Factor Analysis

Factor analysis is a powerful statistical methods technique used to uncover hidden patterns in complex datasets by identifying underlying constructs called factors that explain correlations between observed variables.

Core Principles

The fundamental premise of factor analysis rests on the assumption that many observed variables are correlated because they reflect the influence of common underlying factors. For example:

• Multiple test scores might correlate because they all measure general intelligence
• Various economic indicators might correlate due to underlying market conditions
• Different personality traits might correlate due to fundamental personality dimensions

Types of Factor Analysis

Exploratory Factor Analysis (EFA)

• Used when researchers lack strong theoretical predictions
• Discovers the factor structure naturally present in the data
• Helps generate new hypotheses about underlying constructs

Confirmatory Factor Analysis (CFA)

• Tests specific hypotheses about factor structure
• Requires strong theoretical foundation
• Uses structural equation modeling techniques to verify models

Mathematical Foundation

The basic model expresses each observed variable as a linear combination of underlying factors:

X = ΛF + ε

Where:

• X represents observed variables
• Λ (lambda) is the factor loading matrix
• F represents the common factors
• ε (epsilon) represents unique factors/error

Applications

Factor analysis finds wide application across multiple fields:

1. Psychology

   • psychometrics test development
   • Personality assessment
   • Intelligence research

2. Social Sciences

   • Survey development
   • Attitude measurement
   • construct validity assessment

3. Market Research

   • Consumer behavior analysis
   • Product preference studies
   • Market segmentation

Key Considerations

Sample Size

• Generally requires large samples
• Minimum N often cited as 300
• Subject-to-variable ratio typically 5:1 to 10:1

Assumptions

• multivariate normality
• Linear relationships between variables
• Meaningful correlations among variables

Interpretation

• Requires subject matter expertise
• Factor rotation for clearer structure
• Balance between parsimony and explanation

Limitations and Criticisms

• Subjective elements in factor selection
• Multiple possible solutions
• measurement error can affect results
• Requires careful theoretical grounding

Modern Developments

Recent advances include:

• Robust methods for non-normal data
• Bayesian approaches to factor analysis
• Integration with machine learning techniques
• dimensional reduction methods

Factor analysis remains a cornerstone of multivariate statistics, providing researchers with tools to understand complex patterns in their data and develop more refined measurement instruments.