Survey Methods & Design in Psychology

1. Survey Methods & Design in Psychology Lecture 5 (2007) Factor Analysis 1 Lecturer: James Neill

2. Readings • Bryman, A. & Cramer, D. (1997). Concepts and their measurement (Ch. 4). • Fabrigar, L. R. ...[et al.]. (1999). Evaluating the use of exploratory factor analysis in psychological research. • Tabachnick, B. G. & Fidell, L. S. (2001). Principal components and factor analysis. • Francis 5.6

3. Overview • What is FA? (purpose) • History • Assumptions • Steps • Reliability Analysis • Creating Composite Scores

4. What is Factor Analysis? • A family of techniques to examine correlations amongst variables. • Uses correlations among many items to search for common clusters. • Aim is to identify groups of variables which are relatively homogeneous. • Groups of related variables are called ‘factors’. • Involves empirical testing of theoretical data structures

5. Purposes The main applications of factor analytic techniques are: • to reduce the number of variables and • (2) to detect structure in the relationships between variables, that is to classify variables.

6. Factor 1 Factor 2 Factor 3 Conceptual Model for a Factor Analysis with a Simple Model e.g., 12 items testing might actually tap only 3 underlying factors

7. Conceptual Model for Factor Analysis with a Simple Model

8. Extraversion/ introversion Neuroticism Psychoticism loner anxious tense talkative fun nurturing gloomy harsh shy sociable relaxed unconventional Eysenck’s Three Personality Factors e.g., 12 items testing three underlying dimensions of personality

9. Conceptual Model for Factor Analysis (with cross-loadings)

10. Conceptual Model for Factor Analysis (3D)

11. Conceptual Model for Factor Analysis

12. Conceptual Model for Factor Analysis One factor Independent items Three factors

13. Conceptual Model for Factor Analysis

14. Conceptual Model for Factor Analysis

15. Factor Analysis Process

16. Purpose of Factor Analysis? • FA can be conceived of as a method for examining a matrix of correlations in search of clusters of highly correlated variables. • A major purpose of factor analysis is data reduction, i.e., to reduce complexity in the data, by identifying underlying (latent) clusters of association.

17. History of Factor Analysis? • Invented by Spearman (1904) • Usage hampered by onerousness of hand calculation • Since the advent of computers, usage has thrived, esp. to develop: • Theory– e.g., determining the structure of personality • Practice– e.g., development of 10,000s+ of psychological screening and measurement tests

18. Examples of Commonly Used Factor Structures in Psychology • IQ viewed as related but separate factors, e.g,. • verbal • mathematical • Personality viewed as 2, 3, or 5, etc. factors, e.g., the “Big 5” • Neuroticism • Extraversion • Agreeableness • Openness • Conscientiousness

19. Example: Factor Analysis of Essential Facial Features • Six orthogonal factors, represent 76.5 % of the total variability in facial recognition. • They are (in order of importance): • upper-lip • eyebrow-position • nose-width • eye-position • eye/eyebrow-length • face-width.

20. Problems Problems with factor analysis include: • Mathematically complicated • Technical vocabulary • Results usually absorb a dozen or so pages • Students do not ordinarily learn factor analysis • Most people find the results incomprehensible

21. Exploratory vs. Confirmatory Factor Analysis EFA = Exploratory Factor Analysis • explores & summarises underlying correlational structure for a data set CFA = Confirmatory Factor Analysis • tests the correlational structure of a data set against a hypothesised structure and rates the “goodness of fit”

22. Data Reduction 1 • FA simplifies data by revealing a smaller number of underlying factors • FA helps to eliminate: • redundant variables • unclear variables • irrelevant variables

23. Steps in Factor Analysis • Test assumptions • Select type of analysis(extraction & rotation) • Determine # of factors • Identify which items belong in each factor • Drop items as necessary and repeat steps 3 to 4 • Name and define factors • Examine correlations amongst factors • Analyse internal reliability

24. Garbage.In.Garbage.Out

25. Assumption Testing – Sample Size • Min. N of at least 5 cases per variable • Ideal N of at least 20 cases per variable • Total N of 200+ preferable

26. Assumption Testing – Sample Size Comrey and Lee (1992): • 50 = very poor, • 100 = poor, • 200 = fair, • 300 = good, • 500 = very good • 1000+ = excellent

27. Assumption Testing – Sample Size

28. Example Factor Analysis – Classroom Behaviour Example (Francis 5.6) • 15 classroom behaviours of high-school children were rated by teachers using a 5-point scale • Use FA to identify groups of variables (behaviours) that are strongly inter-related & represent underlying factors.

29. Classroom Behaviour Items 1 • Cannot concentrate --- can concentrate • Curious & enquiring --- little curiousity • Perseveres --- lacks perseverance • Irritable --- even-tempered • Easily excited --- not easily excited • Patient --- demanding • Easily upset --- contented

30. Classroom Behaviour Items 2 • Control --- no control • Relates warmly to others --- provocative,disruptive • Persistent --- easily frustrated • Difficult --- easy • Restless --- relaxed • Lively --- settled • Purposeful --- aimless • Cooperative --- disputes

31. Francis 5.6 – Victorian Quality Schools Project

32. Assumption Testing – LOM • All variables must be suitable for correlational analysis, i.e., they should be ratio/metric data or at least Likert data with several interval levels.

33. Assumption Testing – Normality • FA is robust to assumptions of normality(if the variables are normally distributed then the solution is enhanced)

34. Assumption Testing – Linearity • Because FA is based on correlations between variables, it is important to check there are linear relations amongst the variables (i.e., check scatterplots)

35. Assumption Testing - Outliers • FA is sensitive to outlying cases • Bivariate outliers(e.g., check scatterplots) • Multivariate outliers (e.g., Mahalanobis’ distance) • Identify outliers, then remove or transform

36. Assumption Testing – Factorability 1 • It is important to check the factorability of the correlation matrix (i.e., how suitable is the data for factor analysis?) • Check correlation matrix for correlations over .3 • Check the anti-image matrix for diagonals over .5 • Check measures of sampling adequacy (MSAs) • Bartlett’s • KMO

37. Assumption Testing - Factorability 2 • The most manual and time consuming but thorough and accurate way to examine the factorability of a correlation matrix is simply to examine each correlation in the correlation matrix • Take note whether there are SOME correlations over .30 – if not, reconsider doing an FA– remember garbage in, garbage out

38. Francis 5.6 – Victorian Quality Schools Project

40. Assumption Testing - Factorability 3 • Medium effort, reasonably accurate • Examine the diagonals on the anti-image correlation matrix to assess the sampling adequacy of each variable • Variables with diagonal anti-image correlations of less that .5 should be excluded from the analysis – they lack sufficient correlation with other variables

41. Francis 5.6 – Victorian Quality Schools Project

42. Anti-Image correlation matrix

43. Assumption Testing - Factorability 4 • Quickest method, but least reliable • Global diagnostic indicators - correlation matrix is factorable if: • Bartlett’s test of sphericity is significant and/or • Kaiser-Mayer Olkin (KMO) measure of sampling adequacy > .5

44. Francis 5.6 – Victorian Quality Schools Project

45. Summary: Measures of Sampling Adequacy • Are there several correlations over .3? • Are the diagonals of anti-image matrix > .5? • Is Bartlett’s test significant? • Is KMO > .5 to .6?(depends on whose rule of thumb)

46. Extraction Method:PC vs. PAF • There are two main approaches to EFA based on: • Analysing only shared variancePrinciple Axis Factoring (PAF) • Analysing all variancePrinciple Components (PC)

47. Principal Components (PC) • More common • More practical • Used to reduce data to a set of factor scores for use in other analyses • Analyses all the variance in each variable

48. Principal Components (PAF) • Used to uncover the structure of an underlying set of p original variables • More theoretical • Analyses only shared variance(i.e. leaves out unique variance)

49. Total variance of a variable

50. PC vs. PAF • Often there is little difference in the solutions for the two procedures. • Often it’s a good idea to check your solution using both techniques • If you get a different solution between the two methods • Try to work out why and decide on which solution is more appropriate