PCA中输入变量与主成分的关系及RC2解释与因子数量疑问
Let’s walk through your PCA results and tackle your questions clearly:
Key Context from Your Output
First, let’s recap the critical numbers from your analysis:
- Your 7 variables have eigenvalues:
2.350,1.417,1.266,0.815,0.561,0.344,0.247 - You ran a varimax-rotated PCA with 3 factors, resulting in RC1, RC2, RC3 explaining 32%, 20%, and 20% of variance respectively (cumulative 72%)
Can We Retain Only 2 Factors?
Short answer: It’s not recommended, and here’s why:
- Variance explained: Dropping RC3 would cut your cumulative explained variance from 72% down to 52%—that’s a huge loss of information about your data’s underlying structure.
- Fit statistics: The test for 3 factors already shows a significant chi-square (
63.33, p < 1.1e-13), which means even 3 factors don’t fully capture the covariance in your data. Using only 2 would make this fit even worse, leaving far more unaccounted variance. - Eigenvalue rule: The standard Kaiser rule (retain factors with eigenvalues >1) supports keeping all 3 top factors, since the fourth eigenvalue drops below 1.
That said, if you must reduce to 2 factors for a specific use case, you’d lose the distinct variance captured by RC3 (which strongly correlates with variables C and D).
What Variables Should RC2 Be Associated With?
Looking at the standardized loadings (pattern matrix), RC2 has clear, meaningful associations with these variables:
- Variable B: Loading of
-0.83(the strongest single correlation with RC2) - Variable A: Loading of
0.69(a strong positive correlation) - Variable E: Loading of
0.42(moderate positive correlation)
RC2 essentially represents a contrast between B (negative loading) and A/E (positive loadings). Variables F and G have negligible loadings on RC2, while C and D barely correlate with it at all.
If RC2 feels "unexplainable," it might be because its variance contribution (20%) is lower than RC1’s 32%, but it still captures distinct variation tied to A, B, and E.
内容的提问来源于stack exchange，提问作者S. Oh
