Hypothesis Tests for Principal Component Analysis When Variables are Standardized

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. Supplementary materials accompanying this paper appear online