By a simplex is meant a set of statistical variables whose interrleations reveal a simple order pattern. For the case of quantitative variables, an order model was analyzed previously which allowed only for positive correlations among the variables and a limited type of gradient among the correlation coefficients. The present paper analyzes a more general model and shows how it is more appropriate to empirical data. Among the novel features emerging from the analysis are: (a) the “factoring” implied of the correlation matrix; (b) the use of a non-Euclidean distance function; and (c) the possible underlying psychological theories.

Certain problems of equating are discussed. The maximum likelihood solution is presented for the following special equating problem: Two tests, U and V, are to be equated, making use of a third “anchor” test, W. The examinees are divided into two random halves. Tests U and W are administered to one half; tests V and W are administered to the other half. It is assumed that any practice effect or other effect, exerted by U and V on W, is the same for U and for V.

Analysis of an empirical theory into a formal system with specified primitive notions and axioms has the advantage of making it clear what deductions from the theory are permissible, and clarifying the internal structure of the theory. An example of such analysis is presented in this paper.

Requirements for an objective definition of simple structure are investigated and a number of proposed objective criteria are evaluated. A distinction is drawn between exploratory factorial studies and confirmatory factorial studies, with the conclusion drawn that objective definition of simple structure depends on study design as well as on objective criteria. A proposed definition of simple structure is described in terms of linear constellations. This definition lacks only a statistical test to compare with possible chance results. A computational procedure is also described for searching for linear constellations. This procedure is very laborious and might best be accomplished on high-speed automatic computers. There is no guarantee that the procedure will find all linear constellations, but it probably would yield satisfactory results for well-designed studies.

Reference is made to Neyman's study of F-test bias for the randomized blocks and Latin square designs employed in agriculture, and some account is given of later statistical developments which sprang from his work—in particular, the classification of model-types and the technique of variance component analysis. It is claimed that there is a need to carry out an examination of F-test bias for experimental designs in education and psychology which will utilize the method and, where appropriate, the known' results of this new branch of variance analysis. In the present paper, such an investigation is carried out for designs which may be regarded as derivatives of the agricultural randomized blocks design. In a paper to follow, a similar investigation will be carried out for experimental designs of the Latin square type.

A simple algebraic development is given showing that criterion estimates derived by usual multiple regression procedures are optimal for personnel classification. It is also shown that, for any assignment of men to jobs, the sum of the multiple regression criterion estimates will equal the sum of the actual criterion scores.

From the ratio of the cross-products of a fourfold table, with the application of two tabled corrections, tetrachoric r's can be estimated with a mean discrepancy of less than .005 even when splits vary greatly from the medians. The necessary calculations can be handled by slide rule and the correction tables used without interpolation.