Hypothesis Tests for Principal Component Analysis When Variables are Standardized
Hypothesis Tests for Principal Component Analysis When Variables are Standardized
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DOI：
10.1007/s13253-019-00355-5
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发表时间：
2019-06-01
影响因子：
1.4
通讯作者：
Piepho, Hans-Peter
中科院分区：
文献类型：
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作者：
Forkman, Johannes;Josse, Julie;Piepho, Hans-Peter
在主成分分析（PCA）中，前几个主成分可能揭示数据中有趣的系统模式，而最后一个可能反映随机噪声。研究人员可能想知道有多少主成分具有统计学意义。已经提出了许多方法来确定在模型中保留多少主成分，但其中大多数假设非标准化数据。然而，在农业、生物和环境应用中，往往需要标准化。本文提出了在变量标准化的情况下主成分假设检验的参数自助法。与以前提出的方法不同，所提出的参数自举方法不依赖于任何需要大尺寸的渐近结果。在模拟研究中，建议的参数引导方法的标准化数据进行了比较，并行分析PCA和方法使用的Tracy-Widom分布。平行分析在测试第一主成分时表现良好，但在测试不反映随机噪声的高阶主成分时过于保守。当变量被标准化时，Tracy-Widom分布可能不近似最大特征值的分布。所提出的参数引导方法保持了近似的显著性水平，并且是使用Tracy-Widom分布的方法的两倍。为推荐的方法提供了SAS和R计算机代码。
In principal component analysis (PCA), the first few principal components possibly reveal interesting systematic patterns in the data, whereas the last may reflect random noise. The researcher may wonder how many principal components are statistically significant. Many methods have been proposed for determining how many principal components to retain in the model, but most of these assume non-standardized data. In agricultural, biological and environmental applications, however, standardization is often required. This article proposes parametric bootstrap methods for hypothesis testing of principal components when variables are standardized. Unlike previously proposed methods, the proposed parametric bootstrap methods do not rely on any asymptotic results requiring large dimensions. In a simulation study, the proposed parametric bootstrap methods for standardized data were compared with parallel analysis for PCA and methods using the Tracy-Widom distribution. Parallel analysis performed well when testing the first principal component, but was much too conservative when testing higher-order principal components not reflecting random noise. When variables are standardized, the Tracy-Widom distribution may not approximate the distribution of the largest eigenvalue. The proposed parametric bootstrap methods maintained the level of significance approximately and were up to twice as powerful as the methods using the Tracy-Widom distribution. SAS and R computer code is provided for the recommended methods.