This theory, developed for the testing of military equipment during World War II, is summarized in Wald’s book Sequential Analysis (1947). Bartky (1943) generalized this idea in his “multiple sampling,” which allowed many stages, and his procedure was very closely related to a particular case of the general theory of sequential analysis that Wald was developing simultaneously. The first stage of sampling would always be used, but the second stage would be used only if the results of the first were equivocal furthermore, the size of the second sample and the acceptance criteria might depend on the first stage results. Their problem was to specify sampling inspection schemes that discriminated between batches of good and bad quality. An important precursor of the modern theory of sequential analysis was the work done in 1929 by Dodge and Romig (1929–1941) on double sampling schemes. An example might be a social survey in which data could be collected rather quickly but in which a full analysis would be long and costly. In other situations it may be possible to scrutinize the results as they are obtained, but only at very great cost. It would not be possible, for example, to do a sequential analysis of the effect of some social or medical policy if this effect could not be assessed until five years had elapsed. To make effective sequential use of observations they must become available without too great a delay. Sometimes a sequential investigation, although desirable, may not be practicable. Some details of particular methods are given in later sections.
Economy in number of observations is typically important for sequential design. The statistical formulation of that purpose may take one of a number of forms, usually either estimation of some quantity to a given degree of precision or testing a hypothesis with given size and given power against a given alternative hypothesis. The most appropriate design and method of analysis of a sequential investigation depend on the purpose of the investigation. A given degree of discrimination between good and bad batches could be achieved in various ways, but a sequential scheme will often be more economical than one in which a sample of constant size is taken from each batch. (3) A manufacturer carrying out inspection of batches of some product may be able to pass mostof his batches with little inspection but may carryout further inspection of batches of doubtful quality. (2) A physician wishing to compare the effects of two drugs in the treatment of some disease may wish to stop the investigation if at some stage a convincing difference can already be demonstrated using the available data. In order to determine the sample size he would need an estimate of the variability of the expenditure from household to household, and this might be obtainable only from the survey itself. (1) An investigator may wish to estimate to within 10 per cent the mean weekly expenditure on tobacco per household. Three examples will illustrate these points: Alternatively, he may have no intrinsic interest in the intermediate results but may be able to achieve economy in sample size by taking them into account. The investigator might wish to have an up-to-date record at any stage, either for general information or because the appropriate sample size depends on quantities that he can estimate only from the data themselves. A sequential procedure might be desirable for various reasons. In a sequential investigation observations must be examined either one by one as they are collected or at certain stages during collection. In this article attention will be restricted mainly to the usual situation in which termination of a single investigation is the point at issue.
For example, preliminary experience in an experiment may suggest changes in the treatments being compared in a social survey a small pilot survey may lead to modifications in the design of the main investigation.
The term “sequential” is occasionally extended to cover also investigations in which various aspects of the design may be changed according to the observations made. The motivation for most sequential investigations is that when the ends achieved are measured against the costs incurred (including the cost of making observations), sequential designs are typically more efficient than nonsequential designs some disadvantages of the sequential approach are discussed later. Sequential analysis is the branch of statistics concerned with investigations in which the decision whether or not to stop at any stage depends on the observations previously made.
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