What Is Sequential Analysis?

Sequential analysis is a branch of mathematical statistics in which data are evaluated as they are collected and a decision to stop or continue gathering observations is made at each step based on the accumulated evidence. Unlike classical fixed-sample hypothesis testing, which specifies the sample size before data collection begins, sequential methods allow the sample size to vary depending on how quickly the data yield a conclusive result. This adaptive approach can substantially reduce the expected number of observations needed to reach a decision while maintaining specified type I and type II error rates.

The field originated with the work of Abraham Wald during World War II, when military inspection problems demanded a way to minimize testing costs without sacrificing statistical rigor. Wald's 1945 paper introducing the sequential probability ratio test (SPRT) established the theoretical foundation and demonstrated that the method requires, on average, substantially fewer samples than equivalent fixed-sample tests. Statistical decision theory, optimal stopping theory, and martingale methods from probability provide the mathematical framework that underpins modern sequential procedures.

The Sequential Probability Ratio Test

The SPRT is the foundational procedure in sequential analysis. At each observation, the cumulative log-likelihood ratio of the data under two competing hypotheses is computed and compared against two boundaries: an upper threshold at which the null hypothesis is rejected and a lower threshold at which it is accepted. As long as the statistic remains between these boundaries, testing continues. When the statistic crosses either boundary, sampling stops and the corresponding decision is made.

Wald and Wolfowitz (1948) proved that the SPRT is optimal in the sense that no other test with the same type I and type II error probabilities can achieve a smaller expected sample size under either hypothesis. This optimality result makes the SPRT the natural starting point for sequential design, and it remains the reference against which more complex sequential procedures are measured. A detailed treatment of Wald's original formulation appears in the Springer chapter on sequential tests of statistical hypotheses.

Group Sequential Methods and Multiple Hypotheses

Applied sequential testing often departs from the one-observation-at-a-time structure of the classic SPRT. Group sequential designs specify a fixed number of planned interim analyses at which accumulated data are reviewed; between analyses, the trial proceeds without modification. This structure suits clinical trials and industrial experiments where data arrive in batches or where administrative constraints make continuous monitoring impractical.

Controlling error rates across multiple interim analyses requires adjusted significance boundaries. Methods such as those developed by O'Brien and Fleming, Pocock, and the alpha-spending function approach of Lan and DeMets allocate the overall type I error budget across interim looks in ways that preserve overall error control while giving strong stopping opportunities when effects are large. The PMC paper on sequential tests of multiple hypotheses controlling familywise error rates extends these ideas to settings with multiple endpoints.

Applications in Surveillance and Quality Control

Sequential analysis finds extensive application in post-market safety surveillance for drugs and vaccines, where regulators and manufacturers monitor adverse event counts in real time as a product accumulates exposure in the population. The PMC paper on exact sequential tests for clinical trials and post-market surveillance with Poisson and binary data presents methods used by regulatory programs including the FDA's Sentinel System and the CDC's Vaccine Safety Datalink.

In industrial quality control, the cumulative sum (CUSUM) chart is a sequential monitoring scheme that accumulates deviations from a target process level and signals when the cumulative sum exceeds a threshold. CUSUM charts detect small sustained shifts in process mean more quickly than Shewhart control charts because they use the entire history of deviations rather than only the current observation.

Applications

Sequential analysis has applications in a wide range of fields, including:

• Clinical trials with adaptive stopping for efficacy or futility
• Post-market drug and vaccine safety surveillance
• Industrial process monitoring and quality control charting
• Signal detection in communications and radar systems
• Ecological and environmental monitoring with sequential sampling