This paper studies an information criterion for selecting covariance structure models using the generalized least squares (GLS) procedure. A risk assessed by the predictive GLS discrepancy function is introduced and used to determine the quality of a model. By correcting the biases in the sample GLS discrepancy function, four GLS discrepancy based information criteria are proposed. Monte Carlo results illustrate the merits of each criterion in model selection and in minimizing the risk.
The performance of the Gibbs sampling procedure for the three-parameter normal ogive (3PNO) IRT model was investigated using Monte Carlo simulations. Model parameters were estimated for tests with 10, 20, and 40 items and samples of 100, 300, 500, and 1000 examinees, where different actual values and prior specifications were considered for the item parameters. Summary statistics showed that this procedure was more affected by the choice of the prior distributions for the three-parameter model than the two-parameter model. For the 3PNO model, appropriate informative priors with relatively small spread should be adopted for the slope and intercept parameters to obtain more efficient and accurate MCMC estimates when sample sizes are not large and/or tests are not long enough. When it is not clear whether the prior information is appropriate, informative priors with small prior variances are not recommended.
Multiple correspondence analysis (MCA) is a useful tool for exploring the interdependencies among multiple-choice variables. However, MCA is not geared for explicitly investigating whether or not heterogeneous subgroups of respondents exist in the population with qualitatively distinct patterns of choice behaviour. In this paper, we extend MCA to capture such cluster-level heterogeneity. Specifically, the proposed method combines MCA with fuzzy k-means simultaneously. Consequently, it can provide a single map of displaying variable-level and cluster-level structures so as to facilitate the interpretation of the underlying structures. The performance of the proposed method in recovering true coordinates is investigated based on a Monte Carlo study involving synthetic data. In addition, two empirical applications are presented which compare the proposed method to two extant approaches that combine MCA and cluster analysis.
The Bradley-Terry model has been widely and effectively used to rank stimuli from paired comparison data. Existing approaches for paired comparison data analysis, however, have a number of limitations. First, among applied Bradley-Terry models in which multidimensionality is assumed, the effects of individual differences are not considered. Second, in these multidimensional Bradley-Terry models, the number of dimensions is generally evaluated only after analyzing several models with different dimensions separately, thus causing computational inconvenience. In this study, a multidimensional Bradley-Terry model that considers the effects of individual differences is proposed. The proposed model allows estimation of parameters for both scale values of stimuli and individual differences in multidimensional space. A procedure of parameter estimation is presented that uses a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm to estimate the optimal number of dimensions as well as associated parameters simultaneously. Simulation studies for examining the utility of the RJMCMC algorithm are performed, and real sports data from sumo are used in a representative example, where the time periods of tournaments are used as parameters for individual differences, in order to verify the validity of the proposed model.