An iterative method is proposed for constructing a weak order from a partial order on a set of stimuli that is based on individual pairwise comparison data. The method generalizes Duncan Luce's construction of the weak order induced by a semiorder. Various aspects of the iterative procedure are discussed, including its rationale, the number of iterations required to obtain a weak order, and the extent to which the data support additions to the initial partial order as a function of the number of iterations performed before the additions occur.

This paper presents a new methodology for estimating the weights or saliences of subcriteria (attributes) in a composite criterion measure. The inputs to the estimation procedure consist of (i) a set of stimuli or objects with each stimulus defined by its subcriteria profile (set of attribute values) and (ii) the set of paired comparison dominance (e.g., preference) judgments on the stimuli made by a single judge (expert) in terms of the global criterion. A criterion of fit is developed and its optimization via linear programming is illustrated with an example. The procedure is generalized to estimate a common set of weights when the pairwise judgments on the stimuli are made by more than one judge. The procedure is computationally efficient and has been applied in developing a composite criterion of managerial success yielding high concurrent validity.

This methodology can also be used to perform ordinal multiple regression—i.e., multiple regression with an ordinally scaled dependent variable and a set of intervally scaled predictor variables. The approach is further extended to “internal analysis” (unfolding) using the vector model of preference and to the additive model of “conjoint measurement.”

Generalized image analysis is considered as a logical algebraic extension of Guttmanian image analysis. Under the assumption that a reduced-rank description of the images is desired, a procedure is developed which achieves the scale-free property by simultaneously rescaling in the metrics of both the images and anti-images, and which produces images and anti-images that are maximally independent in terms of the dimensions needed to account for them.

The precision of scale value estimates using maximum likelihood estimation is examined for varying numbers of categories, varying discriminal dispersions and with and without category boundary estimation. It is concluded that this type of estimation is somewhat unsatisfactory if boundaries are estimated unless these are of specific interest. If they are not estimated, it is concluded that using seven or more categories provides very nearly as much precision of estimate as a corresponding continuous judgment task.

In this paper, recent developments in empirical Bayes procedures are tied in with current work in mental test theory. Point estimators of “true scores” are derived for the binomial and Rasch test models. These estimators are shown to be asymptotically optimal. Smoothing and an empirical study of the behavior of empirical Bayes estimates are taken up in the final section.

The object of this paper is to clarify Schönemann's unfolding algorithm and, in particular, to make it clear that the equations numbered (3.2) in Schönemann's [1970] article, which define Schönemann's solutions, are not a complete set of restraints for the purpose of defining metric unfoldings. Namely, Schönemann has transformed the original equations which define an unfolding to a set of linear and non-linear equations of which he uses only the linear equations to define his solutions. Given infallible data (solution(s) exist) Schönemann's solutions will include the correct solutions. If enough data are available so that there are enough linear equations to uniquely determine a single solution, then Schönemann's solution will coincide with the correct solution. Let P and Q denote the number of elements in the two sets of points, the interset distances of which are specified by the data in the unfolding problem. Let m denote the dimensionality of the Euclidean space into which these points are to be imbedded. If only the linear equations, numbered (18) herein, are to be used, then Schönemann gives the following data requirement for the solution to be uniquely determined: If the full set of linear and nonlinear equations (18–20) are used, then the amount of data required for a solution to be locally unique is relaxed to Both of these results assume that the equations are independent, which has not been proved.

Evidence is given to indicate that Lawley's formulas for the standard errors of maximum likelihood loading estimates do not produce exact asymptotic results. A small modification is derived which appears to eliminate this difficulty.

Beginning with the results of Girshick on the asymptotic distribution of principal component loadings and those of Lawley on the distribution of unrotated maximum likelihood factor loadings, the asymptotic distribution of the corresponding analytically rotated loadings is obtained. The principal difficulty is the fact that the transformation matrix which produces the rotation is usually itself a function of the data. The approach is to use implicit differentiation to find the partial derivatives of an arbitrary orthogonal rotation algorithm. Specific details are given for the orthomax algorithms and an example involving maximum likelihood estimation and varimax rotation is presented.

In a manner similar to that used in the orthogonal case, formulas for the aymptotic standard errors of analytically rotated oblique factor loading estimates are obtained. This is done by finding expressions for the partial derivatives of an oblique rotation algorithm and using previously derived results for unrotated loadings. These include the results of Lawley for maximum likelihood factor analysis and those of Girshick for principal components analysis. Details are given in cases including direct oblimin and direct Crawford-Ferguson rotation. Numerical results for an example involving maximum likelihood estimation with direct quartimin rotation are presented. They include simultaneous tests for significant loading estimates.