Exploratory Factor Analysis

1. Exploratory Factor Analysis • Purpose • EFA Process • Russell’s (2002) Recommendations • “Take-Home” Practical for next week • No in-class session next Monday

2. EFA Purpose • Data Reduction • Creating smaller number of variables (factors or components) from a larger set • Scale Construction • Does the scale have the appropriate number of factors • Do the items fall on the factors as intended?

3. EFA Purpose • Determine the number and “content” of factors that underlie a set of observed variables • Used when factor structure is unknown or uncertain • “Order out of chaos” • Examples • Intelligence Testing • Personality Testing

4. Raymond Cattell Basic idea: “important” traits will be well-represented in language Lexical criterion 18,000 personality-relevant terms taken from dictionary Reduced (eventually) to 16 trait dimensions Clusters of interrelated traits Determined by factor analysis EFA and Personality

5. Factor Analysis Neat Nervous Happy Polite Outspoken Hardworking Kind Responsible Energetic Friendly Intelligent

6. Factor 3 Factor 1 Factor 2 Factor Analysis Polite Nervous Kind Friendly Intelligent Energetic Hardworking Outspoken Happy Neat Responsible

7. Five Factor Model Personality = five “basic” traits Convergence on same five traits by: different researchers different measures different populations different languages All rely on factor analysis EFA and Personality Modern Consensus Extraversion Agreeableness Neuroticism Conscientiousness Openness

8. Factor Analysis... • Is Useful • Reduces number of dimensions • Has Drawbacks • Content and number of factors are item-dependent • garbage in; garbage out • Labeling of factors is highly subjective • jingle-jangle problem

9. Polite Nervous Kind Friendly Competent Factor 3 Intelligent Smart Factor 1 Factor 2 Energetic Hardworking Outspoken Happy Neat Responsible Number of Factors

10. Labeling Factors Polite Nervous Kind Friendly ????? Intelligent ????? ????? Energetic Hardworking Outspoken Happy Neat Responsible

11. The EFA Process • Correlation Matrix • Represents the amount of covariation among the measured variables • Number of Factors • How many factors should be extracted? • Goal is to reproduce the correlation matrix reasonably well with a small number of factors • Factor Rotation • Is it possible to clarify the meaning of the extracted factors? • Goal is to have variables “load” cleanly on factors

12. Correlation Matrix Relatively easy to determine pattern of relations among small number of variables But, with larger matrices…

13. Factor Extraction • Factor is a set of weights applied to each variable • Factors extracted sequentially based on least squares criterion • First factor = set of weights that best reproduces the original correlation matrix • weights produce an “implied” correlation matrix • Residual matrix is difference between original values in the correlation matrix and the implied values • Process repeated on the residual correlation matrix • Second factor = set of weights that best reproduces the residual correlation matrix • Etc.

14. EFA Example 1 • Open Bigfive.sav (N = 64) • Six Items from a Big Five Personality Measure • Step 1: Factor Extraction • Analyze: Data Reduction: Factor • Select all six items • Descriptives • click on Coefficients and Reproduced • Extraction • click on Scree plot

15. Factor Extraction Example Factor 1 weights Variance Explained by Factor 1

16. Original Correlation Matrix Factor 1 weights (a vector) are multiplied Factor Weights or Loadings .704 .628 .560 .637 .568 .576 -.132 .117 -.119 .025 .265 -.236 -.240 .050 .010 .140 -.125 -.127 .026 .053 .028 Implied Correlation Matrix for Factor 1

17. Original Correlation Matrix minus Implied Correlation Matrix for Factor 1 .704 .628 .560 .637 .568 .576 -.132 .117 -.119 .025 .265 -.236 -.240 .050 .010 .140 -.125 -.127 .026 .053 .028 equals .296 -.114 .440 -.053 -.222 .424 .144 -.179 .217 .975 -.316 .112 .197 .386 .990 -.082 .133 .096 .308 .406 .972 Residual Matrix after extracting Factor 1

18. .296 -.114 .440 -.053 -.222 .424 .144 -.179 .217 .975 -.316 .112 .197 .386 .990 -.082 .133 .096 .308 .406 .972 Residual Matrix after extracting Factor 1 Factor 2 weights (a vector) are multiplied Factor Weights or Loadings Implied Correlation Matrix for Factor 2 .067 .027 .011 .064 .026 .062 .190 .076 .183 .543 .197 .079 .190 .564 .585 .191 .076 .184 .546 .567 .549

19. Residual Matrix after extracting Factor 1 .296 -.114 .440 -.053 -.222 .424 .144 -.179 .217 .975 -.316 .112 .197 .386 .990 -.082 .133 .096 .308 .406 .972 minus Implied Correlation Matrix for Factor 2 .067 .027 .011 .064 .026 .062 .190 .076 .183 .543 .197 .079 .190 .564 .585 .191 .076 .184 .546 .567 .549 equals Residual Matrix after extracting Factor 2 .229 -.141 .429 -.117 -.248 .362 -.046 -.255 .034 .432 -.513 .033 .007 -.178 .405 -.109 .057 -.088 -.238 -.161 .423

20. How Many Factors • When decide that an optimal number of factors has been found? • Eigenvalues > 1 rule • Scree test

21. Number of Factors • Kaiser-Guttman Rule • Number of eigenvalues greater than one • eigenvalues basically represent the proportion of variance accounted for by each factor (vector) • Note that the sum of the eigenvalues will be equal to number of variables • Thus, even with (nearly) complete independence, some will be greater than one • Often, pretty messy Data from our working example are pretty clear. Two Factors > 1

22. Scree Test • Plot of the eigenvalues • Cut-off at the point where the curve changes from flat to rapid • Walking up the scree (trivial factors) until you reach the base of the mountain (real factors) A 60-item Social Desirability measure Our working example

23. Simple Structure in EFA • Goal is Parsimony in Data Description • Extraction: Account adequately for the data with the smallest number of factors • Rotation: Transform extracted factors to minimize the number of nonzero loadings • So, each factor clearly represents certain variables, but not others

24. EFA Example 1 • Step 2: Factor Rotation • Analyze: Data Reduction: Factor • Rotation • Click on Varimax • An “orthogonal rotation” • Does it make patter of loadings more clear? Original Varimax Yes. More loadings closer to zero

25. Promotion / Prevention Scale 1. In general, I am focused on preventing negative events in my life. 2. I am anxious that I will fall short of my responsibilities and obligations. 3. I frequently imagine how I will achieve my hopes and aspirations. 4. I often think about the person I am afraid I might become in the future. 5. I often think about the person I would ideally like to be in the future. 6. I typically focus on the success I hope to achieve in the future. 7. I often worry that I will fail to accomplish my academic goals. 8. I often think about how I will achieve academic success. 9. I often imagine myself experiencing bad things that I fear might happen to me. 10. I frequently think about how I can prevent failures in my life. 11. I am more oriented toward preventing loess than I am toward achieving gains. 12. My major goal in school right now is to achieve my academic ambitions. 13. My major goal in school right now is to avoid becoming an academic failure. 14. I see myself as someone who is primarily striving to reach my 'ideal self' - to fulfill my hopes, wishes, and aspirations. 15. I see myself as someone who is primarily striving to become the self I 'ought' to be - to fulfill my duties, responsibilities, and obligations. 16. In general, I am focused on achieving positive outcomes in my life. 17. I often imagine myself experiencing good things that I hope will happen to me. 18. Overall, I am more oriented toward achieving success than preventing failure. A different example

26. EFA Example 2 • Open pro_prev.sav • 18 Items from a Regulatory Focus measure • Step 1: Factor Extraction • How many factors • Step 2: Factor Rotation • Does Varimax rotation “improve” loadings? • How about Promax rotation? • Which allows factors to be correlated

27. . . . 18 Looks like 2 factors

28. Unrotated Factor Plot Not very many zero loadings

29. 0.0 0.0 Varimax Rotation Plot Quite a few more zero loadings

30. Promax Rotation Plot 0.0 0.0 By relaxing orthogonal requirement, it gets a little better (factor r = .13)

31. Promotion / Prevention Scale 1. In general, I am focused on preventing negative events in my life. 2. I am anxious that I will fall short of my responsibilities and obligations. 3. I frequently imagine how I will achieve my hopes and aspirations. 4. I often think about the person I am afraid I might become in the future. 5. I often think about the person I would ideally like to be in the future. 6. I typically focus on the success I hope to achieve in the future. 7. I often worry that I will fail to accomplish my academic goals. 8. I often think about how I will achieve academic success. 9. I often imagine myself experiencing bad things that I fear might happen to me. 10. I frequently think about how I can prevent failures in my life. 11. I am more oriented toward preventing loess than I am toward achieving gains. 12. My major goal in school right now is to achieve my academic ambitions. 13. My major goal in school right now is to avoid becoming an academic failure. 14. I see myself as someone who is primarily striving to reach my 'ideal self' - to fulfill my hopes, wishes, and aspirations. 15. I see myself as someone who is primarily striving to become the self I 'ought' to be - to fulfill my duties, responsibilities, and obligations. 16. In general, I am focused on achieving positive outcomes in my life. 17. I often imagine myself experiencing good things that I hope will happen to me. 18. Overall, I am more oriented toward achieving success than preventing failure. Promote Prevent .185 .353 -.018 .674 .665 -.002 -.052 .722 .666 .198 .770 -.065 .013 .577 .706 .135 -.059 .734 .301.517 -.182 .604 .554 .027 .058 .397 .702 -.024 .416 .220 .709 -.054 .588 -.033 .663-.310

32. Next Week • Read EFA part of Russell (2002) • Posted on blackboard • Factor Analyze Ambivalent Sexism Invintory (sexism.sav) • Follow guidelines in “Factor Analysis Practical” on Blackboard • Use Russell’s recommendations throughout