An analytic criterion for rotation is defined. The scientific advantage of analytic criteria over subjective (graphical) rotational procedures is discussed. Carroll's criterion and the quartimax criterion are briefly reviewed; the varimax criterion is outlined in detail and contrasted both logically and numerically with the quartimax criterion. It is shown that the normal varimax solution probably coincides closely to the application of the principle of simple structure. However, it is proposed that the ultimate criterion of a rotational procedure is factorial invariance, not simple structure –although the two notions appear to be highly related. The normal varimax criterion is shown to be a two-dimensional generalization of the classic Spearman case, i.e., it shows perfect factorial invariance for two pure clusters. An example is given of the invariance of a normal varimax solution for more than two factors. The oblique normal varimax criterion is stated. A computational outline for the orthogonal normal varimax is appended.

The specification of sample size is an important aspect of the planning of every experiment. When the investigator intends to use the techniques of analysis of variance in the study of treatments effects, he should, in specifying sample size, take into consideration the power of the F tests which will be made. The charts presented in this paper make possible a simple and direct estimate of the sample size required for F tests of specified power.

Three methods of rotation (the graphical, the Thurstone analytical, and the direct-rotational) were applied to the matrix of centroid loadings for 35 variables, to determine which method is the most efficient from theoretical and practical standpoints. The direct-rotational method provided the most information for determining the rank of the configuration and was most economical with respect to time required to reach a rotational solution. The analytical method required the least number of judgmental decisions and was the most objective. The graphical method was the most laborious but had a slight advantage with regard to the number of near-zero loadings in the rotational solution.

Two different linear models are presented for the four-dimensional classification system in which correlations exist between certain pairs of observations. Except for the assumption of correlated observations, classical assumptions associated with classification systems are made. The models considered are modifications of those which underlie the split-plot design and the split-split-plot design. In the first model the correlations between observations of the levels of one dimension are all set equal to ρ. In the second model the observations of the levels of one dimension are assumed correlated to degree ρ1, whereas the observations of a second dimension are correlated to degree ρ2. Analyses for the two models and tests of hypotheses for various parameters are indicated.

The analytical method for simple structure proposed by Thurstone is applied to four separate cases and found to yield satisfactory results. The simple structure obtained by Thurstone's method is found to match closely that obtained by other methods and corresponds to the true structure of the matrix in those cases where true structure is known. Difficulties about the choice of the correct trial vector led the writer to develop a modification of Thurstone's method, useful where high speed computational facilities are available. Instructions are given for this so-called mass modification, and the procedure is illustrated with a 5-factor, 14-variable example. While the results do not fully correspond to a previous graphical solution, it can be argued that the results obtained by the new method show an improved simple structure. The modified method is applied to three other correlation matrices, yielding in each case a satisfactory simple structure.

A significance test is proposed for determining whether a correlation coefficient is less than unity by an amount greater than that attributable to errors of measurement.

Case IV of the Kuder-Richardson series, their formula (21), is derived as a generalized split-half Spearman-Brown coefficient. The basic assumption employed is shown to be sufficient to justify the various assumptions used in derivations by other authors. Some of the implications of this assumption are discussed.

This note presents the average Spearman rank correlation between m independent rankings and an untied criterion ranking, corrected for ties in any or all of the independent rankings.