Andrich David, Luo Guanzhong
School of Education, Murdoch University, Murdoch, WA 6150, Australia.
J Appl Meas. 2003;4(3):205-21.
In the Rasch model for items with more than two ordered response categories, the thresholds that define the successive categories are an integral part of the structure of each item in that the probability of the response in any category is a function of all thresholds, not just the thresholds between any two categories. This paper describes a method of estimation for the Rasch model that takes advantage of this structure. In particular, instead of estimating the thresholds directly, it estimates the principal components of the thresholds, from which threshold estimates are then recovered. The principal components are estimated using a pairwise maximum likelihood algorithm which specialises to the well known algorithm for dichotomous items. The method of estimation has three advantageous properties. First, by considering items in all possible pairs, sufficiency in the Rasch model is exploited with the person parameter conditioned out in estimating the item parameters, and by analogy to the pairwise algorithm for dichotomous items, the estimates appear to be consistent, though unlike for the dichotomous case, no formal proof has yet been provided. Second, the estimates of each item parameter is a function of frequencies in all categories of the item rather than just a function of frequencies of two adjacent categories. This stabilizes estimates in the presence of low frequency data. Third, the procedure accounts readily for missing data. All of these properties are important when the model is used for constructing variables from large scale data sets which must account for structurally missing data. A simulation study shows that the quality of the estimates is excellent.
在具有两个以上有序响应类别的项目的拉施模型中，定义连续类别的阈值是每个项目结构的一个组成部分，因为任何类别的响应概率是所有阈值的函数，而不仅仅是任意两个类别之间的阈值的函数。本文描述了一种利用这种结构的拉施模型估计方法。具体而言，该方法不是直接估计阈值，而是估计阈值的主成分，然后从主成分中恢复阈值估计。主成分使用成对最大似然算法进行估计，该算法专门用于二分项目的著名算法。该估计方法具有三个优点。首先，通过考虑所有可能的项目对，在估计项目参数时，以人员参数为条件利用了拉施模型中的充分性，并且类似于二分项目的成对算法，估计似乎是一致的，尽管与二分情况不同，尚未提供正式证明。其次，每个项目参数的估计是项目所有类别的频率的函数，而不仅仅是两个相邻类别的频率的函数。这在低频数据存在的情况下稳定了估计。第三，该过程很容易处理缺失数据。当该模型用于从大规模数据集中构建变量时，所有这些属性都很重要，因为大规模数据集必须考虑结构上的缺失数据。一项模拟研究表明，估计的质量非常好。
