Semiparametric estimation in the Rasch model

A method for estimating the parameters of the Rasch model is examined. The unknown quantities in this method are the item parameters and the distribution function of the latent trait over the population. In this sense, the method is equivalent to marginal maximum likelihood estimation. The new procedure is based on a method suggested by J. Kiefer and J. Wolfowitz (1956). Their conclusions are reviewed, and links to the Rasch model are specified. In marginal maximum likelihood estimation, the item parameters are estimated first, and then the prior distribution of the person parameters is estimated using these estimates. The proposed method illustrates that it is possible to estimate these two quantities together and arrive at consistent estimates

Modified Profile Likelihood for Fixed-Effects Panel Data Models

We show how modified profile likelihood methods, developed in the statistical literature, may be effectively applied to estimate the structural parameters of econometric models for panel data, with a remarkable reduction of bias with respect to ordinary likelihood methods. Initially, the implementation of these methods is illustrated for general models for panel data including individual-specific fixed effects and then, in more detail, for the truncated linear regression model and dynamic regression models for binary data formulated along with different specifications. Simulation studies show the good behavior of the inference based on the modified profile likelihood, even when compared to an ideal, although infeasible, procedure (in which the fixed effects are known) and also to alternative estimators existing in the econometric literature. The proposed estimation methods are implemented in an R package that we make available to the reader

Violations of ignorability in computerized adaptive testing

Using auxiliary information and allowing item review in computerized adaptive testing produces a violation of the ignorability principle for missing data (Rubin, 1976) that may bias parameter estimates in IRT models. However, the violation of ignorability does not automatically lead to bias. In this report, two situations are distinguished. 1. Estimation of the proficiency parameters in computerized adaptive testing using auxiliary information about proficiency and allowing item review, where the item parameters are considered known. Both analytically and through simulation studies, it is shown that the violation of ignorability does not lead to a gross inflation of bias. 2. Calibration of item and population parameters using maximum marginal likelihood estimation. Through simulation studies it is shown that violation of ignorability does result in bias. An analytical explanation of the result is given

Testing linear models for ability parameters in item response models

Methods for testing hypotheses concerning the regression parameters in linear models for the latent person parameters in item response models are presented. Three tests are outlined: A likelihood ratio test, a Lagrange multiplier test and a Wald test. The tests are derived in a marginal maximum likelihood framework. They are explicitly formulated for the 3-parameter logistic model, but it is shown that the approach applies to a broad class of item response models. Since the distributions of the test statistics are derived asymptotically, simulation studies were performed to assess the Type I error rates of the tests for small realistic sample sizes. Overall, the Type I error rates for the null hypothesis that a regression coefficient equals zero, were close to the nominal significance level. A number of power studies were conducted. It is argued that on theoretical grounds the power of the Lagrange multiplier test might be less than the power of the other two tests, but this expectationwas not corroborated. The robustness of the tests to violation of the item response model was investigated with simulation studies of the power and Type I error rate. The results showed that the performance of the tests was acceptable in the cases where local independence and the constancy of the discrimination parameters over treatment groups were violated to the same extent for all treatment groups. The simulation studies also showed that the tests were biased if local independence was violated for one of the treatment groups

A Comparison of Two MCMC Algorithms for Estimating the 2PL IRT Models

The fully Bayesian estimation via the use of Markov chain Monte Carlo (MCMC) techniques has become popular for estimating item response theory (IRT) models. The current development of MCMC includes two major algorithms: Gibbs sampling and the No-U-Turn sampler (NUTS). While the former has been used with fitting various IRT models, the latter is relatively new, calling for the research to compare it with other algorithms. The purpose of the present study is to evaluate the performances of these two emerging MCMC algorithms in estimating two two-parameter logistic (2PL) IRT models, namely, the 2PL unidimensional model and the 2PL multi-unidimensional model under various test situations. Through investigating the accuracy and bias in estimating the model parameters given different test lengths, sample sizes, prior specifications, and/or correlations for these models, the key motivation is to provide researchers and practitioners with general guidelines when it comes to estimating a UIRT model and a multi-unidimensional IRT model. The results from the present study suggest that NUTS is equally effective as Gibbs sampling at parameter estimation under most conditions for the 2PL IRT models. Findings also shed light on the use of the two MCMC algorithms with more complex IRT models