Higher-order latent traits are proposed for specifying the joint distribution of binary attributes in models for cognitive diagnosis. This approach results in a parsimonious model for the joint distribution of a high-dimensional attribute vector that is natural in many situations when specific cognitive information is sought but a less informative item response model would be a reasonable alternative. This approach stems from viewing the attributes as the specific knowledge required for examination performance, and modeling these attributes as arising from a broadly-defined latent trait resembling the ϑ of item response models. In this way a relatively simple model for the joint distribution of the attributes results, which is based on a plausible model for the relationship between general aptitude and specific knowledge. Markov chain Monte Carlo algorithms for parameter estimation are given for selected response distributions, and simulation results are presented to examine the performance of the algorithm as well as the sensitivity of classification to model misspecification. An analysis of fraction subtraction data is provided as an example.

The regression trunk approach (RTA) is an integration of regression trees and multiple linear regression analysis. In this paper RTA is used to discover treatment covariate interactions, in the regression of one continuous variable on a treatment variable withmultiple covariates. The performance of RTA is compared to the classical method of forward stepwise regression. The results of two simulation studies, in which the true interactions are modeled as threshold interactions, show that RTA detects the interactions in a higher number of cases (82.3% in the first simulation study, and 52.3% in the second) than stepwise regression (56.5% and 20.5%). In a real data example the final RTA model has a higher cross-validated variance-accounted-for (29.8%) than the stepwise regression model (12.5%). All of these results indicate that RTA is a promising alternative method for demonstrating differential effectiveness of treatments.

This paper presents a new hierarchical classes model, called Tucker2-HICLAS, for binary three-way three-mode data. As any three-way hierarchical classes model, the Tucker2-HICLAS model includes a representation of the association relation among the three modes and a hierarchical classification of the elements of each mode. A distinctive feature of the Tucker2-HICLAS model, being closely related to the Tucker3-HICLAS model (Ceulemans, Van Mechelen & Leenen, 2003), is that one of the three modes is minimally reduced and, hence, that the differences among the association patterns of the elements of this mode are maximally retained in the model. Moreover, as compared to Tucker3-HICLAS, Tucker2-HICLAS implies three rather than four different types of parameters and as such is simpler to interpret. Two types of Tucker2-HICLAS models are distinguished: a disjunctive and a conjunctive type. An algorithm for fitting the Tucker2-HICLAS model is described and evaluated in a simulation study. The model is illustrated with longitudinal data on interpersonal emotions.

This paper proposes a multidimensional generalization of Coombs' (1964) parallelogram model for “pick any/n” data, which result from each of a number of subjects having selected a number of objects (s)he likes most from a prespecified set of n objects. In the model, persons and objects are represented in a low dimensional space defined by a set of ordinal variables with a prespecified number of categories; objects are represented as points and persons as intervals on each dimension. A conjunctive combination rule is assumed implying that a person selects an object if and only if the object is within the subject's interval on each dimension. An algorithm for fitting the model to a data set is presented and evaluated in a simulation study. The model is illustrated with data on preferences regarding holiday trips.

Data in social and behavioral sciences typically possess heavy tails. Structural equation modeling is commonly used in analyzing interrelations among variables of such data. Classical methods for structural equation modeling fit a proposed model to the sample covariance matrix, which can lead to very inefficient parameter estimates. By fitting a structural model to a robust covariance matrix for data with heavy tails, one generally gets more efficient parameter estimates. Because many robust procedures are available, we propose using the empirical efficiency of a set of invariant parameter estimates in identifying an optimal robust procedure. Within the class of elliptical distributions, analytical results show that the robust procedure leading to the most efficient parameter estimates also yields a most powerful test statistic. Examples illustrate the merit of the proposed procedure. The relevance of this procedure to data analysis in a broader context is noted.

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes.

Intraclass correlation and Cronbach's alpha are widely used to describe reliability of tests and measurements. Even with Gaussian data, exact distributions are known only for compound symmetric covariance (equal variances and equal correlations). Recently, large sample Gaussian approximations were derived for the distribution functions.

New exact results allow calculating the exact distribution function and other properties of intraclass correlation and Cronbach's alpha, for Gaussian data with any covariance pattern, not just compound symmetry. Probabilities are computed in terms of the distribution function of a weighted sum of independent chi-square random variables.

New F approximations for the distribution functions of intraclass correlation and Cronbach's alpha are much simpler and faster to compute than the exact forms. Assuming the covariance matrix is known, the approximations typically provide sufficient accuracy, even with as few as ten observations.

Either the exact or approximate distributions may be used to create confidence intervals around an estimate of reliability. Monte Carlo simulations led to a number of conclusions. Correctly assuming that the covariance matrix is compound symmetric leads to accurate confidence intervals, as was expected from previously known results. However, assuming and estimating a general covariance matrix produces somewhat optimistically narrow confidence intervals with 10 observations. Increasing sample size to 100 gives essentially unbiased coverage. Incorrectly assuming compound symmetry leads to pessimistically large confidence intervals, with pessimism increasing with sample size. In contrast, incorrectly assuming general covariance introduces only a modest optimistic bias in small samples. Hence the new methods seem preferable for creating confidence intervals, except when compound symmetry definitely holds.

A simple modification substantially simplifies the use of the gradient projection (GP) rotation algorithms of Jennrich (2001, 2002). These algorithms require subroutines to compute the value and gradient of any specific rotation criterion of interest. The gradient can be difficult to derive and program. It is shown that using numerical gradients gives almost precisely the same results as using exact gradients. The resulting algorithm is very easy to use because the only problem specific code required is that needed to define the rotation criterion. The computing time is increased when using numerical gradients, but it is still very modest for most purposes. While used extensively elsewhere, numerical derivatives seem to be underutilized in statistics.

There are two main theories with respect to the development of spelling ability: the stage model and the model of overlapping waves. In this paper exploratory model based clustering will be used to analyze the responses of more than 3500 pupils to subsets of 245 items. To evaluate the two theories, the resulting clusters will be ordered along a developmental dimension using an external criterion. Solutions for three statistical problems will be given: (1) an algorithm that can handle large data sets and only renders non-degenerate clusters; (2) a goodness of fit test that is not affected by the fact that the number of possible response vectors by far out-weights the number of observed response vectors; and (3) a new technique, data expunction, that can be used to evaluate goodness-of-fit tests if the missing data mechanism is known.