Grup İçi Varyanslar Heterojen Olduğunda Çok Düzeyli Madde Tepki Modelinin Bayes Yaklaşımı ile Modellenmesi

Abstract

In educational and behavioral sciences, it is very common to work with hierarchical data structures where individuals are nested in upper-level units such as schools, neighborhoods etc. Hierarchical data are even inevitable in large-scale assessments. Employing hierarchal linear models would be more efficient in such multilevel data conditions. However, this model family is appropriate only if the dependent variable is manifest. Thus, it would be more efficient to employ multilevel item response models (MIRMs), if the relevant variable is latent like “ability levels of examinees”. Conventional MIRMs assume homogeneity of within-group variances, just as conventional multilevel models in general. Nonetheless, assuming a constant variance for ability distributions of different classes or schools does not sound realistic. Moreover, heterogeneity of within-group variances should be regarded as an important source of information for school effectiveness studies rather than a violation of a model assumption. In this study, three-level conventional MIRM was re-modeled in order to estimate unique within-group variances. Bayesian approach was adopted for estimation, and parameter recovery was evaluated under varying levels of within-group heterogeneity through a simulation study. The proposed model and its conventional counterpart then were applied to a real dataset that were obtained from 2009 Turkish test of the Student Achievement Determination Exam administered in Turkey. Model-data fits, as well as, parameter estimates were compared across models. Lastly, conditional extension of the proposed model was applied to the data in order to explain possible sources of within-group variance heterogeneity. Depending on results of the simulation study, it was concluded that the proposed model’s parameter estimation was mostly efficient under moderate level heterogeneity, while it was acceptable under high level. When heterogeneity was low, especially the highest within-school variance parameters and variance of the within-school variances were not recovered efficiently. Nonetheless, there was no significant problems in recovery of the other model parameters. In the light of the mentioned simulation results, it is recommended to employ the conventional homogeneous MIRM in case of low within-group heterogeneity. Analyses on real data showed that the proposed model had a better model fit than the conventional MIRM. Moreover, schools that require an entry exam for admission had lower within-school variances and higher school-level abilities compared to the other ones. Such an inference should be regarded as a good sign of better school effectiveness for examination-required high schools.