The numbers in each column of an n × m matrix of multivariate data are interpreted as giving the measured values of all n of the objects studied on one of m different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show (a) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but (b) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.

Browne [1967] has given a method of solving the problem (originally stated by Mosier, [1939]) of finding a least squares fit to a specified factor structure. The problem is one of minimizing the sum of squared residuals of Φ — FT with Diag (T'T)= I. Browne's solution involves the eigenvectors and values of F'F and leads to an iterative solution.

This paper gives a form of the solution which does not involve solution of an eigenvalue problem but does require an iteration similar to Browne's. It suggests the possible existence of a singularity, and a simple modification of Browne's computational procedure is proposed which deals with this case. A better starting value for the iteration is also proposed for which convergence is guaranteed using the ordinary Newton iteration.

This paper discusses the general problem of measuring the association between an independent nominal-scaled variable X and a dependent variable Y whose scale of measurement may be interval, ordinal or nominal. The theoretical foundations of a wide range of asymmetric association measures are discussed. Some new measures are also suggested. Fifteen of these association measures, some previously suggested, some new, are singled out for a computer-assisted numerical study in which we compute the value actually taken by each measure under a wide variety of conditions. This comparative study provides important insights into the behavior of the measures.

Maximum likelihood estimates of the free parameters, and an asymptotic likelihood-ratio test, are given for the hypothesis that one or more elements of a covariance matrix are zero, and/or that two or more of its elements are equal. The theory applies immediately to a transformation of the covariance matrix by a known nonsingular matrix. Estimation is by Newton's method, starting conveniently from a closed-form least-squares solution.

Numerical illustrations include a test for equality of diagonal blocks of a covariance matrix, and estimation of quasi-simplex structures.

Results obtained by Guttman [1955] on the determinacy of common factors have been thought to have disturbing consequences for the common factor model. It is argued that factors must be thought of as unobservable, and uniquely defined but numerically indeterminate. It follows that Guttman's measure of indeterminacy is inconsistent with the foundations of the factor model in probability theory, and the traditional measures of factor indeterminacy used by earlier writers should be reinstated. These yield no disturbing conclusions about the model.

An approach to the analysis of multivariate time series is presented in which linear structural relationships among multiple stochastic variables are investigated. A number of alternative structural models are considered for the case of two stochastic variables. Each model represents a possible hypothesis concerning the relationship of growth in one variable to growth in the second. Both symmetric and asymmetric models are considered. Extensions of two of the models to three variables are illustrated by means of a numerical example. Implications of the models for the problem of detecting change in multivariate time series are discussed.

Omitted items cannot properly be treated as wrong when estimating ability and item parameters. A convenient method for utilizing the information provided by omissions is presented. Theoretical and empirical justifications are presented for the estimates obtained by the new method.