A simple and very general algorithm for oblique rotation is identified. While motivated by the rotation problem in factor analysis, it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length. The algorithm has two steps. The first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns. For this reason it is called a gradient projection algorithm. Because the projection step is very simple, implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple. It is proven that the algorithm is strictly monotone, that is as long as it is not already at a stationary point, each step will decrease the value of the criterion. Examples from a variety of areas are used to demonstrate the algorithm, including oblimin rotation, target rotation, simplimax rotation, and rotation to similarity and simplicity simultaneously. While it may be, the algorithm is not intended for use as a standard algorithm for well established problems, but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort.

In this paper robustness properties of the maximum likelihood estimator (MLE) and several robust estimators for the logistic regression model when the responses are binary are analysed. It is found that the MLE and the classical Rao's score test can be misleading in the presence of model misspecification which in the context of logistic regression means either misclassification's errors in the responses, or extreme data points in the design space. A general framework for robust estimation and testing is presented and a robust estimator as well as a robust testing procedure are presented. It is shown that they are less influenced by model misspecifications than their classical counterparts. They are finally applied to the analysis of binary data from a study on breastfeeding.

The U.S. National Assessment of Educational Progress (NAEP), the Third International Mathematics and Science Study (TIMSS), and the U.S. Adult Literacy Survey collect probability samples of students (or adults) who are administered brief examinations in subject areas such as mathematics and reading (cognitive variables), along with background demographic (primary) and educational environment (secondary) questions. The demographic questions are used in the primary reporting, while the numerous “explanatory” secondary variables, or “covariates”, are only directly utilized in subsequent secondary analyses. The covariates are also used indirectly to create the plausible values (multiple imputations) that are an integral part of analyses because of the use of sparse matrix sampling of cognitive items. The improvement in the precision of the primary reporting due to the inclusion of the covariates is assessed here and contrasted with the precision of reporting using plausible values created using only the primary demographic variables.

The results demonstrate that the improvement in precision depends on the matrix sampling designs for the cognitive assessments. The improvements range from essentially none for the most common designs, to moderate for some less common designs. Consequently, two potential changes in the reporting procedures that could improve the statistical and operational efficiency of primary reporting are (a) eliminate or reduce the collection of covariates and increase the number of cognitive items, (b) to avoid delays, eliminate the covariates from the creation of plausible values used for the primary reports, but include them later when creating public-use files for secondary analyses. The potential improvements in statistical and operational efficiency must be weighed against the intrinsic interest in the covariates, and the potential for small discrepancies in the primary and secondary reporting.

Multilevel factor analysis models are widely used in the social sciences to account for heterogeneity in mean structures. In this paper we extend previous work on multilevel models to account for general forms of heterogeneity in confirmatory factor analysis models. We specify various models of mean and covariance heterogeneity in confirmatory factor analysis and develop Markov Chain Monte Carlo (MCMC) procedures to perform Bayesian inference, model checking, and model comparison.

We test our methodology using synthetic data and data from a consumption emotion study. The results from synthetic data show that our Bayesian model perform well in recovering the true parameters and selecting the appropriate model. More importantly, the results clearly illustrate the consequences of ignoring heterogeneity. Specifically, we find that ignoring heterogeneity can lead to sign reversals of the factor covariances, inflation of factor variances and underappreciation of uncertainty in parameter estimates. The results from the emotion study show that subjects vary both in means and covariances. Thus traditional psychometric methods cannot fully capture the heterogeneity in our data.

For any given number of factors, Minimum Rank Factor Analysis yields optimal communalities for an observed covariance matrix in the sense that the unexplained common variance with that number of factors is minimized, subject to the constraint that both the diagonal matrix of unique variances and the observed covariance matrix minus that diagonal matrix are positive semidefinite. As a result, it becomes possible to distinguish the explained common variance from the total common variance. The percentage of explained common variance is similar in meaning to the percentage of explained observed variance in Principal Component Analysis, but typically the former is much closer to 100 than the latter. So far, no statistical theory of MRFA has been developed. The present paper is a first start. It yields closed-form expressions for the asymptotic bias of the explained common variance, or, more precisely, of the unexplained common variance, under the assumption of multivariate normality. Also, the asymptotic variance of this bias is derived, and also the asymptotic covariance matrix of the unique variances that define a MRFA solution. The presented asymptotic statistical inference is based on a recently developed perturbation theory of semidefinite programming. A numerical example is also offered to demonstrate the accuracy of the expressions.

Factor analysis is regularly used for analyzing survey data. Missing data, data with outliers and consequently nonnormal data are very common for data obtained through questionnaires. Based on covariance matrix estimates for such nonstandard samples, a unified approach for factor analysis is developed. By generalizing the approach of maximum likelihood under constraints, statistical properties of the estimates for factor loadings and error variances are obtained. A rescaled Bartlett-corrected statistic is proposed for evaluating the number of factors. Equivariance and invariance of parameter estimates and their standard errors for canonical, varimax, and normalized varimax rotations are discussed. Numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures.

This paper is about the Linear Logistic Test Model (LLTM). We demonstrate that there are infinitely many equivalent ways to specify a model. An implication is that there may well be many ways to change the specification of a given LLTM and achieve the same improvement in model fit. To illustrate this phenomenon, we analyze a real data set using a Lagrange multiplier test for the specification of the model. This Lagrange multiplier test is similar to the modification index used in structural equation modeling.

The problem of choosing the correct number of clusters is as old as cluster analysis itself. A number of authors have suggested various indexes to facilitate this crucial decision. One of the most extensive comparative studies of indexes was conducted by Milligan and Cooper (1985). The present piece of work pursues the same goal under different conditions. In contrast to Milligan and Cooper's work, the emphasis here is on high-dimensional empirical binary data. Binary artificial data sets are constructed to reflect features typically encountered in real-world data situations in the field of marketing research. The simulation includes 162 binary data sets that are clustered by two different algorithms and lead to recommendations on the number of clusters for each index under consideration. Index results are evaluated and their performance is compared and analyzed.

A construction method is given for all factors that satisfy the assumptions of the model for factor analysis, including partially determined factors where certain error variances are zero. Various criteria for the seriousness of indeterminacy are related. It is shown that Green's (1976) conjecture holds: For a linear factor predictor the mean squared error of prediction is constant over all possible factors. A simple and general geometric interpretation of factor indeterminacy is given on the basis of the distance between multiple factors. It is illustrated that variable elimination can have a large effect on the seriousness of factor indeterminacy. A simulation study reveals that if the mean square error of factor prediction equals .5, then two thirds of the persons are “correctly” selected by the best linear factor predictor.