Analysis of variance and uncertainty analysis are analogous techniques for partitioning variability. In both analyses negative interaction terms due to negative covariance terms that appear when non-orthogonal predictor variables are allowed may occur. Uncertainties can be estimated directly from variances if the form of distribution is assumed. The decision as to which of the techniques to use depends partly on the properties of the criterion variable. Only uncertainty analysis may be used with a non-metric criterion. Since uncertainties are dimensionless (using no metric), however, uncertainty analysis has a generality which may make it useful even when variances can be computed.

The problem of choosing the correct additive constant to convert relative interstimulus distances to absolute interstimulus distances in multidimensional scaling is investigated. An artificial numerical example is constructed, and various trial values of the constant are inserted to demonstrate the effect on the multidimensional map of making a variety of incorrect choices. Finally, a general solution to the problem, suggested by Dr. Ledyard R Tucker, is presented; each of the computational steps in this solution is set down for easy reference.

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.

Perhaps an appropriate subtitle for this paper would be “Some Speculations on the Interrelations of Psychological Methodologies.” The Scale Grid is a name I have given to a model which presumes to define the underlying continuities between such diverse areas as psychophysics, objective testing, attitude studies including questionnaire and interview techniques, learning experiments, rating scale methods, essay examinations, and projective instruments. The intent of the Scale Grid is to make explicit the fundamental similarities and differences of the methodologies among these various areas of psychological research.

Four different value laws are developed and tested by using them to predict the scale values of composite stimuli from the scale values of their components. These four laws are: an additive law, a square-root law, a logarithmic, and a negative exponential law. They are tried out on a set of food preferences by means of Pearson's Method of False Position. The negative exponential law of diminishing returns gave the best fit to the data but was not markedly better than any of the other laws.

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.

A technique is presented for obtaining a rapid estimate of reliability between judges, with special reference to qualitative judgments. It is shown that reliability and discrimination are independent and that estimates of both are needed. A method of obtaining an independent estimate of multi-judge discrimination is developed. It is shown that the size of item-samples is specified by the latter method. Tests of significance for both reliability and discrimination are described.

Speeded and unspeeded tests of vocabulary, spatial relations, and arithmetic reasoning were factorially analyzed, together with certain reference tests and academic grades. Lawley's maximum likelihood method was used, the computations being carried out on the Whirlwind electronic computer. Four different speed factors were isolated, together with a second-order general speed factor. Consistent small positive correlations between the academic grades and the speed factors were found.

Estimators which are efficient in the sense of having minimum asymptotic variance are obtained for the structural parameters of Lazarsfeld's latent class model of latent structure analysis. Sufficient conditions for the local identification of the structural parameters are also presented.

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.

A procedure is developed for computing optimum regression weights under the restriction that they be non-negative. The weights maximize, subject to the restrictions, the multiple correlation between several predictors and a criterion. A numerical example 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.

For the case of a single criterion a method is already available for determining the optimal distribution of testing time for a battery of predictors, assuming that intercorrelation, validity, and reliability data are available for predictors of arbitrary lengths. In this article a modification and generalization of the method is presented for the case of differential prediction involving a number of criterion variables. A numerical example is given to illustrate the method, after which the mathematical rationale is outlined.

Tables of the exact distributions of number of matches are given for small decks having the same number of cards in each suit. Several approximate distributions are considered for use with larger decks, and some indication of the goodness of the approximations is given.

A scheme for choosing a few individuals whose preferences for given objects are most representative of those of a larger group of individuals is proposed. The method involves (a) quantifying the preferences of each individual so as to discriminate optimally among objects, (b) testing statistically whether or not a common preference continuum may be assumed for the quantified preferences, (c) constructing a linear estimator of values for the objects on this continuum, if it may be assumed, and (d) selecting as judges the least number of individuals whose quantified preferences, when used with this estimator, determine values for the objects with acceptable accuracy. A numerical example based on food preferences is presented.

The multidimensional method of successive intervals and the method of complete triads are applied to similarity judgments of Munsell colors varying in brightness, saturation, and hue. Both methods yield configurations that correlate highly with the Munsell color structure. This validation of these scaling methods in an area of known dimensionality indicates their applicability for exploration in areas of unknown dimensionality.

It is shown that Estes' formula for the asymptotic behavior of a subject under conditions of partial reinforcement can be derived from the assumption that the subject is behaving rationally in a certain game-theoretic sense and attempting to minimax his regret. This result illustrates the need for specifying the frame of reference or set of the subject when using the assumption of rationality to predict his behavior.

A method of estimating the regression of gains upon initial scores is suggested and compared with two other methods which have been used in recent investigations.

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.