The concepts of multiple differential prediction and multiple absolute prediction are developed in earlier papers (2, 3). The problem of determining the optimal distribution of testing time for multiple differential prediction has been previously considered (4). This paper develops an analogous procedure for multiple absolute prediction. A numerical example illustrating the procedure is presented. The mathematical rationale underlying the procedure is given.

A precise and rapid procedure has been devised for dealing with a matrix of incomplete data in paired comparisons. This method should increase the general applicability of paired comparisons since experiments involving large numbers of stimuli may now be shortened to feasible experimental proportions. Also, we may now use sets of stimuli which cover a wide range, resulting in a considerable number of 100 per cent vs. 0 per cent judgments, and still give a precise solution depending equally on each of the observations.

The identity of problem and solution in Lazarsfeld's latent structure analysis and Cattell's proportional profiles is pointed out. Anderson's latent structure solution is adapted to proportional profiles to yield a possible solution for the communality and rotational problems in factor analysis. A numerical example of the latter is provided.

Kendall's rank order test for association between two variables is generalized to the case where the total sample is made up of several subgroups and the data on one or both variables consist of the rank order within each subgroup. The test involves no assumptions concerning scales of measurement, shapes of distributions, or relative level of excellence or amount of variability of the different subgroups. Two empirical examples indicate that the normal approximation to the exact test of significance can be considered adequate for most practical situations. Special consideration is given to the case of tied ranks. If ties occur in but one variable within any given subgroup, only a slight modification in procedure is needed. Extensive ties in both variables within subgroups lead to difficulties in determining the appropriate correction for continuity.

The exact condition for discarding k items from a test in order to obtain a residual test with higher validity is derived. A proof that validity always increases is given for the case k = 1. The lack of uniqueness of maximum validity when achieved by use of the condition is discussed. With the use of additional restrictions on items to be included in the initial test, a practical test construction procedure which has several advantages over previous methods is developed. The homogeneity of tests constructed by the method is discussed, and applications are given.

The degree to which scale values computed by the method of successive intervals diverge from theoretically “true” values is seen to be due to three types of error: error due to inequalities in variances of the distributions from which the scale values are computed, error due to non-normality of the distributions, and sampling error. The contribution of each type of error to the total error is evaluated; the latter is seen to be surprisingly small under appropriate conditions. Certain aspects of the formal methodology underlying scaling procedures are also briefly considered.

Considering only population values, it is shown that the complete set of factors of a correlation matrix with units in the diagonal cells may be transformed into the factors derived by factoring these correlations with communalities in the diagonal cells. When the correlations are regarded as observed values, the common factors derived as a transformation of the complete set of factors of the correlation matrix with units in the diagonal cells satisfy Lawley's requirement for a maximum likelihood solution and are a first approximation to Rao's canonical factors.

The growth of a population of knowers of a message was studied to test a human interactance hypothesis. The conditions investigated involved people interacting in time, with the population pairing off randomly (i.e., determined by many, small, different influences) and transferring an attribute (i.e., an all-or-none act) at either a steady rate or a waning rate, subsequent to the originating stimulus. The mathematical expressions for these pre-conditions were the differential equations for the linear logistic for steady acting and the harmonic logistic for waning acting. Variant forms of these curves were developed. Two exploratory experiments, or pretests, comprised launching a coffee slogan in a town and imitating a badge wearer in a boys' camp. Since the activity rate waned harmonically in both cases, the harmonic logistic fit best in both the town and the camp as expected by the hypothesis.

A method is presented for collecting data which will yield a scale on which the entities are ranked in preference (ordinality), the distances between the entities on the scale are ranked (ordered metric), and all combinations of the distances are ranked (higher-ordered metric). The sources drawn upon are von Neumann and Morgenstern (9), and lattice theory. An empirical example is given in which a higher-ordered metric scale is derived.