Sample Size | 11 min read | Beginner
How to Calculate Sample Size for a Single-Arm Phase II Clinical Trial
Learn how sample size is determined for a single-arm Phase II trial, how Simon's two-stage designs calculate it, and how to compare maximum and expected sample size.
What Determines the Sample Size?
How many patients you need depends on three things: the treatment response rate you consider unacceptable, the response rate that would justify further investigation, and the statistical certainty required to make a decision.
Four statistical inputs define the problem.
These values are illustrative, not universal recommendations.
The unacceptable response rate is usually informed by historical evidence or the current standard of care. The promising response rate reflects the improvement considered clinically meaningful.
The difference between p₀ and p₁ is important. A smaller difference generally requires more patients to distinguish an unpromising treatment from a promising one with the specified statistical guarantees.
Choosing these inputs is a clinical and statistical decision, not merely a mathematical calculation.
For more guidance, read how to choose p₀ and p₁ and how to choose α and β for a Simon's two-stage design.
| Input | Meaning | Example |
|---|---|---|
| Unacceptable response rate, p₀ | A response rate at or below which the treatment is considered insufficiently active | 20% |
| Promising response rate, p₁ | A response rate that would justify further investigation | 40% |
| Type I error, α | Maximum permitted probability of declaring the treatment promising when its true response rate is p₀ | 5% |
| Statistical power, 1 − β | Minimum desired probability of declaring the treatment promising when its true response rate is p₁ | 80% |
Why Use a Two-Stage Design?
Consider a study evaluating a treatment with an unacceptable response rate of 20% and a promising response rate of 40%.
With a fixed-sample design, the researcher recruits the planned number of patients before making the final assessment. If the treatment is ineffective, patients may continue to receive it even when the accumulating evidence suggests that further recruitment is unlikely to be worthwhile.
A two-stage design introduces an interim decision.
The stopping rule is specified before the trial begins. This is essential to maintaining the intended statistical operating characteristics.
The principal advantage is that an unpromising treatment may be identified before all planned patients are recruited.
Read more about why Phase II clinical trials need early stopping.
The four steps of a two-stage design
Stage 1: enrol the first group of patients and assess their responses.
Early stopping: if the number of responses is too low, stop the trial for insufficient activity.
Stage 2: if enough responses are observed, recruit additional patients.
Final decision: apply the prespecified response criterion to determine whether the treatment meets the study's statistical objective.
How Simon's Two-Stage Design Calculates Sample Size
Simon's two-stage design searches for combinations of stage sizes and response thresholds that meet the specified type I error and power requirements.
Four design parameters describe a conventional design: n₁, the number of patients enrolled in Stage 1; r₁, the early-stopping threshold for the number of responses; N, the maximum total sample size across both stages; and r, the final response threshold.
Under the usual convention, the trial stops after Stage 1 if the number of responses is at most r₁. Otherwise, it continues to Stage 2. At the end of the trial, the treatment is declared promising if the total number of responses exceeds r.
A candidate design must satisfy both constraints: P(success | p₀) ≤ α and P(success | p₁) ≥ 1 − β. The first controls the type I error. The second ensures the required statistical power.
Step 1: Define the assumptions
Suppose the study specifies p₀ = 0.20, p₁ = 0.40, a maximum type I error of α = 0.05, and a minimum power of 1 − β = 0.80.
These assumptions define the statistical problem. They do not uniquely determine one design.
Step 2: Calculate the operating characteristics
Let X₁ be the number of responses in Stage 1 and X₂ the number of responses in Stage 2. If the true response probability is p, then X₁ follows a binomial distribution with sample size n₁ and response probability p, and X₂ follows a binomial distribution with sample size N − n₁ and response probability p. The two stage response counts are independent under the conventional design.
The probability of early termination is PET(p) = P(X₁ ≤ r₁). The probability of declaring the treatment promising is given above, where the sum runs over the Stage 1 response counts that continue to Stage 2 and accounts for the additional responses needed in Stage 2.
Step 3: Evaluate candidate designs
A search algorithm evaluates possible combinations of n₁, r₁, N, and r. Valid candidates can then be compared according to their expected and maximum sample sizes.
Not every combination satisfies both statistical constraints. If a search returns no feasible design, the result should be investigated rather than interpreted as proof that no design exists under any circumstances. For example, the search may have imposed a maximum sample size that is too restrictive.
Worked Example: A Simon's Two-Stage Design
Consider the following design for the assumptions above. The design parameters are taken from a published example; the achieved operating characteristics that follow are calculated using the exact binomial probabilities.
The decision rules are: enrol 13 patients in Stage 1; stop if 3 or fewer responses are observed; otherwise enrol up to 30 additional patients; and declare the treatment promising if more than 12 responses are observed across both stages.
| Parameter | Value |
|---|---|
| Unacceptable response rate, p₀ | 0.20 |
| Promising response rate, p₁ | 0.40 |
| Maximum type I error | 0.05 |
| Minimum power | 0.80 |
| Stage 1 sample size, n₁ | 13 |
| Early-stopping threshold, r₁ | 3 |
| Maximum total sample size, N | 43 |
| Final response threshold, r | 12 |
Operating Characteristics of the Worked Example
Applying the exact binomial calculations to the design above gives the operating characteristics below. At the unacceptable response rate the design declares the treatment promising only about 5% of the time, which is the type I error. At the promising response rate it declares success about 80% of the time, which is the power.
The expected sample size differs by a factor of nearly two between the two scenarios: about 21 patients under p₀ against about 38 under p₁. This is the practical benefit of the interim look.
| Measure | At p₀ = 0.20 | At p₁ = 0.40 |
|---|---|---|
| Probability of early termination | 74.73% | 16.86% |
| Expected sample size | 20.58 | 37.94 |
| Probability of declaring the treatment promising | 4.96% | 80.02% |
Maximum vs. Expected Sample Size
One of the most important distinctions in two-stage trial design is the difference between maximum and expected sample size.
The maximum sample size, N, is the largest number of patients who can be enrolled under the design. It is reached when the trial continues after Stage 1 and recruits the full second-stage group.
Maximum sample size is useful when planning recruitment capacity, budgeting for the largest possible enrolment, and estimating the maximum workload for study sites.
Expected sample size is the average number of patients enrolled over repeated trials, assuming a particular true response rate. For a conventional Simon's two-stage design with a fixed second-stage sample size, the formula below applies. Every trial enrols the first n₁ patients, and only trials that continue recruit the additional N − n₁ patients, so the probability of continuing is 1 − PET(p).
For the worked example, the expected sample size is approximately 20.58 patients at p = 0.20 and 37.94 patients at p = 0.40. These are averages over repeated trials, not predictions of the number of patients in any individual study.
Expected sample size is therefore not a single number independent of treatment activity. The response-rate assumption must always be stated.
For more detail, see expected sample size in two-stage trials and probability of early termination (PET) explained.
Optimal vs. Minimax: Which Design Should You Choose?
The statistical requirements define a search problem, not a unique answer. Among all designs that satisfy both constraints, several are admissible: they are not dominated by other designs in expected sample size at the null and the alternative.
For a full comparison, see optimal vs. minimax Simon design.
| Design | Optimises | Typical priority |
|---|---|---|
| Optimal | Smallest expected sample size at p₀ | Fewer patients on average when the treatment is inactive |
| Minimax | Smallest maximum sample size | Lower ceiling on total enrolment |
Simon optimal design
The Simon optimal design is the admissible design with the smallest expected sample size under the null hypothesis p₀. It is often attractive when the treatment is unlikely to be active and the goal is to terminate unpromising treatments quickly.
Its maximum sample size may be larger than that of a minimax design.
Minimax design
The minimax design is the admissible design with the smallest maximum sample size. It is often preferred when total enrolment is constrained or when a large Stage 2 is difficult to justify.
Its expected sample size under the null may be larger than that of an optimal design.
Calculating and Comparing Designs in Practice
Some inputs to the calculation are statistical assumptions, while others are operational constraints. It helps to decide the assumptions first and adjust the constraints afterwards, so that changes are documented and justified.
Inputs specified in advance
The unacceptable and promising response rates, together with the type I error and power, should follow from the study rationale and be specified before any design is selected.
Changing these values to obtain a smaller sample size undermines the interpretation of the design, and any change should be documented with a substantive reason.
Constraints and preferences
A maximum Stage 1 size, a maximum total sample size, and a preferred design type are constraints rather than statistical requirements. They can be varied to examine trade-offs, but the effect of each variation should be reported.
What to report
The final protocol should state all four design parameters, the achieved type I error and power, the maximum sample size, and the expected sample size at both p₀ and p₁.
Reporting only the maximum sample size hides the main practical benefit of a two-stage design, while reporting only the expected sample size can mislead recruitment planning.
Common Mistakes
The following issues appear frequently when calculating or reporting sample sizes for single-arm phase II trials.
Reporting only the maximum sample size
A design that can enrol up to 43 patients does not enrol 43 patients on average. Stating the maximum without the expected sample size makes the two-stage design look less efficient than it is.
Confusing p₀ with a historical estimate
The null response rate is the boundary above which the treatment is considered promising, not necessarily the best estimate of current care. Setting it too low makes it easy for an inactive treatment to appear promising.
Ignoring the uncertainty in p₁
The power calculation assumes the promoting response rate is the true value. If the treatment is less active than assumed, power falls quickly, because the design was built to detect the difference between p₀ and p₁.
Summary
Calculating a sample size for a single-arm phase II trial is a structured process. The response rates and error rates define the statistical problem; a two-stage design such as Simon's design converts those requirements into concrete decision rules.
The four design parameters, n₁, r₁, N, and r, should be reported alongside the achieved type I error, power, and expected sample size. Distinguishing maximum from expected sample size is essential for interpreting the design correctly.
To explore these trade-offs, compare optimal and minimax designs and review the operating characteristics of Simon's two-stage designs.
Frequently Asked Questions
How many patients are needed for a single-arm Phase II clinical trial?
There is no universal sample size. It depends on the unacceptable and promising response rates, the chosen type I error and power, and the design criterion. A Simon's two-stage design may allow early termination and reduce expected enrolment when the treatment is insufficiently active.
What if no valid design is found?
First, check the input assumptions and the maximum sample size permitted by the search. Some combinations of response rates and error constraints require a larger sample size than the search allows. If no design is feasible under the intended constraints, revisit the clinical assumptions with the study statistician rather than relaxing statistical requirements without justification.
Can I calculate a Simon's two-stage design without statistical software?
The underlying probabilities can be calculated using binomial distributions, but searching for valid combinations and comparing designs is tedious to do manually. Statistical software such as the R package clinfun, which includes the ph2simon() function, can search for Optimal, Minimax, and admissible designs. Results should still be checked against the intended design conventions and requirements.