貝氏三階層IRT隨機截距之潛在迴歸模式的發展與應用

Implementation and Application of Bayesian Three-Level IRT Random Intercept Latent Regression Model

傳統試題反應理論中的潛在迴歸模式，自變項屬於觀察變項，其測量誤差往往被忽略。本研究旨在建構貝氏三階層IRT隨機截距項之潛在迴歸模式，除了透過一系列的模擬研究發展此模式之外，同時以兩個實徵研究說明此新模式之應用。本研究首先，將潛在自變項納入IRT的潛在迴歸模式中，同時考慮資料具有巢套的特性，並表徵在隨機截距項中。接著，模擬資料產生自貝氏三階層IRT隨機截距項之潛在迴歸模式，針對模式適配度進行探討。模式絕對性適配指標（PPMC）採用的是各受試群體在效標變項上平均觀察分數分布之標準差，而DIC則是用來進行模式競爭，以此兩個指標用來診斷資料與模式的適配程度。此外，在參數回復性方面，在貝氏三階層IRT隨機截距項之潛在迴歸模式分析下，各項參數回復性均佳，但如果忽略隨機截距項的存在，則會造成潛在迴歸方程式中殘差變異數估計值的嚴重偏誤，低估參數估計的分散程度，且高估測驗的信度。本研究也運用台灣教育長期追蹤資料庫（TEPS）的認知能力測驗，以及國中基本學力測驗（BCTEST）中數學科與英文科成續為例，作為兩個實徵研究來以說明貝氏三階層IRT隨機截距項之潛在迴歸模式之應用，最後提出理論與實務上相關的建議。

Multidimensional item response theory (MIRT) has received much attention developing into many different models. Traditionally, the standard IRT or MIRT models have two-level structure. Three-level IRT latent regression models were proposed with a MML algorithm, the predictors in IRT latent regression, however, have been assumed to be error-free. Recently, research has explored the application of a Bayesian IRT approach. This study aims to explore the Bayesian three-level IRT random intercept latent regression model (Bayesian 3L-IRT-RILRM) and assess the accuracy of its parameter recovery and efficiency. All of simulations were based on the one-parameter logistic model under the 3L-IRT- RILRM. Three tests with 20 items in each test were analyzed and 40 clusters, each containing 50 examinees, were simulated. The generated data sets were fitted to the four different models: the proposed model, the two-level latent regression model, the conventional MIRT model, and the conventional unidimensional IRT models respectively. The computer program WinBUGS with Metropolis-Hastings sampling was implemented to estimate model parameters. The Bayesian model-data fit checking techniques, such as posterior predictive model checking (PPMC), pseudo Bayes factor (PsBF) and Bayesian DIC, were implemented to choose which model was better. The results of PPMC produce an analytic index which can identify the 3L-IRT-RILRM as the best model. Furthermore the proposed model was considered best to describe the generated data through model comparison. The model parameter estimates were recovered fairly well in the framework of the Bayesian approach if the generated data was fitted to the proposed model. If the random intercept in latent regression was ignored, the parameter estimates would be biased and the precision of estimation, as well as the test reliability would be overestimated. Finally, two empirical data sets from the TEPS and BCTEST were used to illustrate the use of 3L-IRT-RILRM as the analytic model for comparison with other competitive models. 3L-IRT-RILRM is reliable and provides the most complete description of real data. Further studies and recommendations are addressed by the authors for extending more general models.