14.5 Rotating Factors

  • Find new factors that are easier to interpret
  • For each \(X\), we want some high/large (near 1) loadings and some low/small (near zero)
  • Two common rotation methods: Varimax rotation, and oblique rotation.

Same(ish) goal as PCA, find a new set of axes to represent the factors.

14.5.1 Varimax Rotation

  • Restricts the new axes to be orthogonal to each other. (Factors are independent)
  • Maximizes the sum of the variances of the squared factor loadings within each factor \(\sum Var(l_{ij}^{2}|F_{j})\)
  • Interpretations slightly less clear

Varimax rotation with principal components extraction.

Varimax rotation with maximum likelihood extraction.

  • communalities are unchanged after varimax (part of variance due to common factors). This will always be the case for orthogonal (perpendicular) rotations.

14.5.2 Oblique rotation

  • Same idea as varimax, but drop the orthogonality requirement
  • less restrictions allow for greater flexibility
  • Factors are still correlated
  • Better interpretation
  • Methods:
    • quartimax or quartimin minimizes the number of factors needed to explain each variable
    • direct oblimin standard method, but results in diminished interpretability of factors
    • promax is computationally faster than direct oblimin, so good for very large datasets

Varimax vs oblique here doesn’t make much of a difference, and typically this is the case. You almost always use some sort of rotation. Recall, this is a hypothetical example and we set up the variables in a distinct two-factor model. So this example will look nice.