## Sequential Analysis of Statistical Data: Theory; [with:] Sequential Tests of Statistical Significance; [with:] Sequential Analysis of Statistical Data: Applications.

Washington: National Defense Research Committee, 1943-1944.

First edition, very rare, of Wald’s seminal invention of 'Sequential analysis', developed while he was in charge of Columbia University’s Statistical Research Group in response to the demand for more efficient methods of industrial quality control during World War II. It is here offered in the original ‘restricted’ reports; it was published four years later in his well-known book of the same title. “Wald’s second major achievement in mathematical statistics is sequential analysis. The notion that in some sense it is economical to observe and analyze data sequentially, rather than to observe and analyze a single sample of predetermined fixed size, was not a new one. Intuitive support for this notion is immediate; if the evidence shown in sequentially unfolding data is sharply one-sided, it seems reasonable to believe that the inquiry can be terminated early, with lengthier inquiries reserved for those situations in which the issue at hand appears, via the sequentially unfolding data, to be in greater doubt. This notion and the partial mathematical formulation of it were to be found in the statistical literature; among those who dealt with it before Wald was Walter Bartky of Chicago, and among Wald’s contemporaries, George Barnard, working in England. But again it was Wald, in 1943, who first formulated mathematically and solved quite generally the problem of sequential tests of statistical hypotheses. He introduced the particular method of the sequential probability ratio test and, with Wolfowitz (1948), showed its optimal properties. He found operating characteristic and average sample number functions; he introduced, if he did not completely solve, the problem of sequential tests of composite hypotheses (utilizing weight functions); and he began vital discussions of such basic topics as multivalued decisions and optimal sequential estimation. All this, plus many special problems, were gathered together in Sequential Analysis (1947), a book surprisingly easy to read, less formal and more elementary in structure than his work on decision functions” (DSB). No copies listed on ABPC/RBH.

“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. 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 non-sequential designs …

“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. 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 … In a sequential investigation observations must be examined either one by one as they are collected or at certain stages during collection. A sequential procedure might be desirable for various reasons. 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. 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. Three examples will illustrate these points:

(1) An investigator may wish to estimate to within 10 per cent the mean weekly expenditure on tobacco per household. 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.

(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.

(3) A manufacturer carrying out inspection of batches of some product may be able to pass most of his batches with little inspection but may carryout further inspection of batches of doubtful quality. 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

“The most appropriate design and method of analysis of a sequential investigation depend on the purpose of the investigation. 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. Economy in number of observations is typically important for sequential design” (International Encyclopedia of the Social Sciences).

“The problem of sequential analysis arose in the Statistical Research Group of Columbia University in connection with some comments made by Captain G. L. Schuyler of the Bureau of Ordnance, Navy Department. Milton Friedman and W. Allen Wallis recognized the great potentialities and the far-reaching consequences that sequential analysis might have for the further development of theoretical statistics. In particular, they conjectured that a sequential test procedure might be constructed which would control the possible errors committed by wrong decisions exactly to the same extent as the best current procedure based on a predetermined number of observations, and at the same time would require, on the average, a substantially smaller number of observations than the fixed number of observations needed for the current procedure. Friedman and Wallis also exhibited a few examples of sequential modifications of current test procedures resulting, in some cases, in an increase of efficiency. It was at this stage that they proposed the problem of sequential analysis to the author [i.e., Wald]. This gave the incentive for the author’s investigations which then led to the development of the sequential probability ratio test.

“Because of the usefulness of the sequential probability ratio test in development work on military and naval equipment, it was classified Restricted within the meaning of the Espionage Act. The author was requested to submit his findings in a restricted report dated September, 1943. In this report the sequential probability ratio test and the basic theory is given. To facilitate the use of this new technique by the Army and the Navy, the Statistical Research Group issued a second report in July, 1944, which gives an elementary non-mathematical exposition of the applications of the sequential probability ratio test and contains a considerable number of tables, charts, and computational simplifications to facilitate applications” (Wald, pp. 2-3).

The 10 parts are as follows:

Sequential analysis of statistical data: Theory, September 1943.

Sequential tests of statistical significance, April 1944.

Applications, 15 July 1944:

Sequential analysis in inspection and experimentation (Introduction) (Section 1);

Sequential analysis when the result of a single observation is a classification as good or bad and when the result of the test is acceptance or rejection (Section 2);

Sequential analysis when the result of a single observation is a classification as good or bad and when the result of the test is a decision between two methods or products (Section 3);

First report: Small 4to (230 x 180 mm), original printed wrappers. Upper right corner with a slight bump. Previous owners initials to front wrapper W.L.D. Stamped RESTRICTED. IX, (1)179, (3: blank) pp. Second report: 8vo (244 x 153 mm). Original printed wrappers, fine. 10, (2: blank) pp. Third report: Large 4to (285 x 220 mm). Original ring binder containing the six sections in original printed wrappers together with the two appendicies, aslo in wrappers: [S1:VIII,16]; [S2:II,22]; [S3:II,22]; [S4:II,22]; [S5:II,18]; [S6:II,18]; [A1:II,14]; [A2:II,22] pp. In very good condition. Also included is the mimiographed distribution list of the third report: 1 sheet typescript, printed on both sides. All three reports are in the rare original issue with the text: "This document contains information affecting the national defense of the United States within the meaning of the Espionage Act, 50 U.S.C., 31 and 32, as amended. Its transmission or the revelation of its contents in any manner to an unauthorized person is prohibited by law". OCLC lists 8 copies of the first report, 1 of the second (National Institute of Standards and Technology), and the third report is not located in any library.

Item #5180

Price: \$18,500.00