Lloyd Bond comments here on the Focus article in this issue of "Measurement: Interdisciplinary Research and Perspectives". The Focus article is entitled: "How Task Features Impact Evidence from Assessments Embedded in Simulations and Games" (Russell G. Almond, Yoon Jeon Kim, Gertrudes Velasquez, and Valerie J. Shute). Bond…

The information matrix can equivalently be determined via the expectation of the Hessian matrix or the expectation of the outer product of the score vector. The identity of these two matrices, however, is only valid in case of a correctly specified model. Therefore, differences between the two versions of the observed information matrix indicate…

Cognitive diagnostic models (CDMs) attempt to uncover latent skills or attributes that examinees must possess in order to answer test items correctly. The DINA (deterministic input, noisy "and") model is a popular CDM that has been widely used. It is shown here that a logistic version of the model can easily be fit with standard software for…

A fundamental goal of educational research is identifying students' current stage of skill mastery (complete/partial/none). In recent years a number of cognitive diagnosis models have become a popular means of estimating student skill knowledge. However, these models become difficult to estimate as the number of students, items, and skills grows.…

This article presents a state-space modeling (SSM) technique for fitting process factor analysis models directly to raw data. The Kalman smoother via the expectation-maximization algorithm to obtain maximum likelihood parameter estimates is used. To examine the finite sample properties of the estimates in SSM when common factors are involved, a…

Second-order latent growth curve models (S. C. Duncan & Duncan, 1996; McArdle, 1988) can be used to study group differences in change in latent constructs. We give exact formulas for the covariance matrix of the parameter estimates and an algebraic expression for the estimation of slope differences. Formulas for calculations of the required sample…

The information matrix for the parameters in a latent-variable model is bounded from above by the information that would obtain if the values of the latent variables could also be observed. The difference is the "missing information." This paper discusses the structure of the information matrix, and characterizes the degree to which…

The identification of two factors when one factor is expected is an artifact caused by using factor analysis on data that would be more appropriately analyzed with a unidimensional unfolding model. A numerical illustration is given, and ways to determine whether data conform to the unidimensional unfolding model are reviewed. (SLD)

The criteria for assessing convergent and discriminant validity proposed by D. T. Campbell and D. W. Fiske (1959) are shown to be inadequate for either the additive or multiplicative structures of data in a multitrait-multimethod matrix. Model-specific criteria are more promising for assessing convergent and discriminant validity. (Author/SLD)

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled…

In confirmatory factor analysis (CFA), models exist that are admissible and demonstrably identified where the factor loading matrix from latent to manifest variables is of less than full column rank, even though other models are not identified. Implications for CFA are discussed. (SLD)

Introduces a schema for classifying research in mathematics education that deepens and extends the Romberg/Bourne Model. Employs dual hierarchical systems, one related to the research task and the other to research technique. (Author/CCM)

Generalizing Boolean matrix decomposition to a larger class of matrix decomposition models is demonstrated, and probability matrix decomposition (PMD) models are introduced as a probabilistic version of the larger class. An algorithm is presented for the computation of maximum likelihood and maximum a posteriori estimates of the parameters of PMD…

The Curriculum Matrix developed by A. W. Foshay (1991) is a broad, comprehensive conception of curriculum, incorporating a range of perspectives and meanings often overlooked. This paper situates Foshay's matrix in the field of curriculum studies. By focusing on Foshay's interpretation of five curricular purposes (aesthetic, physical,…

Describes and assesses missing data methods currently used to analyze data from matrix sampling designs implemented by the National Assessment of Educational Progress. Several improved methods are developed, and these models are evaluated using an EM algorithm to obtain maximum likelihood estimates followed by multiple imputation of complete data…