The method of rotation described here is applied to one hyperplane at a time. The method seems to be simple and quite effective and it can be applied by a relatively inexperienced computer. The method does not postulate a positive manifold, and hence it is applicable also to bipolar factors.

The test-retest reliability of qualitative items, such as occur in achievement tests, attitude questionnaires, public opinion surveys, and elsewhere, requires a different technique of analysis from that of quantitative variables. Definitions appropriate to the qualitative case are made both for the reliability coefficient of an individual on an item and for the reliability coefficient of a population on the item. From but a single trial of a large population on the item, it is possible to compute a lower bound to the group reliability coefficient. Two kinds of lower bounds are presented. From two experimentally independent trials of the population on the item, it is possible to compute an upper bound to the group reliability coefficient. Two upper bounds are presented. The computations for the lower and upper bounds are all very simple. Numerical examples are given.

Most existing tests for significance of difference between means require specific assumptions concerning the distribution function in the parent population. The need for a test which can be applied without making any such assumption is stressed. Such a statistical test is derived. The application of the test involves converting scores to rank orders. The exact probabilities may then be calculated for specified differences between samples by means of which the null hypothesis may be tested. The application of the test is simple and requires a minimum of calculation. The test loses in precision because of the conversion to rank orders but gains in generality since it may be safely used with any kind of distribution.

In this paper a preview of the problem is given. Then the mathematical solutions of estimating the sums of squares and products of different sources of variation under different assumptions are presented. Two kinds of populations from which our samples are supposed to be drawn are specified. One is defined as possessing approximately the same stratification as our sample; while the other is defined as having equal frequencies in the subclasses. For the first kind of population, we should use the restrictions of “the weighted means.” For the second kind, we should use the restrictions of “the unweighted means.” The assumptions of zero interactions and significant interactions are also considered. After working out the exact method, two approximate methods with appropriate statistical assumptions to be fulfilled are given.

The aesthetic preferences of a group of persons are obtained from their orders of sets of pictures and patterns according to “liking.” The same pictures are ordered independently by a team of experts, according to certain artistic criteria such as naturalism, composition, color, rhythm, etc. The orders of preference and orders according to the criteria are compared by correlation and matrices of correlation formed from (1) correlations between the persons' orders of preference; (2) correlations between the orders of preference and orders according to artistic criteria; and (3) correlations between the criterion orders. These matrices are symbolised by Rp, R0, and Rc, respectively, and combined to form a single matrix

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\left[ \begin{gathered}R_p R_o\hfill \\R'_o R_c\hfill \\ \end{gathered}\right]$$\end{document}