Dose Expansion Cohorts in Oncology Trials

Introduction

Modern oncology development frequently begins with a dose-escalation study designed to characterize safety, tolerability, pharmacokinetics, pharmacodynamics, and an appropriate dose or dose range for further development.

Once a dose has been identified as sufficiently suitable for additional evaluation, investigators may enroll substantially more patients at one or more selected doses.

These groups are commonly called dose expansion cohorts.

The purpose of expansion is fundamentally different from simply continuing dose escalation.

Dose escalation primarily asks:

Dose expansion asks a broader set of questions:

The larger cohort can provide additional information about the safety profile, antitumor activity, pharmacokinetics, pharmacodynamics, biomarkers, and feasibility of further development.

Where Dose Expansion Fits in an Oncology Trial

A simplified early oncology development program can be represented as:

The actual sequence can be more complicated. Some modern oncology studies use expansion cohorts before formal completion of escalation, and some protocols evaluate multiple doses or schedules in parallel.

Why Expand a Cohort?

A small dose-escalation cohort is usually not large enough to characterize every clinically important property of a treatment.

For example, a dose-escalation study might identify an appropriate dose after only a few patients have been treated at that dose.

That information may be sufficient to support additional evaluation, but it does not provide a highly precise estimate of the response rate or adequately characterize less common adverse events.

Expansion can therefore provide additional information in several domains.

Expansion Is Not Necessarily Confirmatory

One of the most important distinctions is between an expansion cohort and a confirmatory trial.

An expansion cohort may provide substantial evidence about activity and safety, but it is commonly conducted in a relatively small, selected population and may be single-arm.

A later confirmatory trial may instead require a randomized comparison, prespecified estimands, a larger sample size, and a different evidentiary standard.

Types of Dose Expansion Cohorts

Expansion cohorts can be organized in many ways.

The statistical design should follow the scientific question rather than the label attached to the cohort.

Dose Escalation Versus Dose Expansion

The Statistical Question Must Be Defined First

Before choosing the number of patients in an expansion cohort, define the question the cohort is intended to answer.

For example:

• Estimate the objective response rate with a desired confidence interval width.
• Test whether the response rate exceeds a clinically uninteresting value.
• Characterize the safety profile at the selected dose.
• Estimate the incidence of a particular adverse event.
• Evaluate activity in a biomarker-defined population.
• Characterize PK variability.
• Evaluate exposure-response relationships.
• Generate evidence supporting a subsequent randomized study.

These questions do not imply the same sample-size calculation.

Expansion Cohort Sample Size: Three Common Frameworks

Three broad approaches are particularly useful.

Precision-Based Expansion Cohorts

Suppose the primary objective is to estimate the objective response rate rather than formally test a null hypothesis.

Let:

and let:

where \(X\) is the number of responders among \(n\) evaluable patients.

For a large-sample approximation, the standard error is:

An approximate two-sided confidence interval has half-width:

Solving for \(n\):

This formula is useful for intuition, although exact or score-based methods are generally preferable for small oncology cohorts.

Worked Precision Example

Suppose investigators anticipate an objective response rate of approximately 30% and want an approximate 95% confidence interval with a half-width of 10 percentage points.

Using:

gives:

or approximately:

This illustrates an important point: precise estimation can require substantially more patients than a screening-oriented expansion cohort.

Exact Confidence Intervals Are Often Preferable

For a binary response endpoint, the observed response rate is:

The familiar Wald interval:

can perform poorly in small samples or when the response rate is near 0 or 1.

For small expansion cohorts, investigators commonly consider exact binomial or score-based confidence intervals instead.

Hypothesis-Testing Expansion Cohorts

Some expansion cohorts are designed as formal single-arm efficacy studies.

Suppose:

where:

• \(p_0\) is the clinically uninteresting response rate.
• \(p_1\) is the response rate considered sufficiently promising.

The sample size can then be chosen to achieve a specified one-sided type I error and power.

This leads naturally to designs such as:

• Single-stage binomial designs
• Simon two-stage designs
• Fleming-style multi-stage designs
• Other adaptive or group-sequential approaches

Why the Expansion Cohort May Use a Two-Stage Design

An expansion cohort does not necessarily need to enroll its full planned sample size.

If early patients demonstrate very little activity, continuing to enroll may not be scientifically useful.

A two-stage design can therefore allow:

while maintaining prespecified operating characteristics.

For example, an expansion cohort might enroll 15 patients initially and then add another 15 only if a minimum number of responses is observed.

Worked Example: A 30-Patient Expansion Cohort

Consider an oncology drug for which dose escalation has identified a selected dose for further evaluation.

Investigators open a disease-specific expansion cohort in patients with a defined advanced solid tumor population.

Suppose the expansion cohort has the following planning assumptions:

The cohort is intended primarily to determine whether the selected dose shows sufficient antitumor activity to justify further development.

Stage 1 of the Expansion Cohort

The first 15 evaluable patients are assessed for objective response.

Suppose the protocol specifies that the cohort will continue only if at least three responses are observed among the first 15 patients.

The rule is:

and:

This is a simplified illustrative stopping rule. The actual boundary should be selected through formal operating-characteristic calculations rather than chosen arbitrarily.

Why an Interim Rule Can Be Useful

Suppose the treatment is truly inactive and the response probability is only 15%.

The expected number of responses among 15 patients is:

Thus, observing zero, one, or two responses would be quite compatible with a poorly active treatment.

The interim rule can prevent the remaining patients from being exposed to a treatment that is showing little evidence of activity.

Stage 2 of the Expansion Cohort

If at least three responses are observed among the first 15 patients, another 15 patients are enrolled.

The maximum evaluable sample size is:

At the end of the cohort, the total number of responses is:

where:

and, conditional on continuation:

under the simplifying assumption of independent binary outcomes with a common response probability.

A Final Decision Rule

Suppose the final rule is:

while:

Again, the numerical boundaries here are illustrative. A real protocol should derive them from the desired type I error, power, and clinical decision framework.

Interpreting the Final Response Rate

Suppose all 30 patients are evaluable and nine responses are observed.

The observed response rate is:

or:

An exact two-sided 95% binomial confidence interval for nine responses among 30 patients is approximately:

The interval is wide because the cohort is relatively small.

What Does a Positive Expansion Cohort Mean?

A positive expansion cohort generally means that the prespecified decision criterion has been met.

Depending on the study objective, this may support:

• Further investigation of the dose and disease population
• A randomized development program
• Additional biomarker-defined cohorts
• Additional combination studies
• Further dose or schedule evaluation
• Regulatory discussions

It does not necessarily mean that the treatment has demonstrated definitive clinical benefit.

Safety in Dose Expansion

Safety remains a major purpose of dose expansion.

The larger cohort provides more opportunity to characterize:

• Common adverse events
• Grade 3 or higher adverse events
• Serious adverse events
• Adverse events leading to dose interruption
• Adverse events leading to dose reduction
• Adverse events leading to treatment discontinuation
• Treatment-related adverse events
• Delayed or cumulative toxicities

The expansion cohort can therefore complement the limited safety information available from the dose-escalation phase.

Rare Safety Events and Sample Size

Small expansion cohorts are still limited in their ability to detect rare events.

Suppose an adverse event has true probability \(q\).

The probability of observing no events among \(n\) patients is:

For example, if the true event probability is 5%, then among 30 patients:

Thus, there is still approximately a 21% probability of observing zero events even when the true event probability is 5%.

Probability of Observing at Least One Event

The probability of observing at least one event is:

For \(q=0.05\) and \(n=30\):

Therefore, a 30-patient cohort has approximately a 78.5% chance of observing at least one event if the true event probability is 5%.

This simple calculation is useful for explaining why expansion cohorts can improve safety characterization while still being inadequate for very rare toxicities.

Pharmacokinetics in Expansion Cohorts

Dose expansion may also provide substantially more PK information than the dose-escalation phase.

Common PK parameters include:

• Maximum observed concentration (\(C_{\max}\))
• Area under the concentration-time curve (\(AUC\))
• Time to maximum concentration (\(T_{\max}\))
• Half-life (\(t_{1/2}\))
• Trough concentration (\(C_{\text{trough}}\))
• Accumulation ratio
• Clearance and volume parameters when estimable

The expansion population may also differ from the escalation population in body size, organ function, disease burden, concomitant medications, and other characteristics.

Therefore, expansion PK data can help determine whether exposure remains consistent when the treatment is administered to a broader patient population.

Pharmacodynamic Evaluation

If the drug has a measurable pharmacodynamic biomarker, expansion cohorts may provide additional evidence that the selected dose produces the intended biological effect.

For example, investigators might examine:

• Target occupancy
• Receptor inhibition
• Downstream pathway modulation
• Circulating biomarker changes
• Tumor biomarker changes
• Immune activation markers

This can help connect dose, exposure, biological effect, and clinical activity.

The expansion cohort can therefore contribute to a broader exposure-response framework rather than simply generating a single response-rate estimate.

Biomarker-Selected Expansion Cohorts

A particularly important use of expansion cohorts is evaluation of a biomarker-defined population.

Suppose an investigational drug targets a molecular alteration present in a subset of patients.

The expansion strategy might therefore enroll:

Each cohort may have a different expected response rate and therefore a different statistical rationale.

Multiple Expansion Cohorts

A single protocol may contain several expansion cohorts.

This structure creates an important statistical issue: each cohort may represent a separate scientific question.

The analysis should therefore make clear which conclusions are confirmatory, which are exploratory, and how multiplicity is being handled when formal hypothesis testing is performed across multiple cohorts.

Multiplicity Across Expansion Cohorts

Suppose five independent expansion cohorts each test:

at a nominal one-sided significance level of 5%.

If all five hypotheses are tested without multiplicity adjustment, the probability of at least one false positive can exceed 5%.

Under an independence approximation:

For \(m=5\) and \(\alpha=0.05\):

or approximately 22.6%.

Independence Is Often Not a Safe Assumption

The calculation above assumes independence for illustration.

In an actual oncology program, cohorts may share:

• The same investigators
• The same dose
• The same control-free historical benchmark
• The same biomarker assay
• The same patient-selection process
• The same treatment mechanism

Therefore, the joint statistical behavior of multiple cohorts may be more complicated than a simple independent-test calculation suggests.

Historical Controls

Single-arm expansion cohorts are frequently interpreted relative to historical response rates.

For example:

might represent an estimated response rate under available standard therapy or an uninteresting benchmark.

The choice of \(p_0\) is therefore critical.

Eligibility Criteria Affect the Expansion Cohort

The more narrowly an expansion cohort is defined, the more homogeneous the population may become.

For example, an expansion cohort may require:

• A specific tumor type
• A specific molecular alteration
• A specified line of therapy
• Measurable disease
• A minimum performance status
• Prespecified organ-function criteria

These restrictions can increase the relevance of the cohort to a particular clinical question but may also reduce generalizability.

Analysis Populations

The statistical analysis plan should clearly distinguish several populations.

The appropriate denominator depends on the endpoint and the prespecified analysis rules.

Response Assessment and Missing Data

Objective response endpoints require clearly defined assessment rules.

Important considerations include:

• Assessment criteria
• Timing of tumor assessments
• Confirmation requirements
• Handling of stable disease
• Progressive disease definitions
• Missing assessments
• Patients who discontinue treatment before assessment

The statistical analysis plan should specify how these situations are handled before the data are examined.

Unevaluable Patients

Suppose 30 patients are enrolled but only 27 have an evaluable response assessment.

If eight responses are observed, several quantities could be reported:

or:

These are different estimands and can lead to different conclusions.

Therefore, the protocol and statistical analysis plan must define the primary estimand and analysis population rather than leaving the denominator to be chosen after observing the data.

Interim Monitoring in Expansion Cohorts

Expansion cohorts may include interim analyses for:

• Futility
• Excess toxicity
• Strong efficacy
• Recruitment feasibility
• Biomarker prevalence

Each interim rule can affect the operating characteristics of the study.

For efficacy stopping, repeated unadjusted hypothesis testing can inflate the type I error.

For this reason, interim boundaries should be incorporated into the design rather than added informally after the study begins.

Safety Stopping Rules

Safety monitoring can be based on the probability of observing adverse events.

For example, if \(q\) denotes the true probability of a specific toxicity, the probability of observing at least one event among \(n\) patients is:

A protocol might therefore specify that enrollment or treatment should be reviewed if the observed toxicity rate becomes sufficiently high.

The exact rule depends on:

• Severity of the event
• Clinical reversibility
• Expected background incidence
• Exposure duration
• Mechanism of the treatment
• Alternative treatment options

Efficacy and Safety Boundaries Serve Different Purposes

These boundaries should not be treated as interchangeable.

A Bayesian Perspective

Some expansion programs use Bayesian methods to combine prior information with the observed data.

For a response probability \(p\), one simple prior is:

If \(x\) responses are observed among \(n\) patients, the posterior distribution under a beta-binomial model is:

The posterior probability:

can then be used as part of a prespecified decision rule.

Dose Expansion Does Not Always Mean One Dose

Although the term suggests expansion at a single dose, some programs evaluate more than one dose or regimen.

For example:

If multiple doses are being compared, the statistical problem becomes more complex because dose is now a treatment factor rather than merely a fixed design feature.

Randomized Dose Comparison

If the purpose of expansion is to compare two doses, randomization may be appropriate.

For example:

This is fundamentally different from comparing each dose against a historical benchmark.

A randomized dose comparison can reduce confounding from differences in patient characteristics and provides a direct estimate of the relative treatment effect.

Exposure-Response in Expansion

When individual-level PK data are available, investigators may evaluate whether clinical activity or toxicity changes with exposure.

Conceptually:

and:

Depending on the data and objective, these models may involve logistic regression, nonlinear mixed-effects modeling, time-to-event models, or other methods.

Exposure-Response Requires Care

Observed exposure is not randomly assigned.

Patients with different clearance, body size, organ function, disease status, or treatment adherence may have different exposures.

Consequently, an apparent association between exposure and response may reflect confounding.

Sample Size for Safety Characterization

Sample size can also be considered from the perspective of the probability of observing an event.

Suppose investigators want at least a 90% probability of observing at least one event if the true incidence is \(q\).

Then:

Rearranging:

and therefore:

For a 10% event rate:

so approximately 22 patients are needed to have at least a 90% probability of observing one event if the true incidence is 10%.

This is a useful calculation, but it does not characterize the event rate precisely and does not guarantee adequate evaluation of severe or rare toxicities.

Response Rate Versus Duration of Response

An expansion cohort should not necessarily focus only on response rate.

Other efficacy endpoints may include:

• Duration of response
• Progression-free survival
• Time to response
• Overall survival
• Disease control rate

These endpoints have different statistical properties.

For example, duration of response is a time-to-event endpoint and should not generally be analyzed as a simple binary proportion.

Time-to-Event Endpoints

If \(T\) denotes a time-to-event outcome, investigators may estimate the survival function:

The Kaplan-Meier estimator can then be used to summarize the observed time-to-event distribution while accounting for censoring.

This is particularly important in expansion cohorts where follow-up duration may vary substantially across patients.

Enrollment Versus Evaluability

A major operational issue is the difference between:

and:

Suppose the design requires 30 evaluable patients.

If approximately 10% of patients are expected to be nonevaluable, a simple inflation calculation is:

Thus approximately 34 patients might need to be enrolled to obtain 30 evaluable patients, depending on the prespecified definition of evaluability and replacement policy.

Replacement Patients

Replacement rules are particularly important in small expansion cohorts.

For example, suppose a patient withdraws before the first post-baseline response assessment.

The protocol may specify whether:

• The patient remains in the denominator.
• The patient is considered nonevaluable for a specific endpoint.
• The patient can be replaced.
• The patient is included in a separate sensitivity analysis.

Replacement can affect the operating characteristics of a formal two-stage design.

A More Complete Worked Expansion Example

Consider an investigational targeted therapy for a biomarker-positive solid tumor.

Dose escalation has evaluated several dose levels and identified a selected dose for expansion.

The expansion cohort is designed to assess preliminary antitumor activity while continuing to characterize safety and PK.

Stage 1 Decision

After 15 evaluable patients, suppose four objective responses are observed.

The observed response rate is:

Because four responses exceed the illustrative continuation threshold of three, the cohort proceeds to Stage 2.

Another 15 evaluable patients are enrolled.

Final Analysis

Suppose the second stage produces five additional responses.

The total number of responses is:

and the total evaluable sample size is:

Therefore:

Under the illustrative final criterion of at least eight responses, the cohort would meet the predefined activity criterion.

The statistical report should then include the confidence interval, patient disposition, response assessment details, safety findings, and relevant supportive PK/PD and biomarker results.

Do Not Interpret the Point Estimate in Isolation

The result:

looks encouraging relative to the 15% benchmark.

However, the exact 95% confidence interval is approximately:

The lower bound is close to the historical benchmark.

Therefore, the appropriate interpretation is not simply:

"The treatment has a 30% response rate."

A more statistically appropriate interpretation is that the observed response rate was 30%, with considerable uncertainty because only 30 patients were evaluated.

Interim Efficacy Stopping

Some expansion cohorts permit early stopping for strong activity.

For example, suppose an initial cohort of 15 patients produces nine responses.

The observed response rate would be:

That may be sufficiently compelling to justify an early development decision under a prespecified rule.

However, an early efficacy rule should not be introduced merely because the observed response rate looks impressive.

Repeatedly checking accumulating response data without accounting for the monitoring strategy can change the statistical properties of the procedure.

Futility Stopping

The opposite situation occurs when activity is very low.

Suppose the first 15 patients produce zero responses.

The observed response rate is:

If the protocol specifies a futility boundary that includes zero responses, the cohort can close without enrolling additional patients.

This can conserve:

• Patient exposure
• Drug supply
• Study resources
• Recruitment capacity
• Time required for development decisions

Stopping for Safety Can Override Efficacy

Suppose a cohort produces encouraging responses but also develops an unexpected serious toxicity.

The fact that the efficacy boundary has been crossed does not necessarily require continued enrollment.

A protocol can specify that safety review may halt treatment or enrollment independently of the efficacy results.

This is one reason why expansion cohorts should be viewed as integrated clinical decision frameworks rather than isolated statistical tests.

Decision Rules Should Be Hierarchical When Appropriate

A protocol may specify a sequence such as:

Bayesian Predictive Probability for Expansion Decisions

Another approach is to calculate the probability that the final cohort will meet its efficacy criterion given the interim data.

Let \(X_1=x_1\) be the observed interim response count.

A predictive probability can be expressed conceptually as:

This probability incorporates uncertainty about future responses.

A protocol might specify:

• Stop for futility if predictive probability is below a lower threshold.
• Continue if predictive probability lies in an intermediate range.
• Stop early for success if predictive probability exceeds an upper threshold, when such early success stopping is scientifically appropriate.

The exact thresholds should be determined through simulation or analytical operating-characteristic calculations.

Simulation Is Often Essential

Complex expansion designs may include:

• Multiple cohorts
• Multiple doses
• Biomarker selection
• Interim efficacy decisions
• Safety stopping
• Adaptive enrollment
• Unevaluable patients
• Changing cohort sizes
• Borrowing of information across cohorts

In these situations, closed-form calculations may not be sufficient.

Simulation can estimate:

• Type I error
• Power
• Probability of early stopping
• Expected sample size
• Probability of selecting the correct dose
• Probability of selecting the correct cohort
• Probability of falsely expanding an inactive treatment
• Probability of failing to expand an active treatment

Operating Characteristics for an Expansion Cohort

A useful design report may include:

Expected Sample Size

For a simple two-stage design with initial sample size \(n_1\) and additional sample size \(n_2\), the expected sample size is:

If continuation requires \(X_1\ge c\), then:

and:

This quantity can be calculated under the null, alternative, and intermediate response rates.

Expected Sample Size Under the Null

If the treatment is ineffective, a good futility design should result in a relatively high probability of early termination.

Therefore:

can be substantially smaller than the maximum sample size.

This is one of the main efficiency benefits of a staged expansion design.

Expected Sample Size Under the Alternative

If the treatment is active, more patients are likely to satisfy the interim continuation criterion.

Therefore:

will generally be closer to the maximum sample size when early futility stopping is the primary adaptive mechanism.

If early efficacy stopping is also permitted, expected sample size can decrease again when the treatment is extremely active.

Dose Expansion and the Recommended Dose

One important misconception is that the largest tolerated dose is necessarily the correct expansion dose.

Dose selection may consider:

• Safety
• Tolerability
• PK exposure
• PK variability
• Pharmacodynamic effect
• Exposure-response relationships
• Exposure-toxicity relationships
• Feasibility of chronic dosing
• Emerging efficacy

Thus, dose expansion is part of a broader dose-selection process rather than simply a consequence of finding the maximum tolerated dose.

Recommended Phase II Dose Is Not a Purely Statistical Quantity

The selected expansion dose may be informed by statistical analyses, but the decision itself typically integrates multiple sources of evidence.

For example:

This is fundamentally different from selecting a dose solely because it has the largest observed response rate.

Common Statistical Mistakes

1. Calling every larger cohort a dose expansion. The cohort should have a defined scientific purpose.
2. Using an arbitrary sample size. The sample size should be justified by the cohort's objective.
3. Treating an expansion cohort as confirmatory by default. Many expansion cohorts are exploratory or screening-oriented.
4. Ignoring the historical benchmark. The clinical relevance of a response rate depends on the appropriate comparator or benchmark.
5. Using the observed response rate as the only evidence. Confidence intervals and patient-level disposition are essential.
6. Repeatedly checking efficacy without accounting for interim monitoring. This can alter type I error.
7. Ignoring safety stopping rules. Efficacy should not be considered independently of patient safety.
8. Changing the denominator after seeing the data. The analysis population should be prespecified.
9. Ignoring multiplicity across multiple formal cohorts. If several cohorts support formal claims, the inferential framework should address the resulting multiplicity.
10. Assuming a small cohort can characterize rare safety events. Small samples provide limited information about uncommon toxicities.
11. Confusing dose expansion with dose confirmation. Expansion may provide additional evidence about the dose, but the exact development role depends on the protocol.
12. Ignoring biomarker prevalence. A theoretically attractive biomarker cohort may be operationally difficult if the biomarker is rare.

Biomarker Prevalence and Recruitment

Suppose a biomarker is present in only 10% of screened patients and the expansion cohort requires 30 biomarker-positive patients.

If screening were independent and the prevalence were exactly 10%, the expected number of screened patients required would be:

In practice, screening failures, assay failures, competing trials, changing prevalence, and site-level differences can make recruitment substantially more complicated.

Multiplicity From Biomarker Subgroups

Suppose investigators examine responses in several biomarker-defined subgroups.

Even if only one subgroup is the formal primary population, exploratory subgroup analyses can generate many apparent differences by chance.

Therefore, the statistical analysis plan should distinguish:

• Primary biomarker hypothesis
• Key secondary biomarker hypotheses
• Exploratory biomarker analyses

This prevents exploratory findings from being inadvertently interpreted as prespecified confirmatory evidence.

Data Monitoring and Cohort Governance

Expansion cohorts often involve operational decisions that occur before the final database lock.

The protocol should define:

• Who reviews interim safety data
• Who reviews interim efficacy data
• What data must be mature before a decision
• How response assessments are adjudicated
• How dose changes are handled
• How cohort opening and closure are documented

The statistical design should align with the clinical governance structure.

Data Maturity Matters

A response-rate analysis can be performed only when the relevant response assessments are sufficiently mature.

Similarly, duration of response and progression-free survival require adequate follow-up.

Consequently, the phrase "30 patients enrolled" does not necessarily mean that the expansion cohort is ready for its final analysis.

Expansion Cohorts and Randomized Phase II Studies

A dose expansion cohort may eventually transition into a randomized study.

For example:

The expansion cohort therefore frequently serves as a bridge between early dose-finding and later comparative development.

How the Expansion Cohort Informs the Next Trial

The expansion data can inform:

• Selected dose and schedule
• Target patient population
• Biomarker strategy
• Expected response rate
• Expected duration of response
• Safety assumptions
• PK exposure assumptions
• Stratification factors
• Eligibility criteria
• Sample-size assumptions for later trials

However, estimates from small expansion cohorts should be treated as uncertain when used for planning larger studies.

Using Expansion Data for a Later Sample-Size Calculation

Suppose an expansion cohort observes:

It may be tempting to use 30% as the exact expected response rate in a later trial.

This can be risky because the observed rate is itself uncertain.

A more robust planning process may consider:

• The point estimate
• The confidence interval
• Historical data
• External evidence
• Clinical judgment
• Potential differences between the expansion and future-study populations

R Implementation: Exact Response-Rate Confidence Interval

For the worked example with nine responses among 30 patients, R can calculate an exact binomial confidence interval.

x <- 9
n <- 30

binom.test(
  x = x,
  n = n,
  conf.level = 0.95
)

The resulting estimate is:

9 / 30
# 0.30

with an exact 95% confidence interval of approximately:

0.147
0.494

or approximately 14.7% to 49.4%.

R Implementation: Probability of Early Futility

Suppose the interim cohort contains 15 patients and continuation requires at least three responses.

n1 <- 15
c <- 3

p0 <- 0.15

prob_futility <- pbinom(
  c - 1,
  size = n1,
  prob = p0
)

prob_futility

This calculates:

which is the probability of stopping for futility under the assumed null response rate.

R Implementation: Probability of Continuing

prob_continue <- 1 - pbinom(
  c - 1,
  size = n1,
  prob = p0
)

prob_continue

This is:

and it determines how frequently the second stage is expected to open under the null hypothesis.

R Implementation: Expected Sample Size

Suppose the second stage adds another 15 evaluable patients.

n2 <- 15

expected_n <- function(p) {

  continue <- 1 - pbinom(
    c - 1,
    size = n1,
    prob = p
  )

  n1 + continue * n2
}

expected_n(0.15)
expected_n(0.30)

The function evaluates the expected number of evaluable patients at any assumed response probability.

R Implementation: Operating Characteristics

For a more formal two-stage design, define the probability of final success as a function of the true response rate.

prob_success <- function(
  p,
  n1,
  n2,
  continue_boundary,
  final_boundary
) {

  result <- 0

  for (x1 in continue_boundary:(n1)) {

    p_x1 <- dbinom(
      x1,
      size = n1,
      prob = p
    )

    required_x2 <-
      final_boundary - x1

    p_stage2 <- pbinom(
      required_x2 - 1,
      size = n2,
      prob = p,
      lower.tail = FALSE
    )

    result <- result +
      p_x1 * p_stage2
  }

  result
}

The function can then be evaluated over a grid of response probabilities.

p_grid <- seq(
  0,
  0.60,
  by = 0.01
)

success_probability <- sapply(
  p_grid,
  prob_success,
  n1 = 15,
  n2 = 15,
  continue_boundary = 3,
  final_boundary = 8
)

This produces the probability of final success as a function of the true response rate under the illustrative decision rules.

Why Simulation May Be Better Than Simple Binomial Calculations

The previous calculations assume a relatively simple structure.

Real expansion studies may include:

• Multiple cohorts
• Multiple doses
• Different response probabilities across cohorts
• Delayed efficacy assessment
• Safety events
• Patient replacement
• Biomarker assay failure
• Dropout
• Unequal cohort sizes
• Adaptive cohort opening

In these situations, simulation is often the most transparent way to determine the complete operating characteristics.

A Practical Simulation Strategy

Simulation Output

A useful simulation summary might look like:

The exact numerical values should come from the prespecified simulation rather than from qualitative expectations.

Protocol Considerations

A dose expansion section of a protocol should clearly describe:

• Expansion population
• Selected dose and schedule
• Scientific objective
• Primary endpoint
• Secondary endpoints
• Exploratory endpoints
• Planned sample size
• Rationale for sample size
• Interim analyses
• Futility criteria
• Efficacy criteria, if applicable
• Safety stopping rules
• Response assessment schedule
• Analysis population
• Handling of missing data
• Replacement rules
• Biomarker analysis
• PK/PD analysis
• Multiplicity considerations
• Decision criteria for cohort closure or expansion

What Should Be Reported in the Statistical Analysis Plan?

The SAP should make the expansion analysis reproducible.

At minimum, specify:

• Primary estimand
• Primary analysis population
• Definition of response
• Handling of missing response assessments
• Confidence interval method
• Hypothesis-testing procedure, if applicable
• Interim analysis rules
• Final analysis rules
• Multiplicity strategy
• Subgroup analyses
• Safety analysis sets
• PK analysis methods
• PD analysis methods
• Biomarker analysis methods
• Rules for treatment discontinuation and follow-up

Estimands in Dose Expansion

Modern clinical-trial analysis benefits from explicitly defining the estimand.

For example, the efficacy question might be:

What is the probability of achieving an objective response among patients with the prespecified biomarker-positive disease who receive the selected dose, under the treatment-policy strategy for treatment discontinuation?

This is more precise than simply stating:

"The primary endpoint is response rate."

The estimand framework helps clarify:

• Population
• Treatment
• Endpoint
• Intercurrent events
• Summary measure

Expansion Cohorts and Intercurrent Events

Treatment discontinuation, new anticancer therapy, death, or missing assessments can affect the interpretation of an efficacy endpoint.

The analysis should therefore specify how these events are handled.

For example, a patient who stops treatment because of toxicity before the first response assessment may need a prespecified treatment-policy or composite strategy rather than being silently removed from the analysis.

Common Expansion-Cohort Endpoints

Expansion Cohort Versus Basket Trial

A basket study evaluates a treatment across multiple disease or molecular subgroups under a broader master protocol.

An expansion cohort can be one component of such a strategy, but the terms are not synonymous.

The distinction matters because basket-trial designs can require methods for:

• Borrowing information across cohorts
• Heterogeneity across tumor types
• Multiple testing
• Hierarchical decision-making
• Cohort-specific stopping

Borrowing Information Across Cohorts

Suppose two tumor types are biologically related.

Investigators may consider whether information from one cohort should influence the other.

This can increase efficiency when the response rates are genuinely similar.

However, inappropriate borrowing can be harmful if the true response rates differ substantially.

Adaptive Cohort Opening

Some programs open additional expansion cohorts based on emerging evidence.

For example:

This can improve development efficiency, but the resulting decision process must be designed and simulated carefully.

Why Cohort Expansion Can Be Efficient

The fundamental efficiency argument is:

rather than:

If the treatment is clearly unpromising, fewer patients may be enrolled.

If the treatment is promising, additional patients can be concentrated in the population and dose that appear most informative.

But Efficiency Has a Cost

Adaptive expansion can introduce:

• Statistical complexity
• Operational complexity
• Potential selection bias
• Multiplicity
• Interpretation challenges
• Dependence between stages

The design should therefore balance efficiency against transparency and interpretability.

Practical Dose Expansion Workflow

Design Checklist

What the Statistical Report Should Show

A strong dose expansion analysis should allow the reader to reconstruct what happened to every patient.

Useful displays include:

• Patient disposition
• Demographic and baseline characteristics
• Treatment exposure
• Dose interruptions and reductions
• Adverse-event summaries
• Response waterfall plots where appropriate
• Spider plots where appropriate
• Response-rate estimates with confidence intervals
• Duration-of-response curves
• Kaplan-Meier curves
• PK concentration-time profiles
• Exposure distributions
• Biomarker-response relationships

Statistical Interpretation of a Small Expansion Cohort

Suppose 30 patients produce nine responses.

The observed response rate is 30%, but the confidence interval is broad.

The appropriate conclusion may therefore be:

The cohort provides preliminary evidence of antitumor activity at the selected dose, with the magnitude of the response rate remaining uncertain because of the limited sample size.

This is generally more defensible than treating the expansion cohort as if it were a large confirmatory study.

The Most Important Statistical Distinction

A dose expansion cohort can serve at least three different statistical roles:

These objectives can coexist, but the statistical design should identify which one is primary.

Expansion Cohorts Are Decision-Making Experiments

The best way to think about dose expansion is not simply:

"We found the dose, so now we enroll more patients."

Instead:

The expansion cohort is the information-gathering component in the middle.

Its value depends on how well its design connects the treatment dose, patient population, endpoints, and statistical decision criteria.

Worked Example Summary

The numerical stopping boundaries in this example are deliberately illustrative. A real trial should derive its final boundaries using the desired type I error, power, clinical assumptions, and the complete operational design.

Final Takeaways

Dose expansion cohorts occupy an important position between early dose finding and later-stage clinical development.

The central statistical concepts are:

• The expansion dose should be selected using the totality of safety, PK/PD, efficacy, and clinical information.
• The expansion population should be defined around a clear scientific question.
• Sample size should be driven by the objective rather than by an arbitrary number of patients.
• Binary response endpoints can be analyzed using exact binomial methods, single-stage designs, or two-stage designs depending on the objective.
• Confidence intervals are essential when interpreting response rates from small cohorts.
• Interim efficacy and futility decisions should be prespecified.
• Safety monitoring should operate independently of efficacy stopping when appropriate.
• Small cohorts provide limited information about rare safety events.
• Multiple expansion cohorts can introduce multiplicity and heterogeneity issues.
• Biomarker-selected cohorts require consideration of biomarker prevalence and recruitment feasibility.
• PK/PD and exposure-response analyses can provide important information beyond the response rate.
• Simulation is often appropriate for complex adaptive expansion designs.
• An expansion cohort should not automatically be interpreted as a confirmatory trial.

References

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