B AD 6243: Applied Univariate Statistics

1. B AD 6243: Applied Univariate Statistics Factor Analysis Professor Laku Chidambaram Price College of Business University of Oklahoma

2. Factor Analysis • Is typically used to identify a small, stable and simple set of constructs that are not directly measurable or readily observable, but relate to objective or subjective data that can be measured • It involves “reducing” a large number of variables into a small set of factors that are easily interpretable • While a variety of methods exist for extracting factors, principal component analysis (PCA) is the most common • Thus, factors and components are generally used interchangeably BAD 6243: Applied Univariate Statistics

3. Building Blocks of Factor Analysis • Factors can be represented as regression equations such as, Y = 1Xi + 2X2 … + nXn, where the Xs represent predictor variables in the data set • Also note that a factor loading matrix shows the correlations between variables and factors • Further, factor coefficients represent factor loadings divided by correlation coefficients • Variance = Common + Unique + Error • Communality • Factor Analysis • Principal Components BAD 6243: Applied Univariate Statistics

4. “Rotating” to Get a Simple Solution Unrotated F1 F2 X1 .7 .5 X2 .6 .6 X3 .6 -.5 X4 .7 -.6 F2 X2 X1 F1 X3 X4 • Simple structures result when factor vectors pass close to (or preferably through) the coordinates • Orthogonal rotation, such as varimax, results in factors that are uncorrelated and parsimonious, but perhaps less natural • Oblique rotation, such as oblimin, results in factors that are correlated and natural, but perhaps more complicated BAD 6243: Applied Univariate Statistics

5. Orthogonal vs. Oblique Rotation F2 F1* Rotated F1 F2 X1 .7 -.1 X2 .7 .1 X3 .1 .5 X4 .2 .6 X2 X1 Orthogonal Rotation F1 X3 X4 F2* F2 Rotated F1 F2 X1 .7 -.1 X2 .7 .1 X3 0 .4 X4 .1 .5 F1* X2 X1 Oblique Rotation F1 X3 X4 F2* BAD 6243: Applied Univariate Statistics

6. Steps in Factor Analysis • Step 1: Compute correlation matrix (R) • Check inter-correlations (Are they high?) • Bartlett’s test of sphericity (Is it significant?) • KMO (Is it > 0.50?) and MSA (Are they > .50?) • Step 2: Extract factors based on results • Review eigenvalues • Examine percent of variance explained by factors • Inspect scree plot • Step 3: Rotate factors to make them more interpretable • Decide on type of rotation • Step 4: Compute factor scores to use in analysis • Step 5: Calculate reliability of factors BAD 6243: Applied Univariate Statistics

7. Step 1: Correlation Matrix (R) BAD 6243: Applied Univariate Statistics

8. Step 1a: R-1, KMO & Bartlett’s Test BAD 6243: Applied Univariate Statistics

9. Step 2: Extract Factors

10. Step 3a: Rotate Factors (Orthogonal) BAD 6243: Applied Univariate Statistics

11. Step 3b: Rotate Factors (Oblique) BAD 6243: Applied Univariate Statistics

12. Step 4: Compute Factor Scores Factor1 Factor2 Factor3 Factor4 -1.00559 -.95893 -.33848 -1.43688 -.92350 .57507 -.38387 -.26694 .17155 .37103 -.72022 .23342 .55386 -.94577 1.19587 .15282 -.59339 .01151 .24213 -.38924 .34062 .11352 1.48917 -.69486 -.89864 .98348 -.46061 -.12719 -.59846 .67102 -.42846 -.29728 2.06386 .59329 1.38374 3.22495 -.43152 1.73093 -.81954 -.08391 ….. ….. ….. ….. BAD 6243: Applied Univariate Statistics

13. Step 5: Calculate Reliability BAD 6243: Applied Univariate Statistics

14. Some Practical Guidelines • Minimum: • 2 variables are needed to identify one factor; 3 to 5 would be better for both measurement and replication • Sample size of 100 has been recommended by some; others have suggested a minimum sample size of 10 times the number of variables • Check: • For normality and outliers • Determinant of correlation matrix • Results of sphericity test • Number of factors: • Number of eigenvalues > 1 • Scree plot • Rotation: • Use orthogonal, if factors are theoretically unrelated • If not, use oblique • Loadings: • Absolute minimum of 0.30; some suggest 0.40 • Note that smaller loadings can be used with larger samples