Statistical method for testing principal components

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. Parametric bootstrap methods can be used for hypothesis testing of principal components when variables are standardized.

The article Hypothesis tests for principal component analysis when variables are standardized by authors Johannes Forkman, Julie Josse and Hans-Peter Piepho is available at SpringerLink https://link.springer.com/article/10.1007/s13253-019-00355-5.