Sequential analysis
Encyclopedia
In statistics
, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size
is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined stopping rule
as soon as significant results are observed. Thus a conclusion may sometimes be reached at a much earlier stage than would be possible with more classical hypothesis testing or estimation
, at consequently lower financial and/or human cost.
with Jacob Wolfowitz
, W. Allen Wallis, and Milton Friedman
while at Columbia University's
Statistical Research Group
as a tool for more efficient industrial quality control
during World War II
. Another early contribution to the method was made by K.J. Arrow
with D. Blackwell
and M.A. Girshick.
A similar approach was independently developed at the same time by Alan Turing
, as part of the Banburismus
technique used at Bletchley Park
, to test hypotheses about whether different messages coded by German Enigma
machines should be connected and analysed together. This work remained secret until the early 1980s.
that has been studied by, among others, Huyghens in 1657..
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, sequential analysis or sequential hypothesis testing is statistical analysis where the sample size
Sample size
Sample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...
is not fixed in advance. Instead data are evaluated as they are collected, and further sampling is stopped in accordance with a pre-defined stopping rule
Stopping rule
In probability theory, in particular in the study of stochastic processes, a stopping time is a specific type of “random time”....
as soon as significant results are observed. Thus a conclusion may sometimes be reached at a much earlier stage than would be possible with more classical hypothesis testing or estimation
Estimation
Estimation is the calculated approximation of a result which is usable even if input data may be incomplete or uncertain.In statistics,*estimation theory and estimator, for topics involving inferences about probability distributions...
, at consequently lower financial and/or human cost.
History
Sequential analysis was first developed by Abraham WaldAbraham Wald
- See also :* Sequential probability ratio test * Wald distribution* Wald–Wolfowitz runs test...
with Jacob Wolfowitz
Jacob Wolfowitz
Jacob Wolfowitz was a Polish-born American statistician and Shannon Award-winning information theorist. He was the father of former Deputy Secretary of Defense and World Bank Group President Paul Wolfowitz....
, W. Allen Wallis, and Milton Friedman
Milton Friedman
Milton Friedman was an American economist, statistician, academic, and author who taught at the University of Chicago for more than three decades...
while at Columbia University's
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
Statistical Research Group
Applied Mathematics Panel
The Applied Mathematics Panel was created at the end of 1942 as a division of the National Defense Research Committee within the Office of Scientific Research and Development in order to solve mathematical problems related to the military effort in World War II, particularly those of the other...
as a tool for more efficient industrial quality control
Quality control
Quality control, or QC for short, is a process by which entities review the quality of all factors involved in production. This approach places an emphasis on three aspects:...
during World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
. Another early contribution to the method was made by K.J. Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
with D. Blackwell
David Blackwell
-Honors and awards:*President, Institute of Mathematical Statistics, 1956*National Academy of Sciences, 1965*American Academy of Arts and Sciences, 1968*Honorary Fellow, Royal Statistical Society, 1976*Vice President, American Statistical Association, 1978...
and M.A. Girshick.
A similar approach was independently developed at the same time by Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
, as part of the Banburismus
Banburismus
Banburismus was a cryptanalytic process developed by Alan Turing at Bletchley Park in England during the Second World War. It was used by Bletchley Park's Hut 8 to help break German Kriegsmarine messages enciphered on Enigma machines. The process used sequential conditional probability to infer...
technique used at Bletchley Park
Bletchley Park
Bletchley Park is an estate located in the town of Bletchley, in Buckinghamshire, England, which currently houses the National Museum of Computing...
, to test hypotheses about whether different messages coded by German Enigma
Enigma machine
An Enigma machine is any of a family of related electro-mechanical rotor cipher machines used for the encryption and decryption of secret messages. Enigma was invented by German engineer Arthur Scherbius at the end of World War I...
machines should be connected and analysed together. This work remained secret until the early 1980s.
Clinical trials
In a randomized trial with two treatment groups, classical group sequential testing is used in the following manner: If n subjects in each group are available, an interim analysis is conducted on the 2n subjects. The statistical analysis is performed to compare the two groups, and if the alternative hypothesis is accepted, the trial is terminated. Otherwise, the trial continues for another 2n subjects, with n subjects per group. The statistical analysis is performed again on the 4n subjects. If the alternative is accepted, then the trial is terminated. Otherwise, it continues with periodic evaluations until N sets of 2n subjects are available. At this point, the last statistical test is conducted, and the trial is discontinued.Other applications
Sequential analysis also has a connection to the problem of gambler's ruinGambler's ruin
The term gambler's ruin is used for a number of related statistical ideas:* The original meaning is that a gambler who raises his bet to a fixed fraction of bankroll when he wins, but does not reduce it when he loses, will eventually go broke, even if he has a positive expected value on each bet.*...
that has been studied by, among others, Huyghens in 1657..
See also
- Optimal stoppingOptimal stoppingIn mathematics, the theory of optimal stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance...
- Sequential estimationSequential estimationIn statistics, sequential estimation refers to estimation methods in sequential analysis where the sample size is not fixed in advance. Instead, data is evaluated as it is collected, and further sampling is stopped in accordance with a pre-defined stopping rule as soon as significant results are...
- Sequential probability ratio testSequential probability ratio testThe sequential probability ratio test is a specific sequential hypothesis test, developed by Abraham Wald. Neyman and Pearson's 1933 result inspired Wald to reformulate it as a sequential analysis problem...
External links
- Sequential Analysis: Design Methods & Applications Journal
- Course given by Rebecca Betensky at Harvard University, lecture note slides
- Software for conducting sequential analysis and applications of sequential analysis in the study of group interaction in computer-mediated communication by Dr. Allan Jeong at Florida State University