CAJ | 학술논문

考试评分缺失数据较为常见，如何有效利用现有数据进行统计分析是个关键性问题。在考试评分中，题目与评分者对试卷得分的影响不容忽视。根据概化理论原理，按考试评分规则推导出含有缺失数据双侧面交叉设计（P×i×r）方差分量估计公式，用Matlab7．0软件模拟多组缺失数据，验证此公式的有效性。结果发现：（1）推导出的公式较为可靠，估计缺失数据的方差分量偏差相对较小，即便数据缺失率达到50％以上，公式仍能对方差分量进行较为准确地估计；（2）题目数量对概化理论缺失数据方差分量的估计影响最大，评分者次之，当题目数量和评价者数量分别为6和5时，公式能够趋于稳定地估计；（3）学生数量对各方差分量的估计影响较小，无论是小规模考试还是大规模考试，概化理论估计缺失数据的多个方差分量结果相差不大。
고시평분결실수거교위상견，여하유효이용현유수거진행통계분석시개관건성문제。재고시평분중，제목여평분자대시권득분적영향불용홀시。근거개화이론원리，안고시평분규칙추도출함유결실수거쌍측면교차설계（P×i×r）방차분량고계공식，용Matlab7．0연건모의다조결실수거，험증차공식적유효성。결과발현：（1）추도출적공식교위가고，고계결실수거적방차분량편차상대교소，즉편수거결실솔체도50％이상，공식잉능대방차분량진행교위준학지고계；（2）제목수량대개화이론결실수거방차분량적고계영향최대，평분자차지，당제목수량화평개자수량분별위6화5시，공식능구추우은정지고계；（3）학생수량대각방차분량적고계영향교소，무론시소규모고시환시대규모고시，개화이론고계결실수거적다개방차분량결과상차불대。
Missing data are easily found in psychological surveys and experiments such as test scores. For example, in performance as- sessment, a certain group of raters rate a certain group of examinees. By this token, the data from performance assessment compose a sparse data matrix. Researchers are always concerned about how to make good use of the observed data. Brennan （2001） provided the estimating formulas of p x i design of sparse data. But in practice, there is always more than one factor which has an effect on the exper- iment such as the factor of rater and cannot be ignored in the performance assessment. The aim of this article is to find a way that can estimate the variance component of sparse data quickly and effectively. In China, many studies only analyzed complete data and ignored sparse data. There are two merits. Firstly, when facing missing data, researchers usually deleted incomplete records or used imputation before analysis. But using these methods to analyze performance assessment will lose sight of the data which can be used for analysis. Secondly, the estimated value will differ along with different imputation methods. This article provided the estimating formulas of p × i × r design of sparse data, based on the estimating formulas of p x i design of sparse data provided by Brennan （2001）. This article used MATLAB 7.0 to simulate data which were usually encountered in examination, then used the generalizability the- ory to estimate variance components. We simulated two conditions respectively, a small size with 200 students and a large size with 10000 students. We then used the estimating formulas of p × i × r design of sparse data to estimate variance components in order to test the formulas＇ validity. The research showed that these formulas provided a good estimation of variance components. The estimated variance components approach to set values. The accuracy rates of item and rater were the highest. The accuracy rates of interaction of students and items was low. The maximum bias of interaction reached 1.3. The number of items had the most important effect on the estimation. If the number of item increased a little, then the accuracy rate increased by a big margin. These formulas provided a good estimation when the amount of items was moderate. We also found that these formulas could be used in either a small or a large number of data. Either kind of data had a little bias. We can increase the number of items to enhance the accuracy rate of variance components. If researchers cannot in- crease the number of items, they can increase the number of raters instead, which can also enhance the accuracy rate. But the number of raters cannot be too large. It can get a little bias when the number of raters reaches five.