Stefano Garzon

Statistics · Oct 23, 2026

Interim analyses and early stopping: O'Brien-Fleming, Pocock, and Haybittle-Peto

Concept

Long trials include interim analyses. An independent data monitoring committee reviews accumulating data at scheduled meetings. The aim is to stop the study if there is clear benefit, harm, or futility.

Each interim analysis is an additional test on the same outcome. If each analysis uses the 1.96 threshold (two-sided, α=0.05\alpha = 0.05), the probability of a false-positive result at some point during the study exceeds 5%. Group sequential methods define stopping boundaries that keep the overall type I error at the nominal level.

How it works

The trial plans KK analyses. At each analysis kk, the statistic ZkZ_k is computed and compared with a boundary ckc_k. If ∣Zk∣>ck|Z_k| > c_k, the trial may stop.

Pocock. Uses the same critical value at every analysis. With five analyses and two-sided α\alpha of 0.05, the critical value is about 2.413, larger than 1.96. The practical consequence: a moderately large final statistic may be judged non-significant even though it would be significant in a single test.

O’Brien-Fleming. The critical value falls over the course of the study:

ck=cOBFK/kc_k = c_{OBF}\sqrt{K/k}

With five analyses and two-sided α\alpha of 0.05, cOBFc_{OBF} is about 2.04. Early analyses require very large values. These values can be capped (for example at 3.5) without materially changing the later boundaries. The final critical value stays close to that of a single test: about 2.05 versus 1.96 when there are fewer than ten analyses.

Haybittle-Peto. Uses a large fixed critical value, such as 3.0, at every interim analysis. The final analysis uses essentially the usual value. The rule is simple but does not guarantee an exact overall type I error.

Alpha spending functions. Lan and DeMets generalized these methods. Total alpha is distributed over the accumulated information fraction, and analyses do not need to occur at fixed dates. An O’Brien-Fleming-type function spends little alpha early. A Pocock-type function spends it sooner.

Many committees prefer conservative early boundaries. With few events, a handful of additional events can change the result substantially.

Common errors when reading a paper

What to check when reading a trial

  1. The protocol defines the number of interim analyses, the information fraction of each, and the boundary method.
  2. An independent committee performed the interim analyses.
  3. The paper reports the boundary crossed and the value of the statistic at stopping.
  4. The number of accumulated events supports a precise estimate.
  5. There are separate boundaries for harm and futility, which are often asymmetric relative to the benefit boundary.
  6. The observed effect is plausible given prior trials of the same intervention.

Application in cardiology

Event-driven cardiovascular trials often use O’Brien-Fleming-type or Haybittle-Peto boundaries. These boundaries make early stopping for benefit unlikely at the start and keep a final threshold close to 0.05. When a trial stops early for benefit, the reading should take into account the boundary, the number of events, and the width of the confidence interval.

References