Continual Reassessment Method (CRM): A Worked Example

Introduction

Early-phase clinical trials often face a difficult statistical and clinical problem: which dose should be studied next?

In a conventional dose-escalation study, investigators may use a rule-based algorithm such as the familiar 3+3 design. The next dose is determined from a small set of prespecified rules based on the number of dose-limiting toxicities (DLTs) observed in the current cohort.

The Continual Reassessment Method (CRM) takes a fundamentally different approach. Rather than treating each cohort as an isolated decision, CRM continuously updates a statistical model describing the relationship between dose and toxicity.

After each cohort, the observed toxicity data are combined with the prior distribution to produce updated estimates of the probability of DLT at every candidate dose.

Key idea: CRM does not simply ask whether the current dose produced too many or too few toxicities. It asks: given all of the toxicity data observed so far, which dose is most likely to have a toxicity probability near the target?

The Goal of a Phase I Dose-Finding Study

Suppose the primary objective is to identify a dose whose probability of dose-limiting toxicity is close to a prespecified target. Let:

\theta = P(\text{DLT at the target dose})

For example, suppose the trial defines the target toxicity probability as:

\theta=0.25

The objective is then to identify the dose \(d\) for which:

P(\text{DLT}\mid d) \approx0.25

This dose is often referred to as the maximum tolerated dose (MTD), although the precise definition should be specified in the protocol.

Important distinction: The MTD is not necessarily the highest dose administered. It is a dose associated with a toxicity probability near the target specified by the study design.

Why Use CRM?

The major advantage of CRM is that it uses information from all patients enrolled so far, rather than relying only on the current cohort.

Suppose six doses are being investigated. A traditional rule-based design may make its next-dose decision primarily from the observed DLT count at the current dose.

CRM instead estimates the entire dose-toxicity curve:

p_1,p_2,\ldots,p_K

where \(p_j\) is the probability of DLT at dose level \(j\).

The model is updated after each cohort. Consequently, toxicity information observed at one dose can influence the estimated toxicity probabilities at other doses.

The Basic CRM Structure

A CRM trial consists of several important components:

Specify candidate doses. Define the dose levels that can be administered in the trial.

Specify the target toxicity probability. Choose the DLT probability that represents the desired dose.

Construct a dose-toxicity skeleton. Assign prior toxicity probabilities to the candidate doses.

Specify a statistical model and prior. The model connects the candidate doses and allows the toxicity curve to be updated as data accumulate.

Enroll a cohort. Patients are treated at the currently recommended dose.

Update the posterior distribution. The observed DLT outcomes are incorporated into the Bayesian model.

Select the next dose. The dose whose estimated toxicity is closest to the target is identified, subject to protocol-specified safety restrictions.

Candidate Dose Levels

Suppose a Phase I study evaluates six dose levels:

Dose Level	Dose
1	10 mg
2	20 mg
3	40 mg
4	60 mg
5	80 mg
6	100 mg

The actual dose levels in a clinical trial would normally be determined by pharmacology, preclinical data, prior clinical experience, formulation, exposure considerations, and other scientific information.

The Target Toxicity Probability

Assume the study specifies:

\theta=0.25

Thus, the target is a dose associated with approximately a 25% probability of DLT.

CRM does not require the observed proportion of DLTs at the selected dose to equal exactly 25%. With small sample sizes, the observed proportion will often be substantially different from the model-based estimated probability.

The Dose-toxicity Skeleton

A central component of CRM is the dose-toxicity skeleton. The skeleton consists of prior guesses about the probability of DLT at each dose level.

For this example, suppose the skeleton is:

Dose Level	Dose	Skeleton Probability
1	10 mg	0.05
2	20 mg	0.10
3	40 mg	0.20
4	60 mg	0.30
5	80 mg	0.40
6	100 mg	0.50

The skeleton expresses the investigators' initial belief that toxicity increases with dose.

Important: The skeleton is not the final dose-toxicity estimate. It is the starting structure for the Bayesian model. As clinical data accumulate, the model can move away from these initial values.

A Simple One-Parameter CRM Model

One convenient CRM formulation uses the power model:

$$ p_j(a)=s_j^a $$

where:

\(p_j(a)\) is the modeled probability of DLT at dose \(j\)
\(s_j\) is the skeleton probability at dose \(j\)
\(a\) is an unknown model parameter

If \(a=1\), the modeled probabilities are exactly the skeleton probabilities. For example:

$$ p_1(1)=0.05^1=0.05 $$

$$ p_4(1)=0.30^1=0.30 $$

$$ p_6(1)=0.50^1=0.50 $$

The parameter \(a\) allows the entire toxicity curve to shift as evidence accumulates.

The Bayesian Prior

Because \(a\) is unknown, CRM assigns a prior distribution to it. For illustration, suppose:

\log(a)\sim N(0,0.5^2)

This prior centers \(a\) around 1 on the logarithmic scale, corresponding approximately to the original skeleton.

The exact prior distribution is a design choice and should be justified during trial planning rather than selected solely because it produces a convenient simulation result.

Bayesian Updating

Suppose that at dose \(j\), \(y_j\) DLTs are observed among \(n_j\) treated patients. Conditional on \(a\), the likelihood contribution is:

L_j(a) = {n_j\choose y_j} p_j(a)^{y_j} [1-p_j(a)]^{n_j-y_j}

Across all dose levels, the likelihood is proportional to:

L(a) \propto \prod_{j=1}^{K} p_j(a)^{y_j} [1-p_j(a)]^{n_j-y_j}

The posterior distribution is then:

\pi(a\mid\text{data}) \propto L(a)\pi(a)

where \(\pi(a)\) is the prior distribution.

Estimating Toxicity at Each Dose

Once the posterior distribution of \(a\) has been obtained, the posterior mean toxicity probability at dose \(j\) is:

E[p_j(a)\mid\text{data}] = \int p_j(a)\pi(a\mid\text{data})\,da

For the power model:

E[p_j(a)\mid\text{data}] = \int s_j^a\pi(a\mid\text{data})\,da

CRM then compares these model-based probabilities with the target toxicity probability.

The Dose Selection Rule

The basic CRM dose-selection rule is:

d^* = \arg\min_j \left| \widehat p_j-\theta \right|

where:

\(\widehat p_j\) is the posterior estimate of DLT probability at dose \(j\)
\(\theta\) is the target toxicity probability
\(d^*\) is the dose selected as closest to the target

For example, suppose the posterior estimates are:

Dose	Posterior DLT Probability	Distance from 25%
10 mg	0.06	0.19
20 mg	0.11	0.14
40 mg	0.20	0.05
60 mg	0.29	0.04
80 mg	0.38	0.13
100 mg	0.48	0.23

The dose closest to the target is therefore:

d^*=60\text{ mg}

because \(0.29\) is closer to \(0.25\) than the other estimated toxicity probabilities.

A Complete Worked Example

Now consider a hypothetical Phase I trial with:

Parameter	Planning Value
Number of candidate doses	6
Target DLT probability	25%
Cohort size	3 patients
Starting dose	10 mg
Maximum sample size	15 patients
Model	One-parameter power CRM

The skeleton is:

(0.05,\;0.10,\;0.20,\;0.30,\;0.40,\;0.50)

The target is:

\theta=0.25

Step 1: Treat the First Cohort

The first three patients receive the starting dose:

d_1=10\text{ mg}

Suppose none of the three patients experiences a DLT. Thus:

y_1=0,\qquad n_1=3

The observed DLT rate is therefore:

\frac{0}{3}=0

The Bayesian model incorporates this information and updates the posterior distribution of \(a\).

For illustration, the resulting posterior toxicity estimates might be approximately:

Dose	Posterior DLT Probability
10 mg	0.061
20 mg	0.105
40 mg	0.190
60 mg	0.276
80 mg	0.366
100 mg	0.460

The dose closest to the 25% target is 60 mg.

However, a practical trial may impose an additional rule that prevents skipping untested dose levels.

Safety principle: A statistically estimated dose is not automatically an administrable dose. Clinical CRM implementations commonly incorporate restrictions such as no dose skipping, maximum escalation limits, overdose-control criteria, minimum numbers of patients at a dose, and stopping rules.

Step 2: Escalate to the Next Allowed Dose

Because the 60 mg dose would require skipping several untested levels, suppose the protocol specifies that the next cohort can move only one dose level at a time. The next cohort therefore receives:

d_2=20\text{ mg}

Suppose again that zero of three patients experience a DLT. The cumulative data are now:

Dose	Patients	DLTs
10 mg	3	0
20 mg	3	0
40 mg	0	0
60 mg	0	0
80 mg	0	0
100 mg	0	0

Step 3: Update After Six Patients

The model is refit using all six patients. The resulting posterior toxicity estimates might be approximately:

Dose	Posterior DLT Probability
10 mg	0.043
20 mg	0.080
40 mg	0.157
60 mg	0.239
80 mg	0.327
100 mg	0.423

The model now estimates that 60 mg has a toxicity probability of approximately 24%.

Because:

$$ |0.239-0.25|=0.011 $$

60 mg is extremely close to the target.

Step 4: Treat the 40 mg Dose

Suppose the trial continues with the next allowed dose:

d_3=40\text{ mg}

Suppose one of three patients experiences a DLT. The cumulative data become:

Dose	Patients	DLTs
10 mg	3	0
20 mg	3	0
40 mg	3	1
60 mg	0	0
80 mg	0	0
100 mg	0	0

The posterior estimates might now be approximately:

Dose	Posterior DLT Probability
10 mg	0.059
20 mg	0.106
40 mg	0.197
60 mg	0.289
80 mg	0.382
100 mg	0.479

The model still places the target between 40 mg and 60 mg. The posterior estimate closest to 25% is now approximately 60 mg.

Step 5: Treat 60 mg

The trial therefore moves to 60 mg. Suppose one of the next three patients experiences a DLT. The cumulative data are:

Dose	Patients	DLTs
10 mg	3	0
20 mg	3	0
40 mg	3	1
60 mg	3	1
80 mg	0	0
100 mg	0	0

The posterior toxicity estimates might now be approximately:

Dose	Posterior DLT Probability	Distance from Target
10 mg	0.061	0.189
20 mg	0.110	0.140
40 mg	0.204	0.046
60 mg	0.298	0.048
80 mg	0.393	0.143
100 mg	0.490	0.240

The model now places the target approximately between 40 mg and 60 mg. Because 40 mg has an estimated toxicity of 20.4% and 60 mg has an estimated toxicity of 29.8%, either could be very close depending on the exact dose selection rule and safety restrictions.

The key point is that CRM is using the entire accumulated dataset rather than making a decision based only on the most recent cohort.

CRM's Continual Updating Process

The essential CRM loop can now be summarized:

Treat a cohort at the current dose.

Observe DLT outcomes after the relevant assessment window.

Update the posterior distribution of the model parameter.

Calculate posterior toxicity probabilities at all candidate doses.

Identify the dose closest to the target toxicity probability.

Apply protocol-defined safety restrictions.

Treat the next cohort and repeat the process.

Why CRM Is Called "Continual Reassessment"

The term continual refers to the repeated updating of the dose-toxicity model as new patients become available.

The design does not fit a model once and then leave it unchanged. Instead:

\text{Prior} \rightarrow \text{Data} \rightarrow \text{Posterior} \rightarrow \text{Next Dose} \rightarrow \text{New Data} \rightarrow \text{Updated Posterior}

This cycle continues throughout the dose-escalation portion of the trial.

CRM vs. 3+3 Design

CRM is often compared with the traditional 3+3 design. The two approaches differ fundamentally in how dose decisions are generated.

Feature	3+3	CRM
Statistical model	No explicit dose-toxicity model	Explicit dose-toxicity model
Bayesian updating	No	Yes
Uses information across doses	Limited	Yes
Dose selection	Rule based	Model based
Target toxicity	Implicit	Explicit
Adaptation	Discrete rules	Continuous model updating
Statistical efficiency	Generally lower	Potentially higher
Implementation complexity	Low	Higher

Important: CRM is not simply a more complicated version of 3+3. It represents a different statistical philosophy: rather than using a fixed decision table, CRM explicitly models the dose-toxicity relationship and updates that model as evidence accumulates.

Why Safety Restrictions Matter

A purely mathematical CRM algorithm could recommend a dose that would be clinically inappropriate to administer immediately. For example, after observing no DLTs at a low dose, a model might estimate that a substantially higher dose is close to the target.

A protocol may nevertheless prohibit jumping over intermediate dose levels.

Common safety modifications can include:

No dose skipping: do not jump over untested doses.
Maximum escalation: limit the number of dose levels that can be crossed between cohorts.
Overdose control: avoid a dose when the posterior probability that its toxicity exceeds an unacceptable threshold is too high.
Cohort-size requirements: require sufficient information before a dose can be declared the recommended dose.
Stopping rules: stop the trial when the target dose cannot be identified with sufficient confidence or when safety criteria are violated.

Overdose Control

One important refinement is to consider not only which dose is closest to the target, but also the probability that a dose is excessively toxic. Suppose:

\theta=0.25

and the trial defines an overdose threshold of:

\theta_{\mathrm{OD}}=0.40

A candidate dose might be excluded if:

P(p_j>\theta_{\mathrm{OD}}\mid\text{data}) > \gamma

where \(\gamma\) is a prespecified probability threshold.

This creates an important distinction between:

Targeting: Which dose has estimated toxicity closest to the desired target?

Safety: Is the dose sufficiently unlikely to have an unacceptable toxicity probability?

A practical CRM implementation must consider both.

Final Recommended Dose

At the conclusion of the dose-finding portion of the study, the final recommended dose is generally selected using the same model-based principle:

d_{\mathrm{rec}} = \arg\min_j \left| \widehat p_j-\theta \right|

subject to the protocol's safety and eligibility criteria.

The selected dose should therefore be interpreted as a model-based recommendation, not simply as the dose at which the largest number of DLTs occurred.

CRM Does Not Mean "Always Escalate"

A common misconception is that CRM is simply an aggressive escalation algorithm. It is not.

CRM can recommend:

escalation to a higher dose,
remaining at the same dose,
de-escalation to a lower dose, or
stopping the trial.

The decision depends on the updated posterior dose-toxicity curve and the safety restrictions defined in the protocol.

CRM Operating Characteristics

Because CRM is adaptive, its performance is typically evaluated using simulation rather than a single closed-form calculation. Important operating characteristics include:

Operating Characteristic	Question
Probability of selecting the true MTD	How often does the design identify the correct dose?
Probability of overdosing	How often are patients assigned to excessively toxic doses?
Average number of patients	How many patients are treated on average?
Probability of stopping	How often does the design stop without selecting a dose?
DLT rate	What proportion of treated patients experience DLTs?
Allocation by dose	How are patients distributed across dose levels?

Simulation-Based Evaluation

Suppose the true toxicity probabilities are:

(0.05,\;0.10,\;0.20,\;0.25,\;0.40,\;0.55)

If the target is 25%, then dose 4 is the true target dose. A CRM design can be simulated repeatedly under this scenario. For each simulated trial:

Generate DLT outcomes according to the true dose-toxicity curve.

Run the CRM algorithm cohort by cohort.

Record the final recommended dose.

Record the number of patients treated at each dose.

Record the total number of DLTs and any overdose events.

Repeat thousands of times to estimate operating characteristics.

A Simple Simulation Skeleton

A conceptual simulation algorithm looks like this:

for each simulated trial:

    initialize CRM model
    initialize patient outcomes
    assign starting dose

    while trial is not stopped:

        enroll next cohort
        observe DLT outcomes

        update Bayesian posterior

        estimate toxicity probability
        at every candidate dose

        identify dose closest
        to target toxicity

        apply safety restrictions

        assign next allowable dose

    record final recommended dose
    record DLTs and dose assignments

Important Design Choices

CRM is a framework rather than a single universal algorithm. The final design depends on several choices:

Design Choice	Examples
Target toxicity	20%, 25%, 30%, etc.
Skeleton	Prior DLT probabilities by dose
Model	Power model, logistic model, or another monotonic model
Prior	Prior distribution for model parameters
Cohort size	1, 2, 3, or more patients
Escalation restrictions	No skipping, maximum escalation, etc.
Overdose control	Posterior probability criteria
Stopping rules	Safety, precision, sample-size, or posterior criteria

Strengths of CRM

Uses a formal dose-toxicity model.
Incorporates information across dose levels.
Updates continuously as new data become available.
Explicitly targets a prespecified toxicity probability.
Can potentially allocate more patients near the target dose.
Can be evaluated using detailed simulation.

Limitations of CRM

It requires substantially more statistical planning than simple rule-based designs.
The results can depend on the dose skeleton and model specification.
Model misspecification can affect dose selection.
Safety restrictions must be carefully incorporated.
The design can be more difficult to explain operationally to clinical teams.
Simulation is generally necessary to understand the design's operating characteristics.

CRM vs. Bayesian Optimal Interval Designs

CRM belongs to the broader family of model-based dose-finding designs. Other approaches, such as Bayesian interval designs, use different statistical decision rules.

The fundamental distinction is that CRM explicitly estimates a dose-toxicity curve and selects the dose according to that model, whereas interval-based methods generally classify the observed toxicity probability relative to prespecified intervals around the target.

CRM vs. Traditional 3+3: The Conceptual Difference

The difference can be summarized with one question.

3+3 asks: "What does the decision table say we should do after observing this number of DLTs?"

CRM asks: "Given all available data, what does the posterior dose-toxicity model say about the probability of DLT at each dose?"

That difference is the central reason CRM can extract more information from a small dose-finding dataset.

Common Misconceptions

Misconception 1: CRM always chooses the dose with the highest observed DLT rate.

False. CRM chooses a dose based on the model-estimated toxicity probability, not simply the observed DLT proportion.

Misconception 2: CRM ignores clinical safety constraints.

False. A well-designed CRM protocol incorporates explicit restrictions governing which model-recommended doses may actually be administered.

Misconception 3: CRM guarantees identification of the true MTD.

False. CRM is a statistical design intended to improve dose-finding performance; it does not guarantee that the selected dose is the true target dose.

Misconception 4: The skeleton must be exactly correct.

False. The skeleton represents prior information. The model updates its estimates using observed clinical data. However, a poorly chosen skeleton or model can still adversely affect performance, especially in small samples.

Final Summary

The Continual Reassessment Method is a Bayesian model-based approach to dose-finding studies. The central workflow is:

\text{Skeleton} \rightarrow \text{Prior} \rightarrow \text{Patient Data} \rightarrow \text{Posterior} \rightarrow \text{Dose Selection} \rightarrow \text{New Patient Data}

The process repeats throughout the trial.

At each reassessment, CRM estimates the probability of DLT at each candidate dose and identifies the dose whose estimated toxicity is closest to the target.

The fundamental decision rule is:

d^* = \arg\min_j \left| \widehat p_j-\theta \right|

The statistical recommendation is then filtered through prespecified safety rules.

Bottom line: CRM replaces a rigid dose-escalation table with a continuously updated Bayesian model of dose and toxicity. Its major advantage is that every cohort contributes information to the estimated dose-toxicity relationship, allowing the trial to adapt as evidence accumulates.

Key Takeaways

CRM is primarily used for model-based dose finding in early-phase clinical trials.
The design specifies a target probability of dose-limiting toxicity.
A dose-toxicity skeleton provides the initial structure of the model.
Observed DLT data are combined with a prior distribution to obtain a posterior distribution.
Posterior toxicity probabilities are estimated for every candidate dose.
The dose closest to the target is generally selected, subject to safety constraints.
CRM uses information from the entire accumulated dataset rather than only the current cohort.
Simulation is important for evaluating selection probability, overdose probability, sample size, and other operating characteristics.

References

O'Quigley, J., Pepe, M., & Fisher, L. (1990). Continual reassessment method: a practical design for phase I clinical trials. Biometrics, 46(1), 33–48.

O'Quigley, J. & Chevret, S. (1991). Methods for dose finding studies. Statistics in Medicine.

Goodman, S.N., Zahurak, M.L., & Piantadosi, S. (1995). Some practical improvements in the continual reassessment method for phase I studies. Statistics in Medicine, 14, 1149–1161.

Neuenschwander, B., Branson, M., & Gsponer, T. (2008). Critical aspects of the Bayesian approach to phase I cancer trials. Statistics in Medicine, 27, 2420–2439.

Chevret, S. (1993). The continual reassessment method in cancer phase I clinical trials: a simulation study. Statistics in Medicine.