Sample size for a group-sequential trial: Pocock

When to use it

A trial where stopping early is the priority, accepting a larger maximum size.

Not for

Trials where a final analysis close to the conventional one matters; use O’Brien–Fleming.

Notation

$K$: Number of analyses (looks), equally spaced in information
$n_{\text{fixed}}$: Per-group size of the same trial without interim analyses
$R$: Inflation factor: maximum size divided by fixed size
$c_k$: Critical value at look k
$\alpha$: Significance level: the chance of a false positive
$1-\beta$: Power: the chance of detecting the effect if it is real
$q$: Expected dropout

Formulas

The fixed-sample size

n_{\text{fixed}} = \frac{2\sigma^{2}(z_{1-\alpha/2} + z_{1-\beta})^{2}}{\Delta^{2}} \quad\text{or}\quad \frac{\left[p_1(1-p_1) + p_2(1-p_2)\right](z_{1-\alpha/2} + z_{1-\beta})^{2}}{(p_1 - p_2)^{2}}

For means or proportions, by the normal approximation; or entered directly from another calculation.

Pocock’s boundary

c_k = c_P \quad\text{for every look}

Every look uses the same critical value, chosen so that the overall false-positive rate is α. The trial stops early readily, but the maximum size grows by about 20% [1].

Maximum size

n_{\max} = \lceil R \cdot n_{\text{fixed}} \rceil

The boundaries and the probability of stopping at each look are computed by the Armitage–McPherson–Rowe recursion, and R is the information that makes the overall power 1 − β [2, 3]. The expected size, reported alongside, is what the trial uses on average if the effect is real.

Dropout

n_{\text{recruit}} = \left\lceil \frac{n}{1 - q} \right\rceil

Applied last, to the participants actually recruited.

Parameters

Number of analyses $K$

Values: A whole number of at least 1, including the final analysis

Usually 2 to 5. Each extra look costs a little more at the maximum.

Boundary shape $c_k$

Values: O’Brien–Fleming or Pocock

O’Brien–Fleming unless early stopping is the priority.

Difference in means $\Delta$ Optional

Values: Not zero

With the SD, to base the design on means.

Standard deviation $\sigma$ Optional

Values: Above 0

Common to both groups.

Two proportions $p_1,\ p_2$ Optional

Values: Between 0 and 1, different

To base the design on proportions instead.

Fixed-sample size per group $n_{\text{fixed}}$ Optional

Values: Above 0

From any other calculation, such as a survival or cluster design, to inflate it for interim looks.

Significance level $\alpha$

Values: Between 0 and 1; usually 0.05

The false-positive rate you accept. A smaller α, for example to allow for several primary outcomes, needs a larger sample.

Power $1-\beta$

Values: Between 0 and 1; usually 0.80 or 0.90

The chance of a significant result if the assumed effect is real. Funders and ethics committees commonly expect at least 0.80.

Expected dropout $q$ Optional

Values: 0% up to, but not including, 100%

The share of participants expected to provide no outcome, from comparable studies. Recruiting more restores the numbers, not the comparability of the groups: if dropout is related to the outcome, the estimate can still be biased.

Assumptions

The looks are pre-specified and equally spaced in information, and the stopping rule is followed. Any change needs its own statistical assessment.
Stopping is only for benefit: there is no futility boundary.
Dropout is unrelated to the outcome; recruiting more participants restores the numbers but cannot remove bias.

Worked examples

Four looks, from means

The same trial with Pocock boundaries: R = 1.202, so at most 76 per group, but 51 on average if the effect is real.

Group 1: 76 · Group 2: 76 · Total: 152

Protocol statement
To detect a difference in means of 5 with 80% power at a two-sided significance level of 0.05, using a group-sequential design with 4 analyses and Pocock boundaries, and assuming a standard deviation of 10, the required sample size is at most 76 per group (152 in total), against 63 without interim analyses and 51 on average if the effect is real. Calculated with StatWisely (statwisely.com) using: maximum n per group = R × fixed-sample n; boundaries spend α across the looks (Armitage-McPherson-Rowe recursion). These are planning estimates; confirm the final design with a qualified statistician.

Open this example in the calculator

Two looks

With 2 looks R = 1.110: at most 70 per group.

Group 1: 70 · Group 2: 70 · Total: 140

Open this example in the calculator

Validation

Every worked example on this page is recomputed by the StatWisely engine whenever the software is built. If a number or formula here stopped matching the calculator, the release would be stopped until this document was corrected.
An independent implementation in R recomputes this test every night and before each production release, and the results are compared with the engine’s.
The inflation factors agree with Jennison and Turnbull’s tables for α = 0.05 two-sided and 80% power: 1.024 for O’Brien–Fleming and 1.202 for Pocock with four looks, and 1.110 for Pocock with two; the O’Brien–Fleming critical values for four looks are 4.049, 2.863, 2.337 and 2.024 [3].

How to cite

StatWisely. Sample size for a group-sequential trial: Pocock. Method document, engine version 1.0.0, reviewed 2026-10-01. https://statwisely.com/methods/sequential-pocock/

Cite the original references for the formulas your design uses as well.

References

1.Pocock SJ. Group sequential methods in the design and analysis of clinical trials. Biometrika. 1977;64(2):191–199.
2.Armitage P, McPherson CK, Rowe BC. Repeated significance tests on accumulating data. Journal of the Royal Statistical Society, Series A. 1969;132(2):235–244.
3.Jennison C, Turnbull BW. Group Sequential Methods with Applications to Clinical Trials. Boca Raton: Chapman & Hall/CRC; 2000.