层次线性模型中多重共线性的诊断

The Diagnostic of Collinearity in Hierarchical Linear Model

层次线性模型中的多重共线性问题有时是客观存在的。针对该问题,尝试通过对层次线性模型中参数估计的方差进行分解,并使用赖因施形式和奇异值分解的方法对设计阵与转换阵之间的关系进行论述,根据论述结果可知相对于设计阵,转化阵的奇异值会发生收缩,所以当设计阵不存在多重共线性问题时,可推知转化阵也必定不存在多重共线性问题,从而通过这种转化将层次线性模型中多重共线性的诊断转化为用现有的软件计算方差分解比和条件数就可解决的问题。

The multicollinearity in hierarchical linear models sometimes exist objectively. In order to solve this problem, we try to decompose the variance of parameter estimation in hierarchical linear model and discuss the relationship between design matrix and transformation matrix by Reinsch form and singular value decomposition. According to the results, we can know that the singular value of transformation matrix will shrink relative to design matrix, so if it is proved to be that there is no multicollinearity in the design matrix, we can infer that there must be no multicollinearity in the transformation matrix. As a result, the diagnosis of multicollinearity in hierarchical linear model can be transformed into a problem that can be solved through calculating variance decomposition proportions and condition indexes by existing software.