Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the “triple product” [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars—under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotatability characteristic can hold even when the number of factors extracted is greater than every dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments.

Following the work of Tucker, Cooper, and Meredith, image and anti-image covariance matrices from a correlation matrix that may be singular are derived.

Two forms of stationarity prior to criterion in absorbing Markov chains are examined. Both forms require that the probability of a particular response on a particular trial before absorption be independent of trial number. The stronger of these forms holds that this is true independent of starting state; the weaker, only for a specified set of starting probabilities. Simple, necessary and sufficient conditions for both forms are developed and applied to several examples.

A common problem for both principal component analysis and image component analysis is determining how many components to retain. A number of solutions have been proposed, none of which is totally satisfactory. An alternative solution which employs a matrix of partial correlations is considered. No components are extracted after the average squared partial correlation reaches a minimum. This approach gives an exact stopping point, has a direct operational interpretation, and can be applied to any type of component analysis. The method is most appropriate when component analysis is employed as an alternative to, or a first-stage solution for, factor analysis.

Based on the test theory model for ordinal measurements proposed by Schulman and Haden, the present paper considers correlations between tests, attenuation, regressions involving true and observed scores, and prediction of test reliability.

The population correlation between tests is shown to be related to the expected sample correlation for samples of size n1 and n2. Errors of estimation, measurement and prediction are found to be similar to their counterparts in interval test theory, while attenuation is identical to its counterpart. The bias in estimating population reliability from sample data is compared for Kendall's tau and Spearman's rho.

In order to make the parallel analysis criterion for determining the number of factors easy to use, regression equations for predicting the logarithms of the latent roots of random correlation matrices, with squared multiple correlations on the diagonal, are presented. The correlation matrices were derived from distributions of normally distributed random numbers. The independent variables are

log (N − 1) and log {[n(n − 1)/2] − [(i − 1)n]},

where N is the number of observations; n, the number of variables; and i, the ordinal position of the eigenvalue. The results were excellent, with multiple correlation coefficients ranging from .9948 to .9992.

A procedure for ordering object (stimulus) pairs based on individual preference ratings is described. The basic assumption is that individual responses are consistent with a nonmetric multidimensional unfolding model. The method requires data where a numerical response is independently generated for each individual-object pair. In conjunction with a nonmetric multidimensional scaling procedure, it provides a vehicle for recovering meaningful object configurations.

Generalized symmetric means are redefined in a way which allows them to be calculated for any matrix sampling design. It is proved that these sample gsm's are unbiased estimates of the analogous population gsm's. Illustrative examples are given.

Relations between Tucker's three-mode multidimensional scaling and Carroll and Chang's INDSCAL are discussed. The possibility is raised that it may be profitable to attempt to transform a three-mode solution to the general form of an INDSCAL solution. Operationally, this involves transforming the three-mode core matrix so that each section is, as nearly as possible, a diagonal matrix. A technique is developed for accomplishing such a transformation, and is applied to two sets of data from the literature. Results indicate that the process is both feasible and valuable, providing useful information on the relative appropriateness of the two models.

Critical examination is made of the recent controversy over the value of Monte Carlo techniques in nonmetric multidimensional scaling procedures. The case is presented that the major relevance of Monte Carlo studies is not for the local minima problem but for the meaningfulness of the obtained solutions.

An alternative derivation is given for a simple test model which incorporates an ability parameter for the subject and difficulty level and guessing parameters for the problem. The probability of a correct response to the problem is a projective transformation of the problem difficulty. The ability and difficulty parameters separate into additive components.

For analyses with missing data, some popular procedures delete cases with missing values, perform analysis with “missing value” correlation or covariance matrices, or estimate missing values by sample means. There are objections to each of these procedures. Several procedures are outlined here for replacing missing values by regression values obtained in various ways, and for adjusting coefficients (such as factor score coefficients) when data are missing. None of the procedures are complex or expensive.