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doi: 10.1007/s11336-008-9092-xpmid: N/A
It is common practice in IRT to consider items as fixed and persons as random. Both, continuous and categorical person parameters are most often random variables, whereas for items only continuous parameters are used and they are commonly of the fixed type, although exceptions occur. It is shown in the present article that random item parameters make sense theoretically, and that in practice the random item approach is promising to handle several issues, such as the measurement of persons, the explanation of item difficulties, and trouble shooting with respect to DIF. In correspondence with these issues, three parts are included. All three rely on the Rasch model as the simplest model to study, and the same data set is used for all applications. First, it is shown that the Rasch model with fixed persons and random items is an interesting measurement model, both, in theory, and for its goodness of fit. Second, the linear logistic test model with an error term is introduced, so that the explanation of the item difficulties based on the item properties does not need to be perfect. Finally, two more models are presented: the random item profile model (RIP) and the random item mixture model (RIM). In the RIP, DIF is not considered a discrete phenomenon, and when a robust regression approach based on the RIP difficulties is applied, quite good DIF identification results are obtained. In the RIM, no prior anchor sets are defined, but instead a latent DIF class of items is used, so that posterior anchoring is realized (anchoring based on the item mixture). It is shown that both approaches are promising for the identification of DIF.
doi: 10.1007/s11336-008-9071-2pmid: N/A
Samejima (Psychometrika 65:319–335, 2000) proposed the logistic positive exponent family of models (LPEF) for dichotomous responses in the unidimensional latent space. The objective of the present paper is to propose and discuss a graded response model that is expanded from the LPEF, in the context of item response theory (IRT). This specific graded response model belongs to the general framework of graded response model (Samejima, Psychometrika Monograph, No. 17, 1969 and No. 18, 1972; Handbook of modern item response theory, Springer, New York, 1997; Encyclopedia of Social Measurement, Academic Press, San Diego, 2004), and, in particular to the heterogeneous case (Samejima, Psychometrika Monograph, No. 18, 1972). Thus, the model can deal with any number of ordered polytomous responses, such as letter grades (e.g., A, B, C, D, F), etc.
doi: 10.1007/s11336-008-9083-ypmid: N/A
The infinitesimal jackknife provides a simple general method for estimating standard errors in covariance structure analysis. Beyond its simplicity and generality what makes the infinitesimal jackknife method attractive is that essentially no assumptions are required to produce consistent standard error estimates, not even the requirement that the population sampled has the covariance structure assumed. Commonly used covariance structure analysis software uses parametric methods for estimating parameters and standard errors. When the population sampled has the covariance structure assumed, but fails to have the distributional form assumed, the parameter estimates usually remain consistent, but the standard error estimates do not. This has motivated the introduction of a variety of nonparametric standard error estimates that are consistent when the population sampled fails to have the distributional form assumed. The only distributional assumption these require is that the covariance structure be correctly specified. As noted, even this assumption is not required for the infinitesimal jackknife. The relation between the infinitesimal jackknife and other nonparametric standard error estimators is discussed. An advantage of the infinitesimal jackknife over the jackknife and the bootstrap is that it requires only one analysis to produce standard error estimates rather than one for every jackknife or bootstrap sample.
de la Torre, Jimmy; Douglas, Jeffrey
doi: 10.1007/s11336-008-9063-2pmid: N/A
This paper studies three models for cognitive diagnosis, each illustrated with an application to fraction subtraction data. The objective of each of these models is to classify examinees according to their mastery of skills assumed to be required for fraction subtraction. We consider the DINA model, the NIDA model, and a new model that extends the DINA model to allow for multiple strategies of problem solving. For each of these models the joint distribution of the indicators of skill mastery is modeled using a single continuous higher-order latent trait, to explain the dependence in the mastery of distinct skills. This approach stems from viewing the skills as the specific states of knowledge required for exam performance, and viewing these skills as arising from a broadly defined latent trait resembling the θ of item response models. We discuss several techniques for comparing models and assessing goodness of fit. We then implement these methods using the fraction subtraction data with the aim of selecting the best of the three models for this application. We employ Markov chain Monte Carlo algorithms to fit the models, and we present simulation results to examine the performance of these algorithms.
Shedden, Kerby; Zucker, Robert
doi: 10.1007/s11336-008-9077-9pmid: 19956348
Finite mixture models are widely used in the analysis of growth trajectory data to discover subgroups of individuals exhibiting similar patterns of behavior over time. In practice, trajectories are usually modeled as polynomials, which may fail to capture important features of the longitudinal pattern. Focusing on dichotomous response measures, we propose a likelihood penalization approach for parameter estimation that is able to capture a variety of nonlinear class mean trajectory shapes with higher precision than maximum likelihood estimates. We show how parameter estimation and inference for whether trajectories are time-invariant, linear time-varying, or nonlinear time-varying can be carried out for such models. To illustrate the method, we use simulation studies and data from a long-term longitudinal study of children at high risk for substance abuse.
Steinley, Douglas; Hubert, Lawrence
doi: 10.1007/s11336-008-9058-zpmid: N/A
This paper proposes an order-constrained K-means cluster analysis strategy, and implements that strategy through an auxiliary quadratic assignment optimization heuristic that identifies an initial object order. A subsequent dynamic programming recursion is applied to optimally subdivide the object set subject to the order constraint. We show that although the usual K-means sum-of-squared-error criterion is not guaranteed to be minimal, a true underlying cluster structure may be more accurately recovered. Also, substantive interpretability seems generally improved when constrained solutions are considered. We illustrate the procedure with several data sets from the literature.
doi: 10.1007/s11336-008-9067-ypmid: N/A
Methods of incorporating a ridge type of regularization into partial redundancy analysis (PRA), constrained redundancy analysis (CRA), and partial and constrained redundancy analysis (PCRA) were discussed. The usefulness of ridge estimation in reducing mean square error (MSE) has been recognized in multiple regression analysis for some time, especially when predictor variables are nearly collinear, and the ordinary least squares estimator is poorly determined. The ridge estimation method was extended to PRA, CRA, and PCRA, where the reduced rank ridge estimates of regression coefficients were obtained by minimizing the ridge least squares criterion. It was shown that in all cases they could be obtained in closed form for a fixed value of ridge parameter. An optimal value of the ridge parameter is found by G-fold cross validation. Illustrative examples were given to demonstrate the usefulness of the method in practical data analysis situations.
doi: 10.1007/s11336-008-9066-zpmid: N/A
Every set of alternate weights (i.e., nonleast squares weights) in a multiple regression analysis with three or more predictors is associated with an infinite class of weights. All members of a given class can be deemed fungiblebecause they yield identical SSE (sum of squared errors) and R 2 values. Equations for generating fungible weights are reviewed and an example is given that illustrates how fungible weights can be profitably used to evaluate parameter sensitivity in multiple regression.
doi: 10.1007/s11336-008-9062-3pmid: N/A
Uniform sampling of binary matrices with fixed margins is known as a difficult problem. Two classes of algorithms to sample from a distribution not too different from the uniform are studied in the literature: importance sampling and Markov chain Monte Carlo (MCMC). Existing MCMC algorithms converge slowly, require a long burn-in period and yield highly dependent samples. Chen et al. developed an importance sampling algorithm that is highly efficient for relatively small tables. For larger but still moderate sized tables (300×30) Chen et al.’s algorithm is less efficient. This article develops a new MCMC algorithm that converges much faster than the existing ones and that is more efficient than Chen’s algorithm for large problems. Its stationary distribution is uniform. The algorithm is extended to the case of square matrices with fixed diagonal for applications in social network theory.
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