Principal Component Analysis (PCA)
PCA reduces a set of correlated variables into uncorrelated linear combinations (principal components), each capturing maximum variance. The foundational technique for dimension reduction.
Step 1 — Provide your data
Observations x variables (numeric). Each row is one subject, each column one variable.
Supports Excel Import, Sample Data Generator, and Manual Entry.
Step 2 — Results
Suitability diagnostics:
- KMO measure with label (marvelous/meritorious/middling/mediocre/miserable/unacceptable)
- Bartlett's Test of Sphericity — chi-square, df, p-value
Variance explained:
- Eigenvalues, proportion explained, cumulative explained
- Suggested components to retain (Kaiser criterion: eigenvalue > 1)
Component loadings — correlations between variables and components
Communalities — proportion of variance explained per variable
Visualisations:
- Scree plot with Kaiser line
- Loadings table (colour-coded)
- Variance explained bar chart
A UniversalChatBot is available for discussion.
Statistical methods used
Standardisation — variables standardised to mean 0, SD 1 (PCA on correlation matrix).
Eigendecomposition — Jacobi rotation algorithm (max 100*p^2 iterations).
Kaiser criterion — retain components with eigenvalue > 1.
KMO
KMO = sum(R[i][j]^2) / (sum(R[i][j]^2) + sum(Q[i][j]^2)) where Q = partial correlations.
| KMO | Label |
|---|---|
| >= 0.90 | Marvelous |
| 0.80 - 0.90 | Meritorious |
| 0.70 - 0.80 | Middling |
| 0.60 - 0.70 | Mediocre |
| 0.50 - 0.60 | Miserable |
| < 0.50 | Unacceptable |
Bartlett's Test
chi2 = -((n-1) - (2p+5)/6) * ln(det(R)) with df = p(p-1)/2.
H0: correlation matrix = identity. p < 0.05 means PCA is appropriate.