Graduate School of Human Sciences, Osaka University, 1-2 Yamadaoka, Suita, Osaka, 565-0871, Japan.
Open University, Milton Keynes, UK.
Psychometrika. 2019 Dec;84(4):1048-1067. doi: 10.1007/s11336-019-09666-5. Epub 2019 Mar 7.
The factor analysis (FA) model does not permit unique estimation of the common and unique factor scores. This weakness is notorious as the factor indeterminacy in FA. Luckily, some part of the factor scores can be uniquely determined. Thus, as a whole, they can be viewed as a sum of determined and undetermined parts. The paper proposes to select the undetermined part, such that the resulting common factor scores have the following feature: the rows (i.e., individuals) of the common factor score matrix are as well classified as possible into few clusters. The clear benefit is that we can easily interpret the factor scores simply by focusing on the clusters. The procedure is called clustered common factor exploration (CCFE). An alternating least squares algorithm is developed for CCFE. It is illustrated with real data examples. The proposed approach can be viewed as a parallel to the rotation techniques in FA. They exploit another FA indeterminacy, the rotation indeterminacy, which is resolved by choosing the rotation that transforms the loading matrix into the 'most' interpretable one according to a pre-specified criterion. In contrast to the rotational indeterminacy, the factor indeterminacy is utilized to achieve well-clustered factor scores by CCFE. To the best of our knowledge, such an approach to the FA interpretation has not been studied yet.
因子分析（FA）模型不允许唯一估计共同和独特因子得分。这种弱点是众所周知的因子不确定性。幸运的是，因子得分的某些部分可以唯一确定。因此，它们可以整体上被视为确定部分和不确定部分的总和。本文提出选择不确定部分，以使得到的共同因子得分具有以下特征：共同因子得分矩阵的行（即个体）尽可能地分为少数几个聚类。明显的好处是，我们可以通过关注聚类轻松解释因子得分。该过程称为聚类共同因子探索（CCFE）。为 CCFE 开发了交替最小二乘算法。它用实际数据示例进行了说明。该方法可以看作是 FA 中旋转技术的并行方法。它们利用了另一个 FA 不确定性，即旋转不确定性，通过选择根据预定标准将加载矩阵转换为“最”可解释的旋转来解决。与旋转不确定性相反，CCFE 通过利用因子不确定性来实现聚类良好的因子得分。据我们所知，目前还没有研究过这种 FA 解释方法。
