It is shown that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. This solution is readily adapted to the special case treated by Bloxom, Horan, Carroll and Chang. In addition to being very simple, this algebraic solution also permits testing the constraints of these models explicitly. A numerical example is given.

The asymptotic distributions of response probabilities in the Audley-Jonckheere learning model, as applied to the two-choice learning situation, are described for (1) the experimenter-controlled events model, (2) the subject-controlled events model, and (3) the experimenter-subject-controlled events model.

Given three variables, x, y, z, univariate selection on x, usually reduces the correlation ry z. However, in some cases, when rx y or rx z assumes extreme values, ry z may increase. This reversal is explained by comparison to the effects of suppressor variables and predictors of predictability.

A new solution to the additive constant problem in metric multidimensional scaling is developed. This solution determines, for a given dimensionality, the additive constant and the resulting stimulus projections on the dimensions of a Euclidean space which minimize the sum of squares of discrepancies between the formal model for metric multidimensional scaling and the original data. A modification of Fletcher-Powell style functional iteration is used to compute solutions. A scale free index of the goodness of fit is developed to aid in selecting solutions of adequate dimensionality from multiple candidates.

It is shown that McDonald’s generalization of classical Principal Components Analysis to groups of variables maximally channels the total variance of the original variables through the groups of variables acting as groups. A useful equation is obtained for determining the vectors of correlations of the L2 components with the original variables. A calculation example is given.

Performance measures, conditionalized on the last error and other events, have been of central concern in the development of absorbing Markov-chain models for learning and problem solving. Expressions for response probabilities and expected latencies conditional on the occurrence of the last error and other events are derived using matrix methods.

Metric determinacy of nonmetric multidimensional scaling was investigated as a function of the number of points being scaled, the amount of error in the data being scaled, and the accuracy of estimation of the Minkowski distance function parameters, dimensionality and the r-constant. It was found that nonmetric scaling may provide better models if (1) the true structure is of low dimensionality, (2) the dimensionality of recovered structure is not less than the dimensionality of the true structure, (3) degree of error is low, and (4) the degrees of freedom ratio is greater than about 2.5. It was also found that (5) accurate estimation of the Minkowski constant leads to a better model only if the dimensionality has been properly estimated.

It seems intuitively compelling to many investigators that measurements, on the same subjects by different methods, purportedly of the same given trait are somehow better evidenced to be mutually valid measurements of that trait to the degree that they are intercorrelated. It is similarly compelling that measurements on the same subjects of purportedly different and uncorrelated traits are somehow better evidenced to be valid measurements to the degree that they are not intercorrelated. Further, a demonstration of hetero-method mono-trait intercorrelation (convergence) jointly with one of hetero-method, or preferably mono-method, hetero-trait independence (discrimination) is more compelling than either single demonstration alone [see Campbell & Fiske, 1959]. I hope to show in what follows that this intuition is misleading unless certain rather demanding prerequisites are satisfied. Then I hope to show that contrary demonstrations are generally too indecisive to consitute validity disconfirmations. Finally, I shall consider some issues in the practical use of the indecisive multitrait-multimethod data.

A computational short cut suggested by Feldman and Klinger for the one-sided Fisher-Yates exact test is clarified and is extended to the calculation of probability values for certain two-sided tests when sample sizes are unequal.

As a consequence of the complexity of the iterative optimum seeking procedures used by practical nonmetric multidimensional scaling algorithms, many of their computational properties have not been well understood. In particular, the following questions are of interest: (a) from the user’s point of view, are there significant differences between alternative available programs, (b) are suboptimal solutions frequently encountered, and how does this depend on the characteristics of the algorithm? Using Monte Carlo techniques to generate dissimilarity matrices with known underlying configurations, Kruskal’s M-D-SCAL, Guttman-Lingoes’ SSA-I, and Young-Torgerson’s TORSCA-9 programs were compared. It was found: (a) that differences between the solutions obtained by the algorithms were typically so small as to be of little practical importance, (b) deviant solutions were occasionally produced by each of the algorithms, but most often by M-D-SCAL, and furthermore, most frequently in one dimension. The likelihood of being trapped in a nonoptimal position is reduced by a good choice of initial configuration, and also possibly by constructing the iterative process to favor the possibility of stepping over small local depressions when far from the location of the optimum. The conclusions of this study are based on a total of 2160 scaling solutions.

The “longitudinal factor analysis” model, which uniquely resolves factors from two occasions of data representing the same persons measured on the same test battery, is shown to be derivable by application of canonical correlation procedures to factor scores. Interpreted in this light, it is suggested that, in attaining its objective, “longitudinal factor analysis” maximizes temporal stability of factor scores—an emphasis which may be warranted for some types of change but not for others.

Given the complete set R of rank orders derived from some configuration of n stimulus points in r dimensions in accordance with the unfolding model, a stimulus configuration which generates just these orders will be described as a solution for R. The space is assumed to be Euclidean. Necessary and sufficient conditions are determined for a nondegenerate configuration to be a solution for R. The geometrical conditions which are necessary and sufficient to determine the subset of pairs of opposite orders are also identified and constitute the constraint system for the ordinal vector model.