ISSN 0439-755X
CN 11-1911/B

心理学报 ›› 2014, Vol. 46 ›› Issue (12): 1897-1909.doi: 10.3724/SP.J.1041.2014.01897

• 论文 • 上一篇    下一篇



  1. (1华南师范大学心理学院、心理应用研究中心, 广州 510631) (2暨南大学管理学院, 广州 510632) (3华南师范大学教育科学学院, 广州 510631)
  • 收稿日期:2013-07-18 出版日期:2014-12-25 发布日期:2014-12-25
  • 通讯作者: 张敏强, E-mail:; 张文怡,; 黎光明, E-mail:
  • 基金资助:

    全国教育科学“十二五”规划教育部重点课题(GFA111009)、2014年国家自然科学基金面上项目(31470050)、国家社会科学基金“十二五”规划教育学一般课题(BHA130053)、教育部人文社会科学研究青年基金项目(12YJC190016)、广东省教育科学“十二五”规划2011年度研究项目(2011TJK161)、广东省高等院校学科建设专项资金项目育苗工程(人文社科) (2012WYM_0108)、广州市教育科学十二五规划2012年度面上一般课题(12A019)。

Estimating Variance Components of Missing Data for Generalizability Theor

ZHANG Minqiang1; ZHANG Wenyi2; LI Guangming1; LIU Xiaoyu3; Huang Feifei1   

  1. (1 School of Psychology, Center for Studies of Psychological Application, South China Normal University, Guangzhou 510631, China) (2 Management School, Jinan University, Guangzhou 510632, China) (3 School of Education Science, South China Normal University, Guangzhou 510631, China)
  • Received:2013-07-18 Online:2014-12-25 Published:2014-12-25
  • Contact: ZHANG Minqiang, E-mail:; ZHANG Wenyi,; LI Guangming, E-mail:


各种心理调查、心理实验中, 数据的缺失随处可见。由于数据缺失, 给概化理论分析非平衡数据的方差分量带来一系列问题。基于概化理论框架下, 运用Matlab 7.0软件, 自编程序模拟产生随机双面交叉设计p×i×r缺失数据, 比较和探讨公式法、REML法、拆分法和MCMC法在估计各个方差分量上的性能优劣。结果表明:(1) MCMC方法估计随机双面交叉设计p×i×r缺失数据方差分量, 较其它3种方法表现出更强的优势; (2) 题目和评分者是缺失数据方差分量估计重要的影响因素。

关键词: 概化理论, 缺失数据, 方差分量估计, p×i×r设计, MCMC方法


Missing observations are common in operational performance assessment settings or psychological surveys and experiments. Since these assessments are time-consuming to administer and score, examinees seldom respond to all test items and raters seldom evaluate all examinee responses. As a result, a frequent problem encountered by those using generalizability theory with large-scale performance assessments is working with missing data. Data from such examinations compose a missing data matrix. Researchers usually concern about how to make good use of the full data and often ignore missing data. As for these missing data, a common practice is to delete them or make an imputaion for missing records; however, it may cause problems in following aspects. Firstly, deleting or interpolating missing data may result in ineffective statistical analysis. Secondly, it is difficult for researchers to choose an unbiased method among diverse rules of interpolation. As a result of missing data, a series of problems may be caused when estimating variance components of unbalanced data in generalizability theory. A key issue with generalizability theory lies in how to effectively utilize the existing missing data to their maximum statistical analysis capacity. This article provides four methods to estimate variance components of missing data for unbalanced random p×i×r design of generalizability theory: formulas method, restricted maximum likelihood estimation (REML) method, subdividing method, and Markov Chain Monte Carlo (MCMC) method. Based on the estimating formulas of p×i design by Brennan (2001), formulas method is the deduction of estimating variance components formulas for p×i×r design with missing data. The aim of this article is to investigate which method is superior in estimating variance components of missing data rapidly and effectively. MATLAB 7.0 was used to simulate data, and generalizability theory was used to estimate variance components. Three conditions were simulated respectively: (1) persons sample with small size (200 students), medium size (1000 students) and large size (5000 students); (2) item sample with 2 items, 4 items and 6 items; (3) raters sample with 5 raters, 10 raters and 20 raters. The authors also developed some programs for MATLAB, WinBUGS, SAS and urGENOVA software in order to estimate variance components of p×i×r missing data with four methods. Criterions were made for the purpose of comparing the four methods. For example, bias was the criterion when estimating variance components. The reliability of the results increased as the absolute bias decreased. Results indicate that: (1) MCMC method has a strong advantage for estimating variance components of p×i×r missing data over the other three methods. MCMC method is superior to formulas method because of smaller deviation for variance components estimation. It is better than REML method because iteration of MCMC method converge, while REML method does not. Unlike subdividing method, MCMC method does not require variance components to be combined in order to obtain accurate estimations. (2) Item and rater are two important influencing factors for estimating variance components of missing data. If manpower and material resources are limited, priority should be given to increase the number of items in order to increase estimation accuracy. If researchers cannot increase the number of items, the next-best thing is to increase the number of raters. However, the number of raters should be cautiously controlled.

Key words: Generalizability Theory, missing data, estimating variance components, p×i×r design, Markov Chain Monte Carlo (MCMC)