典型相關分析(CCA)是一種用於衡量兩組變數之間線性關係的統計方法。其典型相關係數是透過變數集合的共變異數矩陣與交叉共變異數矩陣所進行的廣義特徵值分解而來，典型變數對則對基於對應的特徵向量建立。為了檢定典型相關是否顯著，目前最常使用的兩種檢定方法皆基於特徵值，分別為傳統的卡方檢定(假設維度固定且樣本數趨近無窮)與適用於高維條件下的 Tracy-Widom 檢定（假設維度與樣本數同時趨近無窮）。近年來，一種基於特徵向量的替代方法──變數填充下的維度 推論方法（DIVA），被發展出來，原先用於充分維度縮減框架下的維度 檢定。文中我們說明該方法也可應用於檢定 CCA 中的典型相關顯著性。為了評估這些方法在有限樣本下的表現，我們在不同的維度設定、樣本大小及相關強度條件下進行了綜合模擬研究。我們發現，當檢定兩組變數是否相關時，若相關性較低，卡方檢定表現較佳；若相關性較高，Tracy-Widom 檢定更為合適。而在估計顯著的典型變數對的個數時，當相關性較低時，卡方檢定效果較好，當相關性較高時，則以 s-DIVA 方法表現較佳。;Canonical Correlation Analysis (CCA) assesses linear relationships between two sets of variables. Canonical correlations are obtained via generalized eigen-decomposition of covariance and cross-covariance matrices, and canonical pairs are based on the corresponding eigenvectors. Two common eigenvalue-based significance tests are the traditional chi-square test (assuming fixed p and n → ∞) and the Tracy-Widom test (for high-dimensional settings where both p and n → ∞). In this work, the eigenvector-based “dimension inference using variable augmentation” (DIVA), originally developed for dimension testing in sufficient dimension reduction framework, is applied to CCA. We evaluates these methods via simulation studies with varying dimensions, sample sizes, and correlation strengths. Our numerical results show that the chi-square test performs better under weak correlations, while Tracy-Widom excels with strong correlations. For selecting number of significant canonical pairs, chi-square test is recommended for weak correlations, whereas DIVA is preferable for strong correlations.
