Sequential analysis

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Sequential analysis is a way of analyzing data as it is collected, not after a fixed sample size is reached. Researchers check the data at planned points and can stop the study as soon as there is strong enough evidence. This can lead to earlier conclusions and lower costs, especially in medical trials or quality control.

Origins and history in brief
- The idea grew in the 1940s with Abraham Wald and colleagues at Columbia, as a tool for wartime quality control.
- Similar early work appeared in Britain and among other researchers, including Arrow, Blackwell, and Girshick.
- Alan Turing also worked on a related approach for breaking German codes.
- In medicine, Peter Armitage helped bring sequential analysis to clinical trials, and Stuart Pocock provided practical guidance for controlling errors when looking at data multiple times.

How it works
- If you repeatedly look at data as it comes in, the chance of a false positive (saying there is an effect when there isn’t one) grows. So you adjust the significance level at each interim look to keep the overall error rate at the desired level.
- Several stopping rules or boundaries help with this:
- Pocock boundary: strengthens the same threshold at every look.
- Haybittle–Peto bounds and O’Brien–Fleming bounds: more conservative early on, looser later.
- Demets & Lan alpha-spending: lets you spend the error allowance in a flexible way.
- E-values and e-processes: newer ideas that avoid some fixed-look restrictions.
- In a two-group trial, you might enroll the same number of participants in each group, analyze after the first batch, and stop if results are convincing enough; otherwise you add more participants and recheck, continuing until a maximum number of looks is reached.

Important notes
- Trials stopped early for positive results often overestimate the true effect size, especially with small samples. This is a known bias, though its impact on meta-analyses can be balanced by including later- stopping studies.
- P-values in sequential analyses don’t have the same meaning as in fixed-sample tests. One approach is stagewise ordering, which ranks p-values by when the data were examined and how strong the result was.

Related ideas and applications
- Sequential analysis connects to step detection and change-point problems, where the goal is to spot abrupt changes in a signal as data arrive.
- In practice, sequential methods support online monitoring and early alerts, helping decisions be made quickly and responsibly.