Let A1, A2, ..., An be any n objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (Ai, Ai), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculating m coordinates {xia} for each Ai for a semi-metric d of preassigned form dij = d(|xi1 -xj1|, |xi2 -xj2|, ..., |xim - xjm|). The dimensionality m is sought to be as small as possible, yet satisfy the monotonicity condition that dij <dkl whenever the observed data indicate that Ai is “closer” to Aj than Ak is to Al. Minkowski and Euclidean spaces are special metric examples of d. A general coefficient of monotonicity μ is defined, whose maximization is equivalent to optimal satisfaction of the monotonicity condition, and which allows various options both for treatment of ties and for weighting error-of-fit. A general rationale for algorithm construction is derived for maximizing μ by gradient-guided iterations; this provides a unified mathematical solution to the basic operational problems of norming the gradient to assure proper convergence, of trading between speed and robustness against undesired stationary values, and of a rational first approximation. Distinction is made between single-phase (quadratic) and two-phase (bilinear) strategies for algorithm construction, and between “hard-squeeze” and “soft-squeeze” tactics within these strategies. Special reference is made to the rank-image and related transformational principles, as executed by current Guttman-Lingoes families of computer programs.

Ogilvie and Creelman have recently attempted to develop maximum likelihood estimates of the parameters of signal-detection theory from the data of yes-no ROC curves. Their method involved the assumption of a logistic distribution rather than the normal distribution in order to make the mathematics more tractable. The present paper presents a method of obtaining maximum likelihood estimates of these parameters using the assumption of underlying normal distributions.

It is shown that the lower bounds for the number of common factors, established by Guttman [1954] and modified by Kaiser [1961], cannot decrease as the number of observed variates is increased. The result implies that the lower bounds cannot become weaker if the number of observed variates is increased and the number of factors remains constant.

Necessary and sufficient conditions for the symmetry of the limiting distributions of response probabilities are derived for the Bush and Mosteller two experimenter-controlled events learning model [Bush and Mosteller, 1955].

When the regression of factored variates on variates defining subpopulations (or experimental groups) is (a) nonlinear or (b) heteroscedastic, an existing solution to the problem of factorial invariance does not apply. This paper presents an approach which can be used in place of the existing solution and derives a method for obtaining maximally similar orthogonal simple structure factor pattern matrices across subpopulations. The method is applied to data from three experimental groups.

A generalization of a cognitive probability model is proposed utilizing the serial position effect to account for decay in probability of recall. The general nature of the model permits the incorporation of almost any of the recent free-recall serial position curve results. Two examples, hypothesizing different serial position curves, are given.