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 Rasch Poisson counts model is a latent trait model for the situation in which "K" tests are administered to "N" examinees and the test score is a count (repeated number of some event). A mixed model is presented that applies the EM algorithm and that can allow for missing data. (SLD)

Missing data analyses have received considerable recent attention in the methodological literature, and two "modern" methods, multiple imputation and maximum likelihood estimation, are recommended. The goals of this article are to (a) provide an overview of missing-data theory, maximum likelihood estimation, and multiple imputation; (b) conduct a…

Less than optimum strategies for missing values can produce biased estimates, distorted statistical power, and invalid conclusions. After reviewing traditional approaches (listwise, pairwise, and mean substitution), selected alternatives are covered including single imputation, multiple imputation, and full information maximum likelihood…

According to Kenny and McCoach (2003), chi-square tests of structural equation models produce inflated Type I error rates when the degrees of freedom increase. So far, the amount of this bias in large models has not been quantified. In a Monte Carlo study of confirmatory factor models with a range of 48 to 960 degrees of freedom it was found that…

A large sample method for obtaining asymptotic simultaneous confidence bands for a three-parameter logistic response curve is described. Simultaneous confidence bands indicate the sampling variation of item response curves relative to a fitted function. A procedure is given which requires as input maximum likelihood parameter estimates and an…

The success of applications of item response theory (IRT) depends upon the properties of the estimates of model parameters. Many theoretical properties of these estimates have been extensively studied. However, the properties of estimates obtained empirically from real data depend not only on the theoretical results, but also on the data and the…

L. V. DiBello, W. F. Stout, and L. A. Roussos (1993) have developed a new item response model, the Unified Model, which brings together the discrete, deterministic aspects of cognition favored by cognitive scientists, and the continuous, stochastic aspects of test response behavior that underlie item response theory (IRT). The Unified Model blends…

This paper investigates methods that avoid using multiple groups to represent the missing data patterns in covariance structure modeling, attempting instead to do a single-group analysis where the only action the analyst has to take is to indicate that data is missing. A new covariance structure approach developed by B. Muthen and G. Arminger is…

In this paper it is shown that various violations of the two parameter logistic (2PL) model can be evaluated using the Lagrange multiplier test (J. Aitchison and S. Silvey, 1958) or the equivalent difference score test. The tests focus on violation of local stochastic independence and insufficient capture of the form of the item characteristic…

Multinomial-response models are available that correspond implicitly to tests in which a total score is computed as the sum of polytomous item scores. For these models, joint and conditional estimation may be considered in much the same way as for the Rasch model for right-scored tests. As in the Rasch model, joint estimation is only attractive if…

A model is presented for item responses when different examinees use different strategies to arrive at their answers and when only those answers, not choice or strategy or subtask results, can be observed. Using substantive theory to differentiate the likelihoods of response vectors under a fixed set of solution strategies, responses are modeled…

Formulas are derived for the bias in the maximum likelihood estimators (MLE) of the item parameters in the logistic item response model when examinee abilities are known. Numerical results are given for a typical verbal test for college admission. Most typically the bias of an MLE is about one-tenth of its standard error. It is very seldom more…

This paper is concerned with estimation and hypothesis testing of treatment effects in nonequivalent control group designs with the assumption that in the absence of treatment effects, natural growth conforms to a particular class of continuous growth models. Point estimation, interval estimation, and hypothesis testing procedures were developed…

A possible method is developed for computing the asymptotic sampling variance-covariance matrix of joint maximum likelihood estimates in item response theory when both item parameters and abilities are unknown. For a set of artificial data, results are compared with empirical values and with the variance-covariance matrices found by the usual…