Choosing Between Varimax and Promax Rotations in Factor Analysis for Biological Data: A Practical Guide

Choosing between Varimax (orthogonal rotation) and Promax (oblique rotation) in factor analysis depends on the relationships between the factors in your dataset. Here's how you can identify which method to use:

Varimax (Orthogonal Rotation):

• Assumes factors are uncorrelated.
• If you believe or want the factors to be independent (i.e., they don't influence each other), Varimax is appropriate.
• Interpretability: Varimax simplifies the factors by maximizing the variance of squared loadings, making it easier to interpret the results (i.e., variables are strongly associated with only one factor).

Promax (Oblique Rotation):

• Assumes factors are correlated.
• If the factors are expected to be related (i.e., some underlying biological processes might influence multiple traits or variables), Promax is better.
• Interpretability: Promax allows factors to be correlated, which may reflect more realistic underlying biological relationships.

Steps to Decide Varimax vs. Promax:

1. Check Factor Correlations: After running an initial Exploratory Factor Analysis (EFA), check the correlations between the factors. If the correlations between factors are close to zero, use Varimax. If the correlations are substantial (e.g., >0.3), consider Promax.
2. Run Initial Factor Analysis Without Rotation: First, run an exploratory factor analysis without rotation to estimate the factors' correlations and see if they are independent or related.

                        # Install the psych package if necessary

                        install.packages("psych")

                        library(psych)

                        # Perform an initial factor analysis without rotation (using example data)

                        fa_result <- fa(your_data, nfactors=3, rotate="none")

                        # Check the factor correlations

                        fa_result$Phi  # This will show the correlations between the factors

• If the values in fa_result$Phi are close to zero, use Varimax.
• If the values are moderate to high (>0.3), consider using Promax.

    3. Compare Interpretability:

• After running both Varimax and Promax, compare the factor loadings. Sometimes Promax provides more interpretable results, especially in biological systems where factors are often correlated.

Example in R for Varimax and Promax Rotations:

Varimax Rotation Example (Orthogonal):

                        # Perform factor analysis with Varimax rotation

                        fa_varimax <- fa(your_data, nfactors=3, rotate="varimax")

                        # Print results

                        print(fa_varimax)

Promax Rotation Example (Oblique):

                        # Perform factor analysis with Promax rotation

                        fa_promax <- fa(your_data, nfactors=3, rotate="promax")

                        # Print results

                        print(fa_promax)

When to Use Varimax or Promax in Biological Data:

• Use Varimax when you expect the biological traits or variables to be independent. For example, in some ecological studies, environmental gradients like soil moisture and nutrient availability might be uncorrelated, so Varimax would simplify the interpretation.

• Use Promax when you expect the factors to be related. For example, in genetic studies, gene expression levels could be correlated, and using Promax would better reflect the underlying biology.

Conclusion:

1. First, check the correlations between the factors after an initial factor analysis without rotation.
2. If the factors are uncorrelated, use Varimax.
3. If the factors are correlated, use Promax for better interpretation of related factors.

By testing both and interpreting the factor loadings, you can determine which rotation provides the most meaningful biological insights from your dataset.