Rank Factor Score in Factor Analysis

Thurstone Regression Scores

Factor scores are used to estimate individuals’ positions on unobserved latent variables based on their responses to observed items. Traditionally, these are computed using:

📘 Traditional Factor Scores

• Thurstone Regression Scores (Ordinary Least Squares, OLS): ˆF(T)i=ΦΛ′Σ−1Yi
  • ˆF(T)i = estimated factor scores for individual i
  • Yi = vector of observed responses
    
  • Λ = factor loading matrix
    
  • Φ = factor covariance matrix
    
  • Σ = model-implied covariance of observed variables
    🔹 Minimizes squared differences between observed and predicted variables, but can be biased if the model is misspecified.

Bartlett Factor Scores

• Bartlett Scores (Maximum Likelihood, ML): ˆF(B)i=(Λ′Ψ−1Λ)−1Λ′Ψ−1Yi
  • Ψ = diagonal matrix of residual variances (uniquenesses)
    🔹 Accounts for measurement error by giving less weight to noisy items.
    🔹 More precise under correct model assumptions, but sensitive to violations (e.g., error correlations or cross-loadings).

Network scores

🌐 Network Scores

• Inspired by network psychometrics: ^NSi=HS⋅Yi
  • HS = hybrid centrality scores (e.g., rank the importance of items in terms of items’ closeness, betweenness, and influences in the network)
  • Ranks items by influence within an item correlation network.

Proposed Solution: Rank Factor Score

🚀 Rank Factor Score: Our Proposed Method

• Use ranked factor loadings to compute factor scores, rather than using their original loading estimates: ˆF(R)i=rank(Λ)′⋅Yi

  • rank(Λ) = a function that ranks factor loadings

  🔹 Highest loading → highest rank → greatest weight
  🔹 Ranking can be global ranking (ranks all loadings) or factor-level ranking (rank loadings within each latent factor)

Proposed Solution: Rank Factor Score (Cont.)

✅ Why Rank?

• Ranking reduces sensitivity to:
  • Estimation noise in Λ and Ψ
  • Violations of normality or residual independence
  • Cross-loading items
• Connects to network-based intuition: “Importance, not exact magnitude”
  
• Easy to compute and interpret
  
• Particularly robust under model misspecification

Traditional vs. Rank Scores

Simulation Overview

Analysis and Evaluation Metrics

All analysis models are simple structured models.

Key Findings

Visual Summary for Study 1

Correlation Between Estimated and True Factor Scores for Uniform Distributions

Correlation Between Estimated and True Factor Scores for Uniform Distributions

Visual Summary for Study 1 (Cont.)

Correlation Between Estimated and True Factor Scores for Normal Distributions

Correlation Between Estimated and True Factor Scores for Normal Distributions

Visual Summary for Study 1 (Cont.)

Model Fit Indices (CFI, TLI, RMSEA) Comparison for Uniform Distributions

Model Fit Indices (CFI, TLI, RMSEA) Comparison for Uniform Distributions

Visual Summary for Study 1 (Cont.)

Model Fit Indices (CFI, TLI, RMSEA) Comparison for Normal Distributions

Model Fit Indices (CFI, TLI, RMSEA) Comparison for Normal Distributions

Visual Summary for Study 2

Correlation Between Estimated and True Factor Scores of factor 1 for Uniform Distributions

Correlation Between Estimated and True Factor Scores of factor 1 for Uniform Distributions

Visual Summary for Study 2 (Cont.)

Correlation Between Estimated and True Factor Scores of factor 1 for Normal Distributions

Correlation Between Estimated and True Factor Scores of factor 1 for Normal Distributions

Visual Summary for Study 2 (Cont.)

Model Fit Indices (CFI, TLI, RMSEA) Comparison of factor 1 for Uniform Distributions

Model Fit Indices (CFI, TLI, RMSEA) Comparison of factor 1 for Uniform Distributions

Visual Summary for Study 2 (Cont.)

Model Fit Indices (CFI, TLI, RMSEA) Comparison of factor 1 for Normal Distributions

Model Fit Indices (CFI, TLI, RMSEA) Comparison of factor 1 for Normal Distributions

Practical Implications

Future Directions

Thank You!

Questions?
Email: jili@uark.edu