1. Psychology 202b Advanced 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.