Definitions are of course arbitrary. Following Lewis Carroll, one can make words mean what one wishes [cf. Gardner, 1960]. Basically, all that is formally required of a definition is that it be clear: that it enable reliable use of the concept concerned. A more informal, heuristic, desideratum is that it actually influence theorists and researchers to progress in their work.

A strategy for attaining reliability is to make a list of related concepts which might be confused with the one intended, and then to define all the concepts simultaneously in one facet framework. Such a mapping helps to make explicit what the target concept has in common with the others, and how it differs from them. This facet strategy has proved to be useful for distinguishing, for example, among varieties of intelligence [Guttman, 1965; Schlesinger and Guttman, 1969], among varieties of attitude [Guttman, 1959; Elizur, 1970; Jordan, 1971], and even between intelligence and attitude [Guttman, in press]. Evidence is accumulating as to the stimulating effect of the approach for the development of new insights and for the discovery of new forms of lawfulness in data.

In problem solving situations, it has been suggested the superiority of groups over individuals is due simply to the fact that groups consist of several individuals. In this paper, the Lorge and Solomon [1955] approach to such situations is reexamined using the method of maximum likelihood. Extensions to trichotomous response situations are also presented, and the resulting models are applied to data gathered by Staub [1970]. The partitioning of the likelihood ratio goodness-of-fit statistic is then discussed in order that the effects of additional variables on the response variate can be assessed. Finally, the small sample behavior of the likelihood ratio statistic is examined.

A multidimensional scaling analysis is presented for replicated layouts of pairwise choice responses. In most applications the replicates will represent individuals who respond to all pairs in some set of objects. The replicates and the objects are scaled in a joint space by means of an inner product model which assigns weights to each of the dimensions of the space. Least squares estimates of the replicates' and objects' coordinates, and of unscalability parameters, are obtained through a manipulation of the error sum of squares for fitting the model. The solution involves the reduction of a three-way least squares problem to two subproblems, one trivial and the other solvable by classical least squares matrix factorization. The analytic technique is illustrated with political preference data and is contrasted with multidimensional unfolding in the domain of preferential choice.

Methods developed by Bernbach [1966] and Millward [1969] permit increased generality in analyses of identifiability. Matrix equations are presented that solve part of the identifiability problem for a class of Markov models. Results of several earlier analyses are shown to involve special cases of the equations developed here. And it is shown that a general four-state chain has the same parameter space as an all-or-none model if and only if its representation with an observable absorbing state is lumpable into a Markov chain with three states.

This paper is concerned with the study of similarities and differences in factor structures between different groups. A common situation occurs when a battery of tests has been administered to samples of examinees from several populations.

A very general model is presented, in which any parameter in the factor analysis models (factor loadings, factor variances, factor covariances, and unique variances) for the different groups may be assigned an arbitrary value or constrained to be equal to some other parameter. Given such a specification, the model is estimated by the maximum likelihood method yielding a large sample x2 of goodness of fit. By computing several solutions under different specifications one can test various hypotheses.

The method is capable of dealing with any degree of invariance, from the one extreme, where nothing is invariant, to the other extreme, where everything is invariant. Neither the number of tests nor the number of common factors need to be the same for all groups, but to be at all interesting, it is assumed that there is a common core of tests in each battery that is the same or at least content-wise comparable.

Considerations of factor score estimates have concentrated on internal characteristics. This report considers external characteristics of four methods for determining factor score estimates; that is, relations of these estimates to measures on attributes not entered into the factor analysis. These external characteristics are important for many uses of factor score estimates. Findings are that different ones of the methods are appropriate for different uses.