Neural Network models are commonly used for cluster analysis in engineering, computational neuroscience, and the biological sciences, although they are rarely used in the social sciences. In this study we compare the classification capabilities of the 1-dimensional Kohonen neural network with two partitioning (Hartigan and Späth k-means) and three hierarchical (Ward's, complete linkage, and average linkage) cluster methods in 2,580 data sets with known cluster structure. Overall, the performance of the Kohonen networks was similar to, or better than, the performance of the other methods.

This paper contains an assessment of the accuracy of Kendall's chi-square approximation to circular triad distributions. Estimated indices are compared to enumerated values across the upper and lower tails of cumulative proportions of circular triad distributions for 5–15 objects. In general, the approximation is found to be quite good. The median difference between the exact distribution and Kendall's approximation is .000692. Critical value approximations are never in error by more than one circular triad and are almost always in the conservative direction. A researcher can be confident that little meaningful error will be present when using Kendall's approximation for experiments outside the range of tabled values. It appears that the chi-square approximation will be accurate to at least two decimal places for distributions beyond 15 objects.

In restricted statistical models, since the first derivatives of the likelihood displacement are often nonzero, the commonly adopted formulation for local influence analysis is not appropriate. However, there are two kinds of model restrictions in which the first derivatives of the likelihood displacement are still zero. General formulas for assessing local influence under these restrictions are derived and applied to factor analysis as the usually used restriction in factor analysis satisfies the conditions. Various influence schemes are introduced and a comparison to the influence function approach is discussed. It is also shown that local influence for factor analysis is invariant to the scale of the data and is independent of the rotation of the factor loadings.

An IRT model based on the Rasch model is proposed for composite tasks, that is, tasks that are decomposed into subtasks of different kinds. There is one subtask for each component that is discerned in the composite tasks. A component is a generic kind of subtask of which the subtasks resulting from the decomposition are specific instantiations with respect to the particular composite tasks under study. The proposed model constrains the difficulties of the composite tasks to be linear combinations of the difficulties of the corresponding subtask items, which are estimated together with the weights used in the linear combinations, one weight for each kind of subtask. Although the model does not belong to the exponential family, its parameters can be estimated using conditional maximum likelihood estimation. The approach is demonstrated with an application to spelling tasks.

In the context of conditional maximum likelihood (CML) estimation, confidence intervals can be interpreted in three different ways, depending on the sampling distribution under which these confidence intervals contain the true parameter value with a certain probability. These sampling distributions are (a) the distribution of the data given the incidental parameters, (b) the marginal distribution of the data (i.e., with the incidental parameters integrated out), and (c) the conditional distribution of the data given the sufficient statistics for the incidental parameters. Results on the asymptotic distribution of CML estimates under sampling scheme (c) can be used to construct asymptotic confidence intervals using only the CML estimates. This is not possible for the results on the asymptotic distribution under sampling schemes (a) and (b). However, it is shown that the conditional asymptotic confidence intervals are also valid under the other two sampling schemes.

Scheiblechner (1995) proposes a probabilistic axiomatization of measurement called ISOP (isotonic ordinal probabilistic models) that replaces Rasch's (1980) specific objectivity assumptions with two interesting ordinal assumptions. Special cases of Scheiblechner's model include standard unidimensional factor analysis models in which the loadings are held constant, and the Rasch model for binary item responses. Closely related are the doubly-monotone item response models of Mokken (1971; see also Mokken & Lewis, 1982; Sijtsma, 1988; Molenaar, 1991; Sijtsma & Junker, 1996; and Sijtsma & Hemker, in press). More generally, strictly unidimensional latent variable models have been considered in some detail by Holland and Rosenbaum (1986), Ellis and van den Wollenberg (1993), and Junker (1990, 1993). The purpose of this note is to provide connections with current research in foundations and nonparametric latent variable and item response modeling that are missing from Scheiblechner's (1995) paper, and to point out important related work by Hemker, Sijtsma, Molenaar, & Junker (1996, 1997), Ellis and Junker (in press) and Junker and Ellis (1997). We also discuss counterexamples to three major theorems in the paper. By carrying out these three tasks, we hope to provide researchers interested in the foundations of measurement and item response modeling the opportunity to give the ISOP approach the careful attention it deserves.