Abstract: The use of computerized testing has enabled us to routinely record the response times (RTs) of test takers on test items. It has long been known that RTs are an important source of information on test takers and test items. For instance, RTs can be used to evaluate the speed of a test, to detect cheating behaviors, to improve the selection of items in a computerized adaptive testing (CAT) and to design better test. However, to make full use of the information contained in RTs, an appropriate statistical treatment of the RTs is required. The log-normal (LN) distribution has been most widely used to model the RTs from various tests. It permits the use of the nice statistical properties of a normal model for the log-transformed RTs. But the log transformed RTs do not always satisfy the normality assumption. Therefore, a more general approach for describing RT distribution would be preferred. One example is the Box-Cox normal (BCN) model proposed by Klein Entink, van der Linden, & Fox (2009), in which a power parameter is introduced to represent a number of different transformations. But the transformation parameter in the BCN model must be restricted to be common to all items in the test. Otherwise, things are different for item-specific transformations, that is, transformations result in item-specific scales. As a result, it is impossible to interpret differences between the item parameter estimates directly as differences between item characteristics. To do so, a more general model for RTs is required. Typically, RTs are non-negative, so their distribution is positively skewed. The skew normal distribution has been shown beneficial in dealing with asymmetric data in various theoretic and applied problems. And, to the author’s knowledge, it has not found application in the psychometric literature of RT modeling. Therefore, a log linear model for RTs is developed based on the skew normal distribution in this paper. The log skew normal (LSN) model is more flexible than BCN model, it permits researcher to choose an appropriate item-specific skewness parameter to describe the distribution of RTs for each item. Furthermore, the new model is embedded in a hierarchical framework in order to model responses and times simultaneously. A Bayesian approach with Markov chain Monte Carlo (MCMC) computation enables straightforward estimation of all model parameters. The deviance information criterion (DIC) and the Bayesian residual analysis methods are developed to compare the model fit between models with different parameter structure. Two simulation studies are constructed to explore the excellence of the LSN model and the performance of the proposed MCMC sampling algorithm. The results obtained in the simulation studies indicated that the LSN model is better than the BCN model and the LN model. The flexibility of the RT model can be improved by using the skew normal distribution, which can give researcher more freedom to fit different distributional shapes to the RT data. Furthermore, it was found that the MCMC algorithm performed well and enabled simultaneous estimation. Finally, our approach is empirically studied by a real-data example in the personality measurement, and the obtained results also indicated that the LSN model as RTs model presented best fit.

Key words: item response times, skew normal distribution, MCMC sampling algorithm, hierarchical model

摘要： 近年来，项目反应时间数据的建模是心理和教育测量领域的热门方向之一。针对反应时间的对数正态模型和Box-Cox正态模型的不足，本文在van der Linden的分层模型框架下基于偏正态分布建立一个反应时间的对数线性模型，并成功给出模型参数估计的马尔科夫链蒙特卡罗（Markov Chain Monte Carlo, MCMC）算法。模拟研究和实例分析的结果均表明，与对数正态模型和Box-Cox正态模型相比，对数偏正态模型表现出更加优良的拟合效果，具有更强的灵活性和适用性。

关键词: 项目反应时间, 偏正态分布, MCMC抽样算法, 分层模型