A factor analysis model consists of a random sequence of variates defined on a probability space and satisfying the usual descriptive equations of the common-factor analysis in which the common-factor scores are dimensionally independent. Necessary and sufficient conditions are given for a model to exist with essentially unique and hence determinate common factor scores. Parallel results are given for the existence of models with nonunique and hence indeterminate scores. It is then próved that two models cannot exist with essentially unique but different scores for the same common factors. The meaning and application of these results are discussed.

Researchers in the past ten years have studied various parameters involved in nonmetric multidimensional scaling by utilizing Monte Carlo procedures. This paper develops stress distributions using Kruskal's second stress formula based upon a null hypothesis of equal likelihood in the ranking of a set of proximities. These distributions can serve to determine whether a set of data has other than random structure.

Cohen's kappa index is reformulated for multiple classifications based on exchangeable random variables. It is found that kappa is between 0 and 1 inclusive. Two characterizations for kappa are stated in terms of the relationship between such random variables. Within the normal test score model, kappa increases with test reliability and test length. Furthermore, when based on binary classifications, kappa is an inverse U-shaped function of the cutoff score. These trends also hold for the beta-binomial test score model.

Several ways of using the traditional analysis of variance to test heterogeneity of spread in factorial designs with equal or unequal n are compared using both theoretical and Monte Carlo results. Two types of spread variables, (1) the jackknife pseudovalues of s2 and (2) the absolute deviations from the cell median, are shown to be robust and relatively powerful. These variables seem to be generally superior to the Z-variance and Box-Scheffé procedures.

A special case of Bloxom's version of Tucker's three-mode model is developed statistically. A distinction is made between modes in terms of whether they are fixed or random. Parameter matrices are associated with the fixed modes, while no parameters are associated with the mode representing random observation vectors. The identification problem is discussed, and unknown parameters of the model are estimated by a weighted least squares method based upon a Gauss-Newton algorithm. A goodness-of-fit statistic is presented. An example based upon self-report and peer-report measures of personality shows that the model is applicable to real data. The model represents a generalization of Thurstonian factor analysis; weighted least squares estimators and maximum likelihood estimators of the factor model can be obtained using the proposed theory.

Problem solving models relating individual and group solution times under time limit censoring are presented. Maximum likelihood estimates of parameters of the resulting censored distributions are derived and goodness of fit tests for the individual-group models are constructed. The methods are illustrated on data previously analyzed by Restle and Davis.

Multidimensional scaling has recently been enhanced so that data defined at only the nominal level of measurement can be analyzed. The efficacy of ALSCAL, an individual differences multidimensional scaling program which can analyze data defined at the nominal, ordinal, interval and ratio levels of measurement, is the subject of this paper. A Monte Carlo study is presented which indicates that (a) if we know the correct level of measurement then ALSCAL can be used to recover the metric information presumed to underlie the data; and that (b) if we do not know the correct level of measurement then ALSCAL can be used to determine the correct level and to recover the underlying metric structure. This study also indicates, however, that with nominal data ALSCAL is quite likely to obtain solutions which are not globally optimal, and that in these cases the recovery of metric structure is quite poor. A second study is presented which isolates the potential cause of these problems and forms the basis for a suggested modification of the ALSCAL algorithm which should reduce the frequency of locally optimal solutions.

A general statistical model for simultaneous analysis of data from several groups is described. The model is primarily designed to be used for the analysis of covariance. The model can handle any number of covariates and criterion variables, and any number of treatment groups. Treatment effects may be assessed when the treatment groups are not randomized. In addition, the model allows for measurement errors in the criterion variables as well as in the covariates. A wide variety of hypotheses concerning the parameters of the model can be tested by means of a large sample likelihood ratio test. In particular, the usual assumptions of ANCOVA may be tested.

A gradient method is used to obtain least squares estimates of parameters of the m-dimensional euclidean model simultaneously in N spaces, given the observation of all pairwise distances of n stimuli for each space. The procedure can estimate an additive constant as well as stimulus projections and the metric of the reference axes of the configuration in each space. Each parameter in the model can be fixed to equal some a priori value, constrained to be equal to any other parameter, or free to take on any value in the parameter space. Two applications of the procedure are described.

Three-dimensional contingency tables are analyzed, with one variable (e.g., sex) as a factor, and with a natural relation between the other variables (e.g. left and right eye vision). Models of special interest, like symmetry and proportional symmetry between the related variables, and homogeneity across the factor levels, are investigated. Maximum likelihood estimators of parameters and partitions of chi-square goodness-of-fit statistics are explicitly presented; the independence of certain models is noted, and an example is discussed.

Under mild assumptions, when appropriate elements of a factor loading matrix are specified to be zero, all orthogonally equivalent matrices differ at most by column sign changes. Here a variety of results are given for the more complex case when the specified values are not necessarily zero. A method is given for constructing reflections to preserve specified rows and columns. When the appropriate k(k − 1)/2 elements have been specified, sufficient conditions are stated for the existence of 2k orthogonally equivalent matrices.

The g1- and g2- bipartial canonical correlation analyses are developed as generalizations of the partial, part, and bipartial canonical correlation analysis. Illustrative examples are provided.

It is reported that (1) a new coordinate estimation routine is superior to that originally proposed for ALSCAL; (2) an oversight in the interval measurement level case has been found and corrected; and (3) a new initial configuration routine is superior to the original.