In this paper, a theoretical index of dimensionality, called the theoretical DETECT index, is proposed to provide a theoretical foundation for the DETECT procedure. The purpose of DETECT is to assess certain aspects of the latent dimensional structure of a test, important to practitioner and research alike. Under reasonable modeling restrictions referred to as “approximate simple structure”, the theoretical DETECT index is proven to be maximized at the correct dimensionality-based partition of a test, where the number of item clusters in this partition corresponds to the number of substantively separate dimensions present in the test and by “correct” is meant that each cluster in this partition contains only items that correspond to the same separate dimension. It is argued that the separation into item clusters achieved by DETECT is appropriate from the applied perspective of desiring a partition into clusters that are interpretable as substantively distinct between clusters and substantively homogeneous within cluster. Moreover, the maximum DETECT index value is a measure of the amount of multidimensionality present. The estimation of the theoretical DETECT index is discussed and a genetic algorithm is developed to effectively execute DETECT. The study of DETECT is facilitated by the recasting of two factor analytic concepts in a multidimensional item response theory setting: a dimensionally homogeneous item cluster and an approximate simple structure test.

When the underlying distribution is discrete with a limited number of categories, methods for interval estimation of the intraclass correlation which assume normality are theoretically inadequate for use. On the basis of large sample theory, this paper develops an asymptotic closed-form interval estimate of the intraclass correlation for the case where there is a natural score associated with each category. This paper employs Monte Carlo simulation to demonstrate that when the underlying intraclass correlation is large, the traditional interval estimator which assumes normality can be misleading. We find that when the number of classes is ≥ 20, the interval estimator proposed here can generally perform reasonably well in a variety of situations. This paper further notes that the proposed interval estimator is invariant with respect to a linear transformation. When the data are on a nominal scale, an extension of the proposed method to account for this case, as well as a discussion on the relationship between the intraclass correlation and a kappa-type measure defined here and on the limitation of the corresponding kappa-type estimator are given.

Factor scores are naturally predicted by means of their conditional expectation given the indicators y. Under normality this expectation is linear in y but in general it is an unknown function of y. It is discussed that under nonnormality factor scores can be more precisely predicted by a quadratic function of y.

It has been shown by Kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation. In this paper, Kaiser's result is generalized. First, the invariance property is shown to hold for any set of orthogonal components. Next, a similar invariance property is derived for the reliability of any set of components. Both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability, respectively. A short-cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix. Finally, the greatest lower bound to reliability and a weighted version are discussed.

In a recent paper by van Buuren (1997) it is concluded that parameter estimates in pure moving-average (MA) models, obtained by software for fitting structural equation models (SEMs), are biased and inefficient. In this comment it is shown that this negative finding may be due to a particular feature of van Buuren's simulation experiment. A modified procedure for fitting MA models by means of SEM software is proposed, and some of its implications are discussed.

This paper generalizes the p* model for dichotomous social network data (Wasserman & Pattison, 1996) to the polytomous case. The generalization is achieved by transforming valued social networks into three-way binary arrays. This data transformation requires a modification of the Hammersley-Clifford theorem that underpins the p* class of models. We demonstrate that, provided that certain (non-observed) data patterns are excluded from consideration, a suitable version of the theorem can be developed. We also show that the approach amounts to a model for multiple logits derived from a pseudo-likelihood function. Estimation within this model is analogous to the separate fitting of multinomial baseline logits, except that the Hammersley-Clifford theorem requires the equating of certain parameters across logits. The paper describes how to convert a valued network into a data array suitable for fitting the model and provides some illustrative empirical examples.