Very general multilinear models, called CANDELINC, and a practical least-squares fitting procedure, also called CANDELINC, are described for data consisting of a many-way array. The models incorporate the possibility of general linear constraints, which turn out to have substantial practical value in some applications, by permitting better prediction and understanding. Description of the model, and proof of a theorem which greatly simplifies the least-squares fitting process, is given first for the case involving two-way data and a bilinear model. Model and proof are then extended to the case of N-way data and an N-linear model for general N. The case N = 3 covers many significant applications. Two applications are described: one of two-way CANDELINC, and the other of CANDELINC used as a constrained version of INDSCAL. Possible additional applications are discussed.

A method for externally constraining certain distances in multidimensional scaling configurations is introduced and illustrated. The approach defines an objective function which is a linear composite of the loss function of the point configuration X relative to the proximity data P and the loss of X relative to a pseudo-data matrix R. The matrix R is set up such that the side constraints to be imposed on X’s distances are expressed by the relations among R’s numerical elements. One then uses a double-phase procedure with relative penalties on the loss components to generate a constrained solution X. Various possibilities for constructing actual MDS algorithms are conceivable: the major classes are defined by the specification of metric or nonmetric loss for data and/or constraints, and by the various possibilities for partitioning the matrices P and R. Further generalizations are introduced by substituting R by a set of R matrices, Ri, i = 1, …r, which opens the way for formulating overlapping constraints as, e.g., in patterns that are both row- and column-conditional at the same time.

A review of the existing techniques for the analysis of three-way data revealed that none were appropriate to the wide variety of data usually encountered in psychological research, and few were capable of both isolating common information and systematically describing individual differences. An alternating least squares algorithm was proposed to fit both an individual difference model and a replications component model to three-way data which may be defined at the nominal, ordinal, interval, ratio, or mixed measurement level; which may be discrete or continuous; and which may be unconditional, matrix conditional, or row conditional. This algorithm was evaluated by a Monte Carlo study. Recovery of the original information was excellent when the correct measurement characteristics were assumed. Furthermore, the algorithm was robust to the presence of random error. In addition, the algorithm was used to fit the individual difference model to a real, binary, subject conditional data set. The findings from this application were consistent with previous research in the area of implicit personality theory and uncovered interesting systematic individual differences in the perception of political figures and roles.

A new method to estimate the parameters of Tucker’s three-mode principal component model is discussed, and the convergence properties of the alternating least squares algorithm to solve the estimation problem are considered. A special case of the general Tucker model, in which the principal component analysis is only performed over two of the three modes is briefly outlined as well. The Miller & Nicely data on the confusion of English consonants are used to illustrate the programs TUCKALS3 and TUCKALS2 which incorporate the algorithms for the two models described.

In measurement studies the researcher may wish to test the hypothesis that Cronbach’s alpha reliability coefficient is the same for two measurement procedures. A statistical test exists for independent samples of subjects. In this paper three procedures are developed for the situation in which the coefficients are determined from the same sample. All three procedures are computationally simple and give tight control of Type I error when the sample size is 50 or greater.

This paper describes an asymptotic inferential procedure for the estimates of the false positive and false negative error rates. Formulas and tables are described for the computations of the standard errors. A simulation study indicates that the asymptotic standard errors may be used even with samples of 25 cases as long as the Kuder-Richardson Formula 21 reliability is reasonably large. Otherwise, a large sample would be required.

We consider the problem of comparing m latent population distributions when the observed values are scores on a test battery with binary items. The latent densities are assumed to be normal densities, and we consider a test for equality of the means as well as a test equality of the variances. In addition, we consider a longitudinal model, where the test battery has been applied to the same individuals at different points in time. This model allows for correlations between the latent variable at different time points, and methods are discussed for estimating the correlation coefficient.

A new algorithm is used to test and describe the set of all possible solutions for any linear model of an empirical ordering derived from techniques such as additive conjoint measurement, unfolding theory, general Fechnerian scaling and ordinal multiple regression. The algorithm is computationally faster and numerically superior to previous algorithms.

Some aspects of the small sample behavior of maximum likelihood estimates in multidimensional scaling are investigated by Monte Carlo. An investigation of Model M2 in the MULTISCALE program package shows that the chi-square test of dimensionality requires a correction of tabled chi-square values to be unbiased. A formula for this correction in the case of two dimensions is estimated. The power of the test of dimensionality is acceptable with as few as two replications for 15 stimuli and as few as five replications for 10 stimuli. The biases in the exponent and standard error estimates in this model are also investigated.