The problem of minimizing a general matrix, trace function, possibly subject to certain constraints, is approached by means of majorizing this function by one having a simple quadratic shape and whose minimum is easily found. It is shown that the parameter set that minimizes the majorizing function also decreases the matrix trace function, which in turn provides a monotonically convergent algorithm for minimizing the matrix trace function iteratively. Three algorithms based on majorization for solving certain least squares problems are shown to be special cases. In addition, by means of several examples, it is noted how algorithms may be provided for a wide class of statistical optimization tasks for which no satisfactory algorithms seem available.

A two-facet measurement model with broad application in the behavioral sciences is identified, and its coefficient of generalizability (CG) is examined. A normalizing transformation is proposed, and an asymptotic variance expression is derived. Three other multifaceted measurement models and CGs are identified, and variance expressions are presented. Next, an empirical investigation of the procedures follows, and it is shown that, in most cases, Type I error control in inferential applications is precise, and that the estimates are relatively efficient compared with the correlation coefficient. Implications for further research and for practice are noted. In an Appendix, four additional models, CGs, and variance expressions are presented.

A Bayes estimation procedure is introduced that allows the nature and strength of prior beliefs to be easily specified and modal posterior estimates to be obtained as easily as maximum likelihood estimates. The procedure is based on constructing posterior distributions that are formally identical to likelihoods, but are based on sampled data as well as artificial data reflecting prior information. Improvements in performance of modal Bayes procedures relative to maximum likelihood estimation are illustrated for Rasch-type models. Improvements range from modest to dramatic, depending on the model and the number of items being considered.

Information functions are used to find the optimum ability levels and maximum contributions to information for estimating item parameters in three commonly used logistic item response models. For the three and two parameter logistic models, examinees who contribute maximally to the estimation of item difficulty contribute little to the estimation of item discrimination. This suggests that in applications that depend heavily upon the veracity of individual item parameter estimates (e.g. adaptive testing or text construction), better item calibration results may be obtained (for fixed sample sizes) from examinee calibration samples in which ability is widely dispersed.

Relations are examined between latent trait and latent class models for item response data. Conditions are given for the two-latent class and two-parameter normal ogive models to agree, and relations between their item parameters are presented. Generalizations are then made to continuous models with more than one latent trait and discrete models with more than two latent classes, and methods are presented for relating latent class models to factor models for dichotomized variables. Results are illustrated using data from the Law School Admission Test, previously analyzed by several authors.

This paper considers total and direct effects in linear structural equation models. Adopting a causal perspective that is implicit in much of the literature on the subject, the paper concludes that in many instances the effects do not admit the interpretations imparted in the literature. Drawing a distinction between concomitants and factors, the paper concludes that a concomitant has neither total nor direct effects on other variables. When a variable is a factor and one or more intervening variables are concomitants, the notion of a direct effect is not causally meaningful. Even when the notion of a direct effect is meaningful, the usual estimate of this quantity may be inappropriate. The total effect is usually interpreted as an equilibrium multiplier. In the case where there are simultaneity relations among the dependent variables in tghe model, the results in the literature for the total effects of dependent variables on other dependent variables are not equilibrium multipliers, and thus, the usual interpretation is incorrect. To remedy some of these deficiencies, a new effect, the total effect of a factor X on an outcome Y, holding a set of variables F constant, is defined. When defined, the total and direct effects are a special case of this new effect, and the total effect of a dependent variable on a dependent variable is an equilibrium multiplier.

The possibility of obtaining locally optimal solutions with categorical data is pointed out for the original version of OSMOD development by Saito and Otsu. A revision of the initialization strategy in OSMOD is suggested, and its effectiveness in diminishing this possibility is demonstrated.

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.

A maximum likelihood approach is described for estimating the validity of a test (x) as a predictor of a criterion variable (y) when there are both missing and censored y scores present in the data set. The missing data are due to selection on a latent variable (ys) which may be conditionally related to y given x. Thus, the missing data may not be missing random. The censoring process in due to the presence of a floor or ceiling effect. The maximum likelihood estimates are constructed using the EM algorithm. The entire analysis is demonstrated in terms of hypothetical data sets.

Given two matrices, A and B, each with n rows, and with p and q columns, respectively, a very common goal is the comparison of these matrices in a subspace of dimension s, 1 ≤ s ≤ min p, q. That is, assuming linear mappings of A and B into the subspace, the comparison involves the choice of two transformation matrices L and M of dimensions p by s and q by s, respectively, and the subsequent comparisons of the images AL and BM in some fashion.