Psychology 202b Advanced Psychological Statistics, II

1. Psychology 202bAdvanced Psychological Statistics, II April 7, 2011

2. The Plan for Today • Introduce exploratory factor analysis. • Historical applications in psychology. • Basic concepts, extraction, rotation. • Determining number of factors. • EFA in SAS. • EFA in Mplus.

3. Introduction • Basic idea: internal (unobservable) attributes exist. • Because they are not observable, they are referred to as “latent” variables. • Such variables are not unique to social sciences. For example, magnetism and gravity are latent variables in physics. • In factor analysis, latent variables are called “factors.”

4. Factors • Factors may be divided into two types: common factors and specific factors. • Example: if we have a test involving addition, subtraction, multiplication and division, a common factor might be arithmetic ability. • If we limit our attention to just addition operations, arithmetic ability would become a specific factor for addition skill.

5. Common and specific factors • The part of behavior on an item or scale that is systematic is determined by common and specific factors. • In addition, error of measurement affects scores on items or scales. • “Unique” behavior is the sum of specific factors and errors.

6. Variance • Just as performance can be broken down into the effects of common factors and unique effects… • …so can variance be divided into variance due to common factors (“communalities”) and variance due to specific factors and errors (“uniquenesses”).

7. Correlations • In factor analytic theory, correlations between observed variables are due entirely to common factors.

8. Historical applications in psychology • Many major theories in psychology have their origins in factor analysis. • Examples: • Intelligence • The MMPI • The Big Five model of personality

9. Basic concepts • Principal components analysis is similar to factor analysis, but all variability is communality. • Principal components analysis uses eigen analysis (or a related approach known as the singular value decomposition) to transform the coordinate system underlying a correlation matrix.

10. Principal components • PCA has a closed-form solution, and results in a re-expression of the variables into linear combinations that are uncorrelated and account for maximum remaining variance. • Often used as a data reduction technique with collinear data sets.

11. Factor analysis • Factor analysis is similar, but theorizes that communalities do not account for all observed variances. • Communalities are estimated. • No closed form solution exists. Rather, iterative estimation procedures are used (often maximum likelihood). • This estimation is often called “extraction”.

12. Factor rotation • Coordinate system is arbitrary. • Sometimes re-expressing it can aid interpretation. • Orthogonal and oblique rotation.

13. Determining the number of factors • Kaiser’s rule. • Scree plots. • Colloquium last year. • Interpretability. • Model fit.

14. Conducting EFA • …in SAS. • …in Mplus.

15. Next time • Confirmatory factor analysis.