What is varimax rotation and why apply it to factor loadings?

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AcadiFi · Accepted Answer

Varimax is an orthogonal factor rotation that preserves orthogonality while maximizing the variance of squared loadings. Goal is to make each variable load heavily on one factor...

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Why rotate after PCA? Raw PCA factors are ordered by variance explained but loadings may spread across many variables, making interpretation fuzzy. For example, Factor 2 might have small positive loadings on tech stocks, medium positive on financials, and small negative on utilities — hard to name. After varimax, tech loads heavily on one rotated factor, financials on another — each factor has a clear identity.

Mechanics: if L is the original loading matrix, we find rotation matrix T (orthogonal, T'T=I) that maximizes Σⱼ [Σᵢ(Lᵢⱼ²)² − (1/n)(Σᵢ Lᵢⱼ²)²]. Rotated loadings = L × T.

Key properties preserved:

1. Factors remain uncorrelated (orthogonal rotation).
2. Total variance explained is unchanged.
3. Communalities (R² per variable) unchanged.

What changes: variance distribution across factors — no longer sorted descending. The factor you call "first" loses its privileged status.

Alternatives: oblique rotations (promax, oblimin) allow correlated factors, often yielding more realistic interpretations for social or economic data. In finance, orthogonality is often valued for hedging and risk decomposition, so varimax or quartimax remain popular.

Example: equity return PCA on 20 stocks. Factor 2 (raw) loads 0.3 on tech, 0.4 on banks, 0.2 on miners. After varimax, this rotates so one factor loads 0.7 on banks and near-zero elsewhere. Interpretation: "banking factor" emerges cleanly.

What is varimax rotation and why apply it to factor loadings?

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