Sequential Tests in Industrial Statistics

Summary After an introductory and an historical note, an elementary problem of simple qualitative inspection of a box of components is treated by using a “lattice diagram representation.” This leads to the consideration of sequential tests for such cases. Procedures for determining “Target-Handicap” forms of inspection, and their operating and sample size properties are given. This leads to a consideration of general linear sequential tests, which are those test procedures which can be formulated in terms of a “score.” Such procedures are shown to be similar to classical games of chance, and to physical diffusion processes. The diffusion analogy leads to a differential equation which gives the approximate characteristics of any such linear test. In many cases, Wald’s “Probability Ratio Sequential Test” takes the form of a linear test. The conditions for this are determined. The P.R.S. test is seen to be “best possible linear test,” in the sense of minimizing average sample size. The effects of deviations from normality, and general distributions are considered. Reference is made to Wald’s work on tests which involve parameters other than those being estimated, and then consideration is restricted to tests for the mean of normal populations where the variance is unknown. Methods of reducing such tests to simple binomial tests are indicated. A number of procedures for use with 2 × 2 comparative trials, and double dichotomies, are given, and their properties discussed. Returning to general inspection problems, the paper indicates that these are not always to be identified with problems involving merely tests of statistical hypotheses. The notions of Consumer’s Lot, Producer’s Batch, the Lot Quality Curve, the Process Curve, are explained, and their importance indicated. A distinction is made between Acceptance Inspection schemes and Rectifying Inspection schemes, and the notions of Operating Characteristic Curve, Operating Characteristic Matrix, and the Sample Size distribution function are explained. The lattice diagram is used to bring out relationships between notions involved in general inspection, and some other uses are also indicated. Finally, some reflections on the relevance of the matters discussed to matters of current debate among statisticians are given.