A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent variables. The response model generalizes GLMMs to incorporate factor structures in addition to random intercepts and coefficients. As in GLMMs, the data can have an arbitrary number of levels and can be highly unbalanced with different numbers of lower-level units in the higher-level units and missing data. A wide range of response processes can be modeled including ordered and unordered categorical responses, counts, and responses of mixed types. The structural model is similar to the structural part of a SEM except that it may include latent and observed variables varying at different levels. For example, unit-level latent variables (factors or random coefficients) can be regressed on cluster-level latent variables. Special cases of this framework are explored and data from the British Social Attitudes Survey are used for illustration. Maximum likelihood estimation and empirical Bayes latent score prediction within the GLLAMM framework can be performed using adaptive quadrature in gllamm, a freely available program running in Stata.

Psychological tests often involve item clusters that are designed to solicit responses to behavioral stimuli. The dependency between individual responses within clusters beyond that which can be explained by the underlying trait sometimes reveals structures that are of substantive interest. The paper describes two general classes of models for this type of locally dependent responses. Specifically, the models include a generalized log-linear representation and a hybrid parameterization model for polytomous data. A compact matrix notation designed to succinctly represent the system of complex multivariate polytomous responses is presented. The matrix representation creates the necessary formulation for the locally dependent kernel for polytomous item responses. Using polytomous data from an inventory of hostility, we provide illustrations as to how the locally dependent models can be used in psychological measurement.

Two psychometric models are presented for evaluating the difficulty of the distractors in multiple-choice items. They are based on the criterion of rising distractor selection ratios, which facilitates interpretation of the subject and item parameters. Statistical inferential tools are developed in a Bayesian framework: modal a posteriori estimation by application of an EM algorithm and model evaluation by monitoring posterior predictive replications of the data matrix. An educational example with real data is included to exemplify the application of the models and compare them with the nominal categories model.

Formulas for the asymptotic biases of the parameter estimates in structural equation models are provided in the case of the Wishart maximum likelihood estimation for normally and nonnormally distributed variables. When multivariate normality is satisfied, considerable simplification is obtained for the models of unstandardized variables. Formulas for the models of standardized variables are also provided. Numerical examples with Monte Carlo simulations in factor analysis show the accuracy of the formulas and suggest the asymptotic robustness of the asymptotic biases with normality assumption against nonnormal data. Some relationships between the asymptotic biases and other asymptotic values are discussed.

Component loss functions (CLFs) are used to generalize the quartimax criterion for orthogonal rotation in factor analysis. These replace the fourth powers of the factor loadings by an arbitrary function of the second powers. Criteria of this form were introduced by a number of authors, primarily Katz and Rohlf (1974) and Rozeboom (1991), but there has been essentially no follow-up to this work. A method so simple, natural, and general deserves to be investigated more completely. A number of theoretical results are derived including the fact that any method using a concave CLF will recover perfect simple structure whenever it exists, and there are methods that will recover Thurstone simple structure whenever it exists. Specific CLFs are identified and it is shown how to compare these using standardized plots. Numerical examples are used to illustrate and compare CLF and other methods. Sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them.

When the underlying responses are on an ordinal scale, gamma is one of the most frequently used indices to measure the strength of association between two ordered variables. However, except for a brief mention on the use of the traditional interval estimator based on Wald's statistic, discussion of interval estimation of the gamma is limited. Because it is well known that an interval estimator using Wald's statistic is generally not likely to perform well especially when the sample size is small, the goal of this paper is to find ways to improve the finite-sample performance of this estimator. This paper develops five asymptotic interval estimators of the gamma by employing various methods that are commonly used to improve the normal approximation of the maximum likelihood estimator (MLE). Using Monte Carlo simulation, this paper notes that the coverage probability of the interval estimator using Wald's statistic can be much less than the desired confidence level, especially when the underlying gamma is large. Further, except for the extreme case, in which the underlying gamma is large and the sample size is small, the interval estimator using a logarithmic transformation together with a monotonic function proposed here not only performs well with respect to the coverage probability, but is also more efficient than all the other estimators considered here. Finally, this paper notes that applying an ad hoc adjustment procedure—whenever any observed frequency equals 0, we add 0.5 to all cells in calculation of the cell proportions—can substantially improve the traditional interval estimator. This paper includes two examples to illustrate the practical use of interval estimators considered here.

Since 1915, statisticians have been applying Fisher's Z-transformation to Pearson product-moment correlation coefficients. We offer new geometric interpretations of this transformation.

This paper discusses a new form of specifying and normalizing a Linear Logistic Test Model (LLTM) as suggested by Bechger, Verstralen, and Verhelst (Psychometrika, 2002). It is shown that there are infinitely many ways to specify the same normalization. Moreover, the relationship between some of their results and equivalent previous results in the literature is clarified, and it is shown that the goals of estimating and testing a single element of the weight matrix, for which they propose new methods, can be reached by means of simple, well-known tools already implemented in published LLTM software.