical limits, and verify that the selected start remains acceptable under reasonable changes to the prior or skeleton. If a small change in prior assumptions moves the start by several levels, that instability deserves review before enrollment. A prior predictive calculation is a design diagnostic, not evidence that the eventual observed toxicity rate will equal $\bar q_j$.

Alternatives, Failure Modes, and Limitations

A fixed lowest-dose start is simple and often highly conservative, but may expose participants to a dose with little prospect of therapeutic activity and can slow learning. Starting at the dose whose skeleton risk is closest to the target uses the target directly but ignores prior uncertainty and other clinical safety constraints. Screening on prior predictive means accounts for parameter uncertainty in an average sense; adding an exceedance-probability limit makes a different, tail-focused concern explicit. These approaches are alternatives with different aims, not interchangeable guarantees of safety.

The principal limitation is that every prior-based screen inherits the model and prior assumptions. If the true dose-toxicity relationship is poorly represented by the skeleton, a precise-looking prior predictive risk can be misleading. These constraints also address toxicity probability, not all dimensions of clinical suitability, such as pharmacology, available nonclinical evidence, or practical administration limits. Those considerations may require a more conservative start than the mathematical screen allows.

Common pitfall: “The prior mean risk at this dose is below the clinical limit, so the dose is safe.” This confuses an average over the prior with a guarantee about the true risk. For example, the mean at $d_2$ in the illustration is below 0.16, yet one third of the prior puts its risk above 0.25. State which safety summary the protocol constrains, inspect the relevant prior tail, and describe the result as a model-based design check rather than a guarantee.

The starting dose also does not determine how many participants must be treated there or exactly what dose may follow an outcome. Those are separate operational decisions that must be specified in the protocol. The role of the starting-dose analysis is narrower: it makes the first assignment defensible under prior uncertainty and shows how its early outcomes can inform the CRM.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In one sentence, distinguish a prior predictive DLT risk from the probability that the dose-specific risk exceeds a clinical threshold.
With equal prior weights on three parameter values, the conditional risks at a dose are 0.18, 0.06, and 0.03. Calculate the prior predictive risk.
Suppose a dose has prior predictive risk 0.12, while the prior probability its risk exceeds a clinical limit is 0.30. Does it pass constraints $\bar q_j\le0.15$ and $P_{\pi}\{p_j(a)>c\}\le0.10$?
Why does a DLT at the initial dose update the predicted risks at other doses as well?
Name one diagnostic that could reveal a fragile starting-dose decision.

Solutions. (1) The prior predictive risk is the prior-averaged probability of a DLT for a future participant; the exceedance probability is the prior probability that the underlying dose-specific risk is above a chosen threshold. (2) $(0.18+0.06+0.03)/3=0.0900$. (3) It passes the mean-risk constraint because $0.12\le0.15$, but fails the exceedance constraint because $0.30>0.10$, so it is ineligible under both requirements. (4) The CRM uses a shared parameter $a$ to determine risks across dose levels; the DLT changes the posterior distribution of $a$, which changes its implied risks at every dose. (5) Recalculate the eligibility set under reasonable alternative priors or skeleton values; a large change in the selected start indicates sensitivity that warrants review.

Dose	Prior predictive risk \(\bar q_j\)	\(P_{\pi}\{p_j(a)>0.25\}\)	Eligible under both limits?
\(d_1\), skeleton 0.05	0.0732	0	Yes
\(d_2\), skeleton 0.12	0.1422	1/3	No
\(d_3\), skeleton 0.25	0.2611	1/3	No
\(d_4\), skeleton 0.35	0.3520	2/3	No