Log-multiplicative association (LMA) models, which are special cases of log-linear models, have interpretations in terms of latent continuous variables. Two theoretical derivations of LMA models based on item response theory (IRT) arguments are presented. First, we show that Anderson and colleagues (Anderson & Vermunt, 2000; Anderson & Böckenholt, 2000; Anderson, 2002), who derived LMA models from statistical graphical models, made the equivalent assumptions as Holland (1990) when deriving models for the manifest probabilities of response patterns based on an IRT approach. We also present a second derivation of LMA models where item response functions are specified as functions of rest-scores. These various connections provide insights into the behavior of LMA models as item response models and point out philosophical issues with the use of LMA models as item response models. We show that even for short tests, LMA and standard IRT models yield very similar to nearly identical results when data arise from standard IRT models. Log-multiplicative association models can be used as item response models and do not require numerical integration for estimation.

Using Lumsden’s Thurstonian fluctuation model as a starting point, this paper attempts to develop a unidimensional item response theory model intended for binary personality items. Under some additional assumptions, a new model is obtained in which the item characteristic curves are defined by a cumulative Pearson-Type-VII distribution, and the person response curves are two-parameter normal ogives. Procedures for fitting the new model are proposed. Furthermore, the relations between individual fluctuation and scalability are discussed, and a scalability index based on the new model is proposed. All the developments in this paper are illustrated using two empirical examples.

The (univariate) isotonic psychometric (ISOP) model (Scheiblechner, 1995) is a nonparametric IRT model for dichotomous and polytomous (rating scale) psychological test data. A weak subject independence axiom W1 postulates that the subjects are ordered in the same way except for ties (i.e., similarly or isotonically) by all items of a psychological test. A weak item independence axiom W2 postulates that the order of the items is similar for all subjects. Local independence (LI or W3) is assumed in all models. With these axioms, sample-free unidimensional ordinal measurements of items and subjects become feasible. A cancellation axiom (Co) gives, as a result, the additive isotonic psychometric (ADISOP) model and interval scales for subjects and items, and an independence axiom (W4) gives the completely additive isotonic psychometric (CADISOP) model with an interval scale for the response variable (Scheiblechner, 1999). The d-ISOP, d-ADISOP, and d-CADISOP models are generalizations to d-dimensional dependent variables (e.g., speed and accuracy of response).

This paper extends the theory of conditional covariances to polytomous items. It has been proven that under some mild conditions, commonly assumed in the analysis of response data, the conditional covariance of two items, dichotomously or polytomously scored, given an appropriately chosen composite is positive if, and only if, the two items measure similar constructs besides the composite. The theory provides a theoretical foundation for dimensionality assessment procedures based on conditional covariances or correlations, such as DETECT and DIMTEST, so that the performance of these procedures is theoretically justified when applied to response data with polytomous items. Various estimators of conditional covariances are constructed, and special attention is paid to the case of complex sampling data, such as those from the National Assessment of Educational Progress (NAEP). As such, the new version of DETECT can be applied to response data sets not only with polytomous items but also with missing values, either by design or at random. DETECT is then applied to analyze the dimensional structure of the 2002 NAEP reading samples of grades 4 and 8. The DETECT results show that the substantive test structure based on the purposes for reading is consistent with the statistical dimensional structure for either grade.

Given that a minor condition holds (e.g., the number of variables is greater than the number of clusters), a nontrivial lower bound for the sum-of-squares error criterion in K-means clustering is derived. By calculating the lower bound for several different situations, a method is developed to determine the adequacy of cluster solution based on the observed sum-of-squares error as compared to the minimum sum-of-squares error.